intellecton/papers/references/Schlosshauer2007.txt

The quantum-to-classical transition and decoherence

Maximilian Schlosshauer
Department of Physics, University of Portland, 5000 North Willamette Boulevard, Portland, Oregon 97203, USA

I give a pedagogical overview of decoherence and its role in providing a dynamical account of the
quantum-to-classical transition. The formalism and concepts of decoherence theory are reviewed,
followed by a survey of master equations and decoherence models.
I also discuss methods for
mitigating decoherence in quantum information processing and describe selected experimental
investigations of decoherence processes.
Note: Please see arXiv:1911.06282 [quant-ph] (published as Phys. Rep. 831, 1–57, 2019) for
a much more extensive and up-to-date review of decoherence.

CONTENTS

I. Introduction
1

II. Basic formalism and concepts
2

A. Decoherence and interference damping
2

B. Environmental monitoring and information
transfer
3

C. Environment-induced superselection and
decoherence-free subspaces
4

1. Pointer states and the commutativity
criterion
5

2. Decoherence-free subspaces
6

D. Proliferation of information and quantum
Darwinism
6

E. Decoherence versus dissipation and noise
7

III. Master equations
7

A. Born–Markov master equations
8

B. Lindblad master equations
8

C. Non-Markovian decoherence
9

IV. Decoherence models
10

A. Collisional decoherence
10

B. Quantum Brownian motion
11

C. Spin–boson models
13

D. Spin-environment models
13

V. Qubit decoherence, quantum error correction,
and error avoidance
14

A. Correction of decoherence-induced quantum
errors
14

B. Quantum computation on decoherence-free
subspaces
15

C. Environment engineering and dynamical
decoupling
16

VI. Experimental studies of decoherence
16

A. Atoms in a cavity
17

B. Matter-wave interferometry
17

C. Superconducting systems
17

VII. Decoherence and the foundations of quantum
mechanics
19

References
19

I.
INTRODUCTION

Realistic quantum systems are never completely iso-
lated from their environment. When a quantum system
interacts with its environment, it will in general become
entangled with a large number of environmental degrees
of freedom. This entanglement influences what we can
locally observe upon measuring the system. In partic-
ular, quantum interference effects with respect to cer-
tain physical quantities (most notably, “classical” quan-
tities such as position) become effectively suppressed,
making them prohibitively difficult to observe in most
cases of practical interest. This is the process of deco-
herence, sometimes also called dynamical decoherence or
environment-induced decoherence [1–10]. Stated in gen-
eral and interpretation-neutral terms, decoherence de-
scribes how entangling interactions with the environment
influence the statistics of results of future measurements
on the system.
Formally, decoherence can be viewed as a dynamical
filter on the space of quantum states, singling out those
states that, for a given system, can be stably prepared
and maintained, while effectively excluding most other
states, in particular, nonclassical superposition states of
the kind popularized by Schr¨odinger’s cat. In this way,
decoherence lies at the heart of the quantum-to-classical
transition. It ensures consistency between quantum and
classical predictions for systems observed to behave clas-
sically. It provides a quantitative, dynamical account of
the boundary between quantum and classical physics. In
any concrete experimental situation, decoherence theory
specifies the physical requirements, both qualitative and
quantitative, for pushing the quantum–classical bound-
ary toward the quantum realm. Decoherence is a pure
quantum effect, to be distinguished from classical dissi-
pation and stochastic fluctuations (noise).
Decoherence processes are extremely efficient.
Even
when the environment does not, from a classical point
of view, impart significant classical perturbations on the
system, quantum-mechanically the system will in most
circumstances become rapidly and strongly entangled
with the environment. Furthermore, due to the many un-
controllable degrees of freedom of the environment, such
entanglement is usually irreversible for all practical pur-
poses. Increasingly realistic models of decoherence pro-

arXiv:1404.2635v2  [quant-ph]  20 Nov 2019


2

cesses have been developed, progressing from toy models
to complex models tailored to specific experiments (see
Sec. IV). Advances in experimental techniques have made
it possible to observe the gradual action of decoherence
in experiments such as matter-wave interferometry [11],
cavity QED [12], and superconducting systems [13] (see
Sec. VI).
The superposition states necessary for quantum in-
formation processing are typically also those most sus-
ceptible to decoherence. Thus, decoherence is a major
barrier to implementing devices for quantum informa-
tion processing such as quantum computers (see Sec. V).
Qubit systems must be engineered to minimize environ-
mental interactions detrimental to the preparation and
longevity of the desired superposition states.
At the
same time, they must remain sufficiently open to al-
low for their control.
Quantum error correction can
undo some of the decoherence-induced degradation of
the superposition state and will be an integral part of
quantum computers (see Sec. V A). Not only is deco-
herence relevant to quantum information, but also vice
versa. An information-centric view of quantum mechan-
ics proves helpful in conveying the essence of the deco-
herence process and is also used in recent explorations
of the role of the environment as an information channel
(see Sec. II B).
It is a curious “historical accident” (Joos’s term [14,
p. 13]) that the role of the environment in quantum me-
chanics was appreciated only relatively late. While one
can find—for example, in Heisenberg’s writings [15]—a
few early anticipatory remarks about the role of environ-
mental interactions in the quantum-mechanical descrip-
tion of systems, it wasn’t until the 1970s that the ubiquity
and implications of environmental entanglement were re-
alized by Zeh [1, 16]. It took another decade for the for-
malism of decoherence to be developed, chiefly by Zurek
[2, 3], and for concrete models and numerical estimates
of decoherence rates to be worked out [17, 18].
Review papers on decoherence include Refs. [4–6, 10,

19].
There are two books on decoherence:
a volume
by Joos et al. [8] (a collection of chapters written by
different authors) and a monograph by this author [9].
Ref. [20] also contains material on decoherence. Foun-
dational implications of decoherence are discussed in
Refs. [6, 7, 9, 21].

II.
BASIC FORMALISM AND CONCEPTS

In the double-slit experiment, we cannot observe an in-
terference pattern if we also measure which slit the parti-
cle went through (that is, if we obtain perfect which-path
information). In fact, there is a continuous tradeoff be-
tween interference (phase information) and which-path
information: the better we can distinguish the two pos-
sible paths, the less visible the interference pattern be-
comes [22]. What is more, for a decrease in interference
visibility to occur it suffices that there are degrees of

freedom somewhere in the world that, if they were mea-
sured, would allow us to make, with a certain degree of
confidence, a statement about the path of the particle
through the slits.
While we cannot say that prior to
their measurement, these degrees of freedom have en-
coded information about a particular, definitive path of
the particle—instead, we have merely correlations involv-
ing both possible paths—no actual measurement is re-
quired to bring about the decrease in interference visibil-
ity. It is enough that, in principle, we could make such
a measurement to obtain which-path information.
This is somewhat loose talk, and conceptual caveats
lurk. But it captures quite well the essence of what is
happening in decoherence, where those “degrees of free-
dom somewhere in the world” are the degrees of freedom
of the system’s environment interacting with the system,
leading to the creation of quantum correlations (entan-
glement) between system and environment. Decoherence
can thus be thought of as a process arising from the con-
tinuous monitoring of the system by the environment;
effectively, the environment is performing nondemolition
measurements on the system (see Sec. II B). We now give
a formal quantum-mechanical account of what we have
just tried to convey in words, and then flesh out the con-
sequences and details.

A.
Decoherence and interference damping

Consider again the double-slit experiment and denote
the quantum states of the particle (call it S, for “sys-
tem”) corresponding to passage through slit 1 and 2 by
|s1⟩ and |s2⟩, respectively. Suppose that the particle in-
teracts with another system E—for example, a detec-
tor or an environment—such that if the quantum state
of the particle before the interaction is |s1⟩, then the
quantum state of E will become |E1⟩ (and similarly for
|s2⟩), resulting in the final composite states |s1⟩ |E1⟩ and
|s2⟩ |E2⟩, respectively. For an initial superposition state
α |s1⟩+β |s2⟩, the final composite state will be entangled,

|Ψ⟩ = α |s1⟩ |E1⟩ + β |s2⟩ |E2⟩ .
(1)

The statistics of all possible local measurements on S
are exhaustively encoded in the reduced density matrix
ρS,

ρS = TrE(ρSE) = TrE|Ψ⟩⟨Ψ|

= |α|2 |s1⟩⟨s1| + |β|2 |s2⟩⟨s2|
+ αβ∗|s1⟩⟨s2|⟨E2|E1⟩ + α∗β|s2⟩⟨s1|⟨E1|E2⟩.
(2)

For example, suppose we measure particle’s position by
letting the particle impinge on a distant detection screen.
Statistically, the resulting particle probability density
p(x) will be given by

p(x) = TrS(ρSx)

= |α|2 |ψ1(x)|2 + |β|2 |ψ2(x)|2

+ 2 Re {αβ∗ψ1(x)ψ∗
2(x)⟨E2|E1⟩} ,
(3)


3

where ψi(x) ≡ ⟨x|si⟩. The last term represents the in-
terference contribution. Thus, the visibility of the inter-
ference pattern is quantified by the overlap ⟨E2|E1⟩, i.e.,
by the distinguishability of |E1⟩ and |E2⟩. In the lim-
iting case of perfect distinguishability, ⟨E2|E1⟩ = 0, no
interference pattern will be observable and we obtain the
classical prediction. Phase relations have become locally
(i.e., with respect to S) inaccessible, and there is no mea-
surement on S that can reveal coherence between |s1⟩ and
|s2⟩. The coherence is now between the states |s1⟩ |E1⟩
and |s2⟩ |E2⟩, requiring an appropriate global measure-
ment (acting jointly on S and E) for it to be revealed.
Conversely, if the interaction between S and E is such
that E is completely unable to resolve the path of the
particle, then |E1⟩ and |E2⟩ are indistinguishable and full
coherence is retained at the level of S, as is also directly
obvious from Eq. (1). In the intermediary regime where
0 < |⟨E2|E1⟩| < 1, meaning that |E1⟩ and |E2⟩ can be
distinguished in a one-shot measurement with nonzero
probability p = 1 − |⟨E2|E1⟩|2 < 1, an interference pat-
tern of reduced visibility is obtained. Equation (3) shows
that the reduction in visibility increases as |E1⟩ and |E2⟩
become more distinguishable.
Here is another way of putting the matter. Looking
back at Eq. (1), we see that E encodes which-way infor-
mation about S in the same “relative-state” sense [23] in
which EPR correlations [24–26] may be said to encode
“information.” That is, if ⟨E2|E1⟩ = 0 and we were to
measure E and found it to be in state |E1⟩, we could, in
EPR’s words [24, p. 777], “predict with certainty” that
we will find S in |s1⟩.1 Whenever such a prediction is
possible were we to measure E, no interference effects be-
tween the components |s1⟩ and |s2⟩ can be measured at
S, even if E is never actually measured. If |⟨E2|E1⟩| > 0,
then E encodes only partial which-way information about
S, in the sense that a measurement of E could not reliably
distinguish between |E1⟩ and |E2⟩; instead, sometimes
the measurement will result in an outcome compatible
with both |E1⟩ and |E2⟩. Consequently, an interference
experiment carried out on S would find reduced visibil-
ity, representing diminished local coherence between the
components |s1⟩ and |s2⟩.
As hinted above, the description developed so far de-
scribes the essence of the decoherence process if we iden-
tify the particle S more generally with an arbitrary quan-
tum system and the second system E with the environ-
ment of S. Then an idealized account of the decoherence

1 Of course, this must not be read as saying that S was already
in |s1⟩ (i.e., “went through slit 1”) prior to the measurement
of E.
Nor does it mean that the result of a subsequent path
measurement on S is necessarily determined, by virtue of the
measurement on E, prior to this S-measurement’s actually be-
ing carried out. After all, as Peres has cautioned us [27], unper-
formed measurements have no outcomes. So while the picture
of E as “encoding which-path information” about S is certainly
suggestive and helpful, it should be used with an understanding
of its conceptual pitfalls.

interaction has form
��

i
ci |si⟩
�
|E0⟩
−→
�

i
ci |si⟩ |Ei(t)⟩ .
(4)

We have here introduced a time parameter t, where t = 0
corresponds to the onset of the environmental interac-
tion, with |Ei(t)⟩ ≡ |E0⟩ for all i; at t < 0 the system
and environment are assumed to be uncorrelated (an as-
sumption common to most decoherence models).
A single environmental particle interacting with the
system will typically only insufficiently resolve the com-
ponents |si⟩ in the system’s superposition state. But be-
cause of the large number of such particles (and, hence,
degrees of freedom), the overlap between their different
joint states |Ei(t)⟩ will rapidly decrease as a result of
the build-up of many interaction events. Specifically, in
many decoherence models an exponential decay of over-
lap is found [3, 5, 9, 17, 20, 28–31],

⟨Ei(t)|Ej(t)⟩ ∝ e−t/τd
for i ̸= j.
(5)

Here τd is the characteristic decoherence timescale, which
can be evaluated for particular choices of the parameters
in each model (see Sec. IV).

B.
Environmental monitoring and information
transfer

We will now motivate, in a different and more rigorous
way, the picture of decoherence as a process of environ-
mental monitoring.
First, we express the influence of
the environment in a completely general way.
We as-
sume that at t = 0 there are no correlations between
system S and environment E, ρSE(0) = ρS(0) ⊗ ρE(0).
We write ρE(0) in its diagonal decomposition, ρE(0) =
�

i pi|Ei⟩⟨Ei|, where �

i pi = 1 and the states |Ei⟩ form
an orthonormal basis of the Hilbert space of E.
If
H denotes the Hamiltonian (here assumed to be time-
independent) of SE and U(t) = e−iHt represents the uni-
tary time evolution operator, then the density matrix of
S evolves according to

ρS(t) = TrE

�

U(t)

�

ρS(0) ⊗

��

i
pi|Ei⟩⟨Ei|

��

U †(t)

�

=
�

ij
pi ⟨Ej| U(t) |Ei⟩ ρS(0) ⟨Ei| U †(t) |Ej⟩ .
(6)

Introducing the Kraus operators [32] defined by Eij ≡
√pi ⟨Ej| U(t) |Ei⟩, we obtain

ρS(t) =
�

ij
EijρS(0)E†
ij.
(7)

It is customary to combine the two indices i and j into a
single index and write the Kraus operators as

Wk ≡ √pik ⟨Ejk| U(t) |Eik⟩ ,
(8)


4

such that

ρS(t) =
�

k
WkρS(0)W †
k.
(9)

This Kraus-operator formalism (also called operator-sum
formalism) represents the effect of the environment as
a sequence of (in general nonunitary) transformations of
ρS generated by the operators Wk. The Kraus operators
exhaustively encode information about the initial state
of the environment and about the dynamics of the joint
SE system. Because the evolution of SE is unitary, the
Kraus operators satisfy the completeness constraint
�

k
WkW †
k = IS,
(10)

where IS is the identity operator in the Hilbert space of
S. Equations (9) and (10) together imply that the Wk are
the generators of a completely positive map Φ : ρS(0) �→
ρS, also known as a quantum operation [32] or quantum
channel.2

We will now use Eq. (9) to formally motivate the view
that decoherence corresponds to an indirect measurement
of the system by the environment, and that it thus re-
sults from a transfer of information from the system to
the environment (see also Ref. [19]).
In such an indi-
rect measurement, we let the system S interact with a
probe—here the environment E—followed by a projec-
tive measurement on E. The probe is treated as a quan-
tum system. This procedure aims to yield information
about S without performing a projective (and thus de-
structive) direct measurement on S. To model such an
indirect measurement, consider again an initial compos-
ite density operator ρSE(0) = ρS(0) ⊗ ρE(0) evolving
under the action of U(t) = e−iHt, where H is the to-
tal Hamiltonian. Consider a projective measurement M
on E with eigenvalues α and corresponding projectors
Pα ≡ |α⟩⟨α|, with P 2
α = P †
α = Pα. The probability of
obtaining outcome α in this measurement when S is de-
scribed by the density operator ρS(t) is

Prob (α | ρS(t)) = TrE (PαρE(t))

= TrE
�
PαTrS
�
U(t) (ρS(0) ⊗ ρE(0)) U †(t)
��
.
(11)

The density matrix of S conditioned on the particular

2 The Kraus formalism is of limited use in calculating decoherence
dynamics for concrete situations of physical interest. This is so
because finding the Kraus operators corresponds to diagonaliz-
ing the full Hamiltonian of SE, usually a prohibitively difficult
task.
Moreover, the Kraus operators contain all contributions
to the evolution of the reduced density matrix, while for con-
siderations of decoherence we are typically interested only in
the nonunitary terms, and certain contributions—such as back-
action effects from the system on the environment—can often be
neglected. (This is where master equations come into play; see
Sec. III.)

outcome α is

ρ(α)
S (t) = TrE {[I ⊗ Pα] ρSE(t) [I ⊗ Pα]}

Prob (α | ρS(t))

= TrE
�
[I ⊗ Pα] U(t) [ρS(0) ⊗ ρE(0)] U †(t) [I ⊗ Pα]
�

Prob (α | ρS(t))
.

(12)

Inserting
the
diagonal
decomposition
ρE(0)
=
�

k pk|Ek⟩⟨Ek|
and
carrying
out
the
trace
gives
[19]

ρ(α)
S (t) =
�

k

Mα,kρS(t)M †
α,k

Prob (α | ρS(t)).
(13)

Here we have introduced the measurement operators

Mα,k ≡ √pk ⟨α| U(t) |Ek⟩ ,
(14)

which
obey
the
completeness
constraint
�

α,k Mα,kM †
α,k = IS.
Equation (12) describes the
effect of the indirect measurement on the state of the
system. If, however, we do not actually inquire about
the result of this measurement, we must assign to the
system a density operator that is a sum over all the
possible conditional states ρ(α)
S (t) weighted by their
probabilities Prob (α | ρS(t)),

ρS(t) =
�

α
Prob (α | ρS(t)) ρ(α)
S (t)

=
�

α,k
Mα,kρS(t)M †
α,k.
(15)

Note that this expression is formally analogous to the
Kraus-operator expression of Eq. (9), which described
the effect of a general environmental interaction on the
state of the system. Recall, further, that the situation we
encounter in decoherence is precisely one in which we do
not actually read out the environment—or, in the present
picture, in which we do not inquire about the result of the
indirect measurement.
This suggests that decoherence
can indeed be understood as an indirect measurement—
a monitoring—of the system by its environment.

C.
Environment-induced superselection and
decoherence-free subspaces

Decoherence can occur in any basis; which observable
is monitored by the environment depends on the spe-
cific form of the system–environment interaction. The
preferred states (or preferred observables) of the system
emerge dynamically as those states that are the most ro-
bust to the interaction with the environment, in the sense
that they become least entangled with the environment;
thus, they are the states most immune to decoherence.


5

This is the stability criterion for the selection of pre-
ferred states, resulting in the dynamical selection of pre-
ferred states (“environment-induced superselection”) [1–
3, 16]. These environment-superselected preferred states
(or observables) are sometimes also called pointer states
(or pointer observables) [2], since they correspond to the
physical quantities that are most easily “read off” at the
level of the system, akin to the pointer on the dial of a
measurement apparatus.

1.
Pointer states and the commutativity criterion

To find the preferred states,
we decompose the
total system–environment Hamiltonian into the self-
Hamiltonians of the system S and environment E rep-
resenting the intrinsic dynamics, and a part Hint repre-
senting the interaction between system and environment,

H = HS + HE + Hint.
(16)

In many cases of practical interest, Hint dominates
the evolution of the system, H ≈ Hint (the quantum-
measurement limit of decoherence). We look for system
states |si⟩ such that the composite system–environment
state, when starting from a product state |si⟩ |E0⟩ at
t = 0, remains in the product form |si⟩ |Ei(t)⟩ for all
t > 0 under the action of Hint (we shall assume here
that Hint is not explicitly time-dependent). That is, we
demand that (setting ℏ ≡ 1 from here on)

e−iHintt |si⟩ |E0⟩ = λi |si⟩ e−iHintt |E0⟩ ≡ |si⟩ |Ei(t)⟩ .
(17)
Thus, the pointer states |si⟩ are the eigenstates of the
part of the interaction Hamiltonian Hint pertaining to the
Hilbert space of the system, with eigenvalues λi. These
states will be stationary under Hint [2]. It follows that the
pointer observable defined by OS = �

i oi|si⟩⟨si| com-
mutes with Hint,
�
OS, Hint
�
= 0.
(18)

This commutativity criterion [2, 3] is particularly easy to
apply when Hint takes the tensor-product form Hint =
S ⊗ E, as is frequently the case. Then the environment-
superselected observables will be those observables that
commute with S.
If S is Hermitian, it represents the
physical quantity monitored by the environment. In gen-
eral, any Hint can be written as a diagonal decomposition
of (unitary but not necessarily Hermitian) system and
environment operators Sα and Eα, Hint = �

α Sα ⊗ Eα.
If the Sα are Hermitian, such a Hamiltonian represents
the simultaneous environmental monitoring of different
observables Sα of the system. A sufficient condition for
{|si⟩} to form a set of pointer states of the system is then
given by the requirement that the |si⟩ be simultaneous
eigenstates of the operators Sα,

Sα |si⟩ = λ(α)
i
|si⟩
for all α and i.
(19)

Interaction Hamiltonians frequently describe the scat-
tering of surrounding particles (photons, air molecules,
etc.), leading to collisional decoherence (see Sec. IV A).
Since the force laws describing such processes typically
depend on some power of distance, the interaction Hamil-
tonian will then commute with the position operator.
Thus, the pointer states will be approximate eigenstates
of position (i.e., narrow position-space wave packets).
This explains why superpositions of mesoscopically and
macroscopically distinct positions are prohibitively diffi-
cult to observe [2, 3, 17, 31, 33–39]. Collisional decoher-
ence can also be dominant in microscopic systems when
these systems occur in distinct spatial configurations that
couple strongly to the surrounding medium. For exam-
ple, chiral molecules such as sugar are always observed to
be in chirality eigenstates (left-handed or right-handed),
which are superpositions of different energy eigenstates.
Any attempt to prepare such molecules in energy eigen-
states leads to immediate decoherence into the environ-
mentally stable chirality eigenstates [40, 41].
The quantum limit of decoherence [42] arises when the
modes of the environment are slow in comparison with
the evolution of the system—that is, when the highest
frequencies (i.e., energies) available in the environment
are smaller than the separation between the energy eigen-
states of the system. Then the environment will be able
to monitor only quantities that are constants of motion.
In the case of nondegeneracy, this quantity will be the en-
ergy of the system, leading to the environment-induced
superselection of energy eigenstates for the system [42].3

In many realistic situations, the commutativity crite-
rion, Eq. (18), can only be fulfilled approximately [43, 44].
In addition, the self-Hamiltonian of the system and the
interaction Hamiltonian may contribute in roughly equal
strengths (e.g., in models for quantum Brownian motion
[4, 45]; see Sec. IV B), rendering neither the quantum-
measurement limit of negligible intrinsic dynamics nor
the quantum limit of decoherence of a slow environ-
ment appropriate. In such cases, more general methods
for determining the preferred states are required. The
predictability-sieve strategy [43, 44, 46] computes the time
dependence of the amount of decoherence introduced into
the system for a large set of initial states of the system
evolving under the total system–environment Hamilto-
nian. Typically, this decoherence is measured using ei-
ther the purity Tr
�
ρ2
S
�
or the von Neumann entropy

3 Textbooks on quantum mechanics usually attribute a special role
to such energy eigenstates (for closed systems) since they are
stationary under the action of the Hamiltonian. In this closed-
system picture, however, arbitrary superpositions of energy
eigenstates should nonetheless be perfectly legitimate. Thus, it
is important to realize that the environment-induced superselec-
tion of energy eigenstates is not equivalent to a situation in which
the presence of the environment could be neglected altogether;
instead, the environment plays the crucial role of continuously
monitoring the energy of the system, leading to a local suppres-
sion of coherence between energy eigenstates.


6

S(ρS) = −Tr (ρS log2 ρS) of the reduced density matrix
ρS. The states most immune to decoherence will be those
which lead to the smallest decrease in purity or the small-
est increase in von Neumann entropy.
Application of
this method leads to a ranking of the possible preferred
states with respect to their robustness to the interac-
tion with the environment. For particular models it has
been explicitly shown that the states picked out by the
predictability sieve are robust to the particular choice of
the measure of decoherence. For example, in the model
for quantum Brownian motion, different measures lead
to the same minimum-uncertainty wave packets in phase
space [5, 8, 16, 44, 47, 48].

2.
Decoherence-free subspaces

The
pointer-state
condition
of
Eq.
(19)
can
be
strengthened to the concept of pointer subspaces [3] or
decoherence-free subspaces (DFS) [49–58]. These are sub-
spaces of the Hilbert space of the system in which every
state in the subspace is immune to decoherence; this is
a nontrivial requirement, since in general superpositions
of pointer states will not be pointer states themselves.
One important condition for this to happen is that the
preferred states |si⟩ defined by Eq. (19) form an orthonor-
mal basis of the subspace, and that the eigenvalues λ(α)
i
in Eq. (19) are independent of i, i.e., that all |si⟩ are
simultaneous degenerate eigenstates of each Sα,

Sα |si⟩ = λ(α) |si⟩
for all α and i.
(20)

This condition states that the action of a given Sα must
be the same for all basis states |si⟩ of the DFS, and thus
the existence of a DFS corresponds to a symmetry in the
structure of the system–environment interaction, i.e., to
a dynamical symmetry. A necessary condition for such a
symmetry to obtain is the absence of terms in Hint that
act jointly on system and environment in a nontrivial
manner.
An arbitrary state |ψ⟩ in the DFS can then be written
as |ψ⟩ = �

i ci |si⟩ and will evolve according to

e−iHintt |ψ⟩ |E0⟩ = |ψ⟩ e−i(
�

α λ(α)Eα)t |E0⟩
≡ |ψ⟩ |Eψ(t)⟩ .
(21)

Thus, the state |ψ⟩ does not become entangled with the
environment and is therefore immune to decoherence.
When the self-Hamiltonian HS of the system cannot be
neglected, one needs to additionally ensure that none of
the basis states |si⟩ of the DFS will drift out of the sub-
space under the evolution generated by HS. Otherwise
an initially decoherence-free state would again become
prone to decoherence. The concept of DFS can be gener-
alized to the formalism of noiseless subsystems (or noise-
less quantum codes) [58–60].

D.
Proliferation of information and quantum
Darwinism

Quantum Darwinism [61–69] builds on the ideas of de-
coherence and environmental encoding of information, by
broadening the role of the environment to that of a com-
munication and amplification channel. Interactions be-
tween the system and its environment lead to the redun-
dant storage of selected information about the system in
many fragments of the environment. By measuring some
of these fragments, observers can indirectly obtain infor-
mation about the system without appreciably disturbing
the system itself. Indeed, this represents how we typi-
cally observe objects. For example, we see an object not
by directly interacting with it, but by intercepting scat-
tered photons that encode information about the object’s
spatial structure [67, 68].

In this sense, quantum Darwinism provides a dynami-
cal explanation for the robustness of states of (especially)
macroscopic objects to observation. It was found that
the observable of the system that can be imprinted most
completely and redundantly in many distinct fragments
of the environment coincides with the pointer observable
selected by the system–environment interaction [62–65];
conversely, most other states do not seem to be redun-
dantly storable. Indeed, it has been shown that the re-
dundant proliferation of information regarding pointer
states is as inevitable as decoherence itself [70]. Quantum
Darwinism has been studied in several concrete models,
for example, in spin environments [64], quantum Brow-
nian motion [71], and photon and photon-like environ-
ments [67, 68, 70]. The efficiency of the amplification pro-
cess described by quantum Darwinism can be expressed
in terms of the quantum Chernoff information [70].

The structure and amount of information that the
environment encodes about the system can be quanti-
fied using the measure of (classical [62, 63] or quantum
[5, 64, 65]) mutual information. Classical mutual infor-
mation is based on the choice of particular observables of
the system S and the environment E and quantifies how
well one can predict the outcome of a measurement of a
given observable of S by measuring some observable on
a fraction of E [62, 63]. Quantum mutual information is
defined as S(ρS)+S(ρE)−S(ρ), where ρS, ρE, and ρ are
the density matrices of S, E, and the composite system
SE, respectively, and S(ρ) = −Tr (ρ log2 ρ) is the von
Neumann entropy associated with ρ. Quantum mutual
information quantifies the degree of quantum correlations
between S and E. Classical and quantum mutual infor-
mation give similar results [5, 62–65] because the differ-
ence between the two measures, known as the quantum
discord [72], disappears when decoherence is sufficiently
effective to select a well-defined pointer basis [72].


7

E.
Decoherence versus dissipation and noise

While the presence of dissipation implies the pres-
ence of decoherence, the converse is not necessarily true.
When dissipation and decoherence are both present, they
typically occur on vastly different timescales; the deco-
herence timescale is typically many orders of magnitude
shorter than the relaxation timescale. A rule-of-thumb
estimate for the ratio of the relaxation timescale τr to the
decoherence timescale τd for a massive object described
by a superposition of two different positions a distance
∆x apart is [18]

τr
τd
∼
� ∆x

λdB

�2
,
(22)

where λdB = (2mkT)−1/2 is the thermal de Broglie wave-
length of the object. For an object of mass m = 1 g at
room temperature in a coherent superposition of two lo-
cations a distance ∆x = 1 cm apart, τr/τd is on the order
of 1040. Thus, for macroscopic objects the dissipative in-
fluence of the environment is usually completely negligi-
ble on the timescale relevant to the decoherence induced
by this environment.
Decoherence is a consequence of environmental entan-
glement. In the literature on quantum computing, how-
ever, the term “decoherence” is often used to refer to
any process that affects the qubits, including perturba-
tions due to classical fluctuations and imperfections. Ex-
amples for sources of such classical noise in the context
of quantum computing are the fluctuations in the inten-
sity [73] and duration [74] of the laser beam incident on
qubits in an ion trap, inhomogeneities in the magnetic
fields in NMR quantum computing [75], and bias fluctu-
ations in superconducting qubits [76]. The distinction be-
tween classical noise and quantum decoherence has been
further blurred in quantum error correction, since the
error-correcting schemes are insensitive to the physical
origin of the qubit errors (see Sec. V A).
Phenomenologically and formally the influence of clas-
sical noise processes may be described in a manner simi-
lar to the effect of environmental entanglement, namely,
in terms of a decay of the off-diagonal elements (in-
terference terms) in the local density matrix (in the
environment-superselected basis).
But in the case of
noise, the decay of the off-diagonal elements occurs be-
cause the system’s density matrix is identified with an
average over a physical ensemble of systems (or, put dif-
ferently, over the different instances of particular noise
processes), while in the case of decoherence the decay is
due to an entanglement-induced delocalization of phase
coherence for individual systems. The fundamental dif-
ference between these physical processes is masked by the
density-matrix description. Indeed, one can always find
an experimental procedure that would, at least in princi-
ple, distinguish between the different physical processes
underlying formally similar density-matrix descriptions.
In contrast with decoherence, noise does not create
system–environment entanglement and can in principle

always be undone using only local operations (witness,
for example, the reversal of ensemble dephasing in NMR
experiments using the spin-echo technique). In any indi-
vidual realization of the noise process the dynamics of the
system are completely unitary, and thus no coherence is
lost from the system. By contrast, if the system becomes
entangled with environmental degrees of freedom, at the
very least we would need to perform a pair of measure-
ments on the environment before and after the interac-
tion with the system in order to gather enough informa-
tion to reverse the effect of decoherence by application
of an appropriate countertransformation. Moreover, as
also seen experimentally [77], these measurements would
not always constitute a sufficient procedure for “undo-
ing” decoherence (see also Sec. IV.C of Ref. [5]).
The loss of phase coherence due to environmental
entanglement is sometimes simulated (with the above
caveats) by classical fluctuations perturbing the system,
i.e., by the addition of certain time-dependent terms to
the self-Hamiltonian of the system. This strategy was
implemented, for example, in theoretical [73, 78] and ex-
perimental [77, 79] studies of the influence of fluctuating
parameters in ion-trap quantum computers.

III.
MASTER EQUATIONS

In the usual approach to modeling decoherence, the
reduced density matrix ρS(t) is obtained from

ρS(t) = TrE ρSE(t) ≡ TrE
�
U(t)ρSE(0)U †(t)
�
,
(23)

where U(t) is the time-evolution operator for the compos-
ite system SE. The task of calculating ρSE(t) is often
computationally cumbersome or even intractable. It is
also unnecessarily detailed, because we are usually only
interested in the dynamics of the system. A master equa-
tion allows us to calculate ρS(t) directly from an expres-
sion of the form

ρS(t) = V(t)ρS(0),
(24)

where the superoperator V(t) is the dynamical map gen-
erating the evolution of ρS(t).
If the master equation
is exact, then we merely have the identity V(t)ρS(0) ≡
TrE
�
U(t)ρSE(0)U †(t)
�
and no computational advantage
is gained.
Therefore, master equations are typically
based on simplifying approximations.
In modeling decoherence, we focus on master equations
that are first-order time-local differential equations of the
form

d
dtρS(t) = L [ρS(t)] ≡ −i [H′
S, ρS(t)] + D[ρS(t)].
(25)

This equation is local in time in the sense that the change
of ρS at time t depends only on ρS evaluated at t. The
superoperator L acting on ρS(t) typically depends on the
initial state of the environment and the different terms
in the Hamiltonian.
We have decomposed L into two


8

parts to distinguish their physical interpretation.
The
first term, −i [H′
S, ρS(t)], is unitary and given by the
Liouville–von Neumann commutator with the “renormal-
ized” Hamiltonian H′
S of the system. (Because the en-
vironment typically leads to a renormalization of the en-
ergy levels of the system, this Hamiltonian does in general
not coincide with the unperturbed free Hamiltonian HS
of S that would generate the evolution of S in absence of
the environment.) The second, nonunitary term D[ρS(t)]
represents decoherence (and often also dissipation) due to
the environment.

A.
Born–Markov master equations

Born–Markov master equations allow for many deco-
herence problems to be treated in a mathematically sim-
ple, and often closed, form. They are based on the fol-
lowing two approximations:

1. The
Born
approximation.
The
system–
environment coupling is sufficiently weak and
the environment is reasonably large such that
changes of the density operator ρE of the environ-
ment are negligible and the system–environment
density operator remains remains approximately
factorized at all times, ρSE(t) ≈ ρS(t) ⊗ ρE.

2. The Markov approximation. Memory effects of the
environment are negligible, in the sense that any
self-correlations within the environment created by
the coupling to the system decay rapidly compared
to the characteristic relaxation timescale of the
open quantum system.

Comparisons between the predictions of models based
on Born–Markov master equations and experimental
data indicate that the Born and Markov assumptions are
reasonable in many physical situations (but see Sec. III C
below for exceptions and non-Markovian models). As-
suming these assumptions hold and writing the inter-
action Hamiltonian as Hint = �

α Sα ⊗ Eα, the Born–
Markov master equation reads [9, 20]

d
dtρS(t) = −i [HS, ρS(t)]

−
�

α
{[Sα, BαρS(t)] + [ρS(t)Cα, Sα]} ,
(26)

where the system operators Bα and Cα are defined as

Bα ≡
� ∞

0
dτ
�

β
cαβ(τ)S(I)
β (−τ),
(27a)

Cα ≡
� ∞

0
dτ
�

β
cβα(−τ)S(I)
β (−τ).
(27b)

Here S(I)
α (−τ) denotes the operator Sα in the interaction
picture. In the following, we will simplify notation by

omitting the superscript “I”; instead we use the conven-
tion that all operators bearing explicit time arguments
are to be understood as interaction-picture operators.
(For density operators, however, we will maintain the
superscript notation in order to distinguish them from
Schr¨odinger-picture density operators, which also carry
a time argument.) The quantities cαβ(τ) appearing in
Eq. (27) are given by

cαβ(τ) ≡ ⟨Eα(τ)Eβ⟩ρE .
(28)

These environment self-correlation functions quantify
how much information the environment retains over time
about its interaction with the system. The Markov ap-
proximation corresponds to the assumption of a rapid
decay of the cαβ(τ) relative to the timescale set by the
evolution of the system.
In many situations of interest, the general form of the
Born–Markov master equation, Eq. (26), simplifies con-
siderably.
For example, typically only a single system
observable S is monitored by the environment, Hint =
S ⊗E. Also, the time dependence of the operators Sα(τ)
and Eα(τ) is often simple, facilitating the calculation of
the quantities Bα and Cα.
Examples are discussed in
Sec. IV.

B.
Lindblad master equations

Lindblad master equations constitute a particular, al-
beit quite general, class of time-local Markovian mas-
ter equations. They arise from the requirement that the
evolution of the reduced density matrix generated by the
master equation must ensure complete positivity [20, 80–
85]. Complete positivity guarantees that the dynamical
map ρS(0) �→ ρS(t) = V(t)ρS(0) described by the master
equation generates physically consistent dynamics even
when S is initially entangled with another system. While
complete positivity is automatically fulfilled if the evo-
lution is exact, approximate master equations will not
necessarily ensure complete positivity [20, 83–86]. The
Lindblad master equation is a special case of the gen-
eral Born–Markov master equation that ensures complete
positivity and takes the general form [81, 82]

d
dtρS(t) = −i [H′
S, ρS(t)]

+ 1

2

�

αβ
γαβ
��
Sα, ρS(t)S†
β
�
+
�
SαρS(t), S†
β
��
,
(29)

where H′
S is the renormalized Hamiltonian of the sys-
tem. The coefficients γαβ are time-independent and ex-
haustively encapsulate information about the physical
parameters of the decoherence processes (and possibly
dissipation processes).
One can show that the matrix
Γ ≡ (γαβ) formed by the coefficients γαβ is positive, i.e.,
all its eigenvalues κµ are ≥ 0. Therefore, Eq. (29) can be


9

simplified by diagonalizing Γ, which results in the diago-
nal form [82, 87]

d
dtρS(t) = −i [H′
S, ρS(t)]

− 1

2

�

µ
κµ
�
L†
µLµρS(t) + ρSL†
µLµ − 2LµρS(t)L†
µ
�
.

(30)

The Lindblad operators Lµ are linear combinations of the
original operators Sα, with coefficients determined by the
diagonalization of Γ. The Lindblad structure of a mas-
ter equation can also be motivated from the requirement
that it gives rise to the most general form of generators
of quantum dynamical semigroups [20, 81, 82, 84, 87–89].
It is possible to bring any Born–Markov master equation
into Lindblad form by imposing the rotating-wave ap-
proximation. This assumption, ubiquituous in quantum
optics, is justified whenever the timescale set by the typ-
ical energy differences ℏ(ω − ω′) of the system Hamilto-
nian is short in comparison with the relaxation timescale
of the system. (See Sec. 3.3.1 of Ref. [20] for details.)
Because the Sα are not necessarily Hermitian, the
Lindblad operators do not always correspond to physical
observables. But when they do, we can rewrite Eq. (30)
in compact double-commutator form,

d
dtρS(t) = −i [H′
S, ρS(t)] − 1

2

�

µ
κµ [Lµ, [Lµ, ρS(t)]] .

(31)
As an example, consider a situation in which the envi-
ronment monitors the position of a system. With L = x
and the “free”-particle Hamiltonian H′
S = HS = p2/2m,
Eq. (31) becomes

d
dtρS(t) = − i

2m
�
p2, ρS(t)
�
− 1

2κ [x, [x, ρS(t)]] .
(32)

Expressing this master equation in the position represen-
tation results in

∂ρS(x, x′, t)

∂t
= − i

2m

� ∂2

∂x′2 − ∂2

∂x2

�
ρS(x, x′, t)

− 1

2κ (x − x′)2 ρS(x, x′, t).
(33)

This is the classic equation of motion for decoherence due
to environmental scattering first derived in Ref. [17].
Lindblad master equations provide an intuitive and
simple way of representing the environmental monitoring
of an open quantum system. Most of the real physics be-
hind this monitoring process is hidden in the coefficients
κµ appearing in Eq. (30). If the Lindblad operators are
chosen to be dimensionless, the κµ can be directly in-
terpreted as decoherence rates, since they have units of
inverse time.
Equation (31) shows that the decoherence term van-
ishes if

[Lµ, ρS(t)] = 0
for all µ, t.
(34)

In this case, ρS(t) evolves unitarily. Since the Lµ are lin-
ear combinations of the Sα, Eq. (34) typically means that
[Sα, ρS(t)] = 0 for all α, t. This implies that simultane-
ous eigenstates of all Sα will be immune to decoherence,
which is precisely the pointer-state criterion of Eq. (19).
In quantum-jump and quantum-trajectory approaches,
the evolution of the reduced density matrix is conditioned
on an explicitly observed sequence of measurement re-
sults in the environment. This allows for the (formal)
description of a single realization of the system evolv-
ing stochastically, conditioned on a particular measure-
ment record. The dynamics are then described by a mas-
ter equation of the Lindblad type, Eq. (31), for the re-
duced density matrix ρC
S conditioned on the measurement
records of the Lindblad operators Lµ,

dρC
S = −i
�
HS, ρC
S
�
dt − 1

2

�

µ
κµ
�
Lµ,
�
Lµ, ρC
S
��
dt

+
�

µ

√κµ W[Lµ]ρC
S dWµ.
(35)

Here, W[L]ρ ≡ Lρ+ρL†−ρ Tr
�
Lρ + ρL†�
, and the dWµ
denote so-called Wiener increments. Equation (35) corre-
sponds to a diffusive unraveling of the Lindblad equation
into individual quantum trajectories, which can then be
expressed by means of a stochastic Schr¨odinger equation
[90–102].

C.
Non-Markovian decoherence

The derivation of the Born–Markov master equation
assumes that the coupling between system and environ-
ment is weak and memory effects of the environment can
be neglected.
These conditions, however, are not met
in certain situations of physical interest.
An example
would be a superconducting qubit strongly coupled to a
low-temperature environment of other two-level systems
[103, 104]. Also, a recent experiment [105] has measured
strongly non-Ohmic spectral densities for the environ-
ment of a quantum nanomechanical system; such densi-
ties lead to non-Markovian evolution.
In many cases, pronounced memory effects in the envi-
ronment will cause strong dependencies of the evolution
of the reduced density operator on the past history of the
system–environment composite and therefore make it im-
possible to describe the reduced dynamics by a differen-
tial equation that is local in time. Surprisingly, however,
one can show that even non-Markovian dynamics some-
times can still be described by a time-local differential
equation of the form

d
dtρS(t) = K(t)ρS(t),
(36)

where the superoperator K(t) depends only on t.
For
example, a non-Markovian master equation for quantum
Brownian motion (see Sec. IV B) can be obtained through


10

a formal modification of the Born–Markov master equa-
tion [4, 5]. In general, it is often possible to arrive at
non-Markovian but time-local master equations via the
so-called time-convolutionless projection operator tech-
nique [106–109].

IV.
DECOHERENCE MODELS

Many physical systems can be represented either by
a qubit if the state space of the system is discrete and
effectively two-dimensional, or by a particle described by
continuous phase-space coordinates. Needless to say, in
the case of quantum information processing the qubit
representation is of particular relevance.
Similarly, a wide range of environments can be modeled
as a collection of quantum harmonic oscillators or qubits.
Harmonic-oscillator environments are of great generality.
At low energies, many systems interacting with an en-
vironment can effectively be represented by one or two
coordinates of the system linearly coupled to an environ-
ment of harmonic oscillators; indeed, sufficiently weak in-
teractions with an arbitrary environment can be mapped
onto a system linearly coupled to a harmonic-oscillator
environment [110, 111].
Environments represented by qubits are often the ap-
propriate model in the low-temperature regine, where de-
coherence is typically dominated by interactions with lo-
calized modes, such as paramagnetic spins, paramagnetic
electronic impurities, tunneling charges, defects, and nu-
clear spins [103, 104, 112]. Each of the localized modes is
represented by a finite-dimensional Hilbert space with a
finite energy cutoff. We can therefore model these modes
as a set of discrete states. Typically, only two such states
are relevant, and thus the localized modes can be mapped
onto an environment of qubits. Since qubits can be for-
mally represented by spin- 1

2 particles, such models are
known as spin-environment models.
In the following, we will discuss four important stan-
dard models, namely, collisional decoherence (Sec. IV A),
quantum Brownian motion (Sec. IV B), the spin–boson
model (Sec. IV C), and the spin–spin model (Sec. IV D).
For details on these and other decoherence models, in-
cluding derivations of the relevant master equations, see
Secs. 3 and 5 of Ref. [9].

A.
Collisional decoherence

Collisional decoherence arises from the scattering of en-
vironmental particles by a massive free quantum particle.
Models of collisional decoherence were first studied in the
classic paper by Joos and Zeh [17]. A more rigorous and
general treatment was later developed by Hornberger and
collaborators [31, 36–39] (see also [34, 35, 113]), which,
among other refinements, remedied a flaw in Joos and
Zeh’s original derivation that had resulted in decoher-
ence rates that were too large by a factor of 2π [31].

If we assume that the central particle is much more
massive than the environmental particles such that its
center-of-mass state is not disturbed by the scattering
events (no recoil), the time evolution of the reduced den-
sity matrix is given by [9, 17, 31, 34, 35]

∂ρS(x, x′, t)

∂t
= −F(x − x′)ρS(x, x′, t).
(37)

This master equation describes pure spatial decoherence
without dissipation. The decoherence factor F(x − x′)
plays the role of a localization rate.
It represents the
characteristic decoherence rate at which spatial coher-
ences between two positions x and x′ become locally sup-
pressed and is given by

F(x − x′) =
� ∞

0
dq ϱ(q)v(q)

×
� dˆn dˆn′

4π

�
1 − eiq(ˆn−ˆn′)·(x−x′)�
|f(qˆn, qˆn′)|2 .
(38)

Here ϱ(q) denotes the number density of incoming par-
ticles with magnitude of momentum equal to q = |q|, ˆn
and ˆn′ are unit vectors (with dˆn and dˆn′ representing the
associated solid-angle differentials), and v(q) denotes the
speed of particles with momentum q. For the scattering
of massive environmental particles we have v(q) = q/m,
where m is each particle’s mass, while for the scatter-
ing of photons and other massless particles v(q) is equal
to the speed of light. The quantity |f(qˆn, qˆn′)|2 is the
differential cross section for the scattering of an environ-
mental particle from initial momentum q = qˆn to final
momentum q′ = qˆn′.
Whenever the mass of the central particle becomes
comparable to the mass of the environmental particles (as
in the case of air molecules scattered by small molecules
and free electrons [114]), the no-recoil assumption does
not hold and more general models for collisional deco-
herence have to be considered [35, 36].
The resulting
dynamics include dissipation, as well as decoherence in
both position and momentum.
To further evaluate the decoherence factor F(x − x′),
Eq. (38), we distinguish two important limiting cases. In
the short-wavelength limit, the typical wavelength of the
scattered environmental particles is much shorter than
the coherent separation ∆x = |x − x′| between the well-
localized wave packets in the spatial superposition state
of the system.
Then a single scattering event will be
able to fully resolve this separation and thus carry away
complete which-path information, leading to maximum
spatial decoherence per scattering event. In this limit,
F(x − x′) turns out to be simply equal to the total scat-
tering rate Γtot [9]. This implies the existence of an upper
limit for the decoherence rate when increasing the sepa-
ration ∆x, in contrast with decoherence rates obtained
from linear models [compare Eqs. (22) and (54)]. Equa-
tion (37) then shows that spatial interference terms will
become exponentially suppressed at a rate set by Γtot,

ρS(x, x′, t) = ρS(x, x′, 0)e−Γtott.
(39)


11

TABLE I. Estimates of decoherence timescales (in seconds)
for the suppression of spatial interferences over a distance ∆x
equal to the size a of the object (∆x = a = 10−3 cm for a
dust grain and ∆x = a = 10−6 cm for a large molecule). See
Ref. [9] for details.

Environment
Dust grain Large molecule

Cosmic background radiation
1
1024

Photons at room temperature
10−18
106

Best laboratory vacuum
10−14
10−2

Air at normal pressure
10−31
10−19

In the opposite long-wavelength limit, the environmen-
tal wavelengths are much larger than the coherent sep-
aration ∆x = |x − x′|, which implies that an individual
scattering event will reveal only incomplete which-path
information. For this case, one can show that spatial co-
herences become exponentially suppressed at a rate that
depends on the square of the separation ∆x [9],

ρS(x, x′, t) = ρS(x, x′, 0)e−Λ(∆x)2t,
(40)

where Λ is a scattering constant that encapsulates the
physical details of the interaction.
Thus, the quantity
Λ(∆x)2 plays the role of a decoherence rate.
The de-
pendence of this rate on ∆x is reasonable: if the envi-
ronmental wavelengths are much larger than ∆x, it will
require a large number of scattering events to encode
an appreciable amount of which-path information in the
environment, and this amount will increase, for a given
number of scattering events, as ∆x becomes larger. Note
that if ∆x is increased beyond the typical wavelength of
the environment, the short-wavelength limit needs to be
considered instead, for which the decoherence rate is in-
dependent of ∆x and attains its maximum possible value.
Numerical values of collisional decoherence rates ob-
tained from Eqs. (39) and (40), with the physically rele-
vant scattering parameters Γtot and Λ appropriately eval-
uated, have shown the extreme efficiency of collisions in
suppressing spatial interferences; Table I shows a few
classic order-of-magnitude estimates [8, 9, 17].
Excel-
lent agreement between theory and experiment has been
demonstrated for the decoherence of fullerenes due to col-
lisions with background gas molecules in a Talbot–Lau
interferometer [31, 115–118] (see Sec. VI B and Fig. 2),
and for the decoherence of sodium atoms in a Mach–
Zehnder interferometer due to the scattering of photons
[119] and gas molecules [120].

B.
Quantum Brownian motion

A classic and extensively studied model of decoherence
and dissipation is the one-dimensional motion of a par-
ticle weakly coupled to a thermal bath of noninteracting
harmonic oscillators, a model known as quantum Brown-
ian motion. The self-Hamiltonian HE of the environment

is given by

HE =
�

i

� 1

2mi
p2
i + 1

2miω2
i q2
i

�
,
(41)

where mi and ωi denote the mass and natural frequency
of the ith oscillator, and qi and pi are the canonical posi-
tion and momentum operators. The interaction Hamilto-
nian Hint describes the bilinear coupling of the system’s
position coordinate x to the positions qi of the environ-
mental oscillators, Hint = x ⊗ �

i ciqi, where the ci de-
note coupling strengths. This interaction represents the
continuous environmental monitoring of the position co-
ordinate of the system.
The Born–Markov master equation describing the evo-
lution of the density matrix ρS(t) of the system is given
by [9, 45]

d
dtρS(t) = −i
�
HS, ρS(t)
�

−
� ∞

0
dτ
�
ν(τ)
�
x,
�
x(−τ), ρS(t)
��

− iη(τ)
�
x,
�
x(−τ), ρS(t)
���
.
(42)

Here, x(τ) denotes the system’s position operator in the
interaction picture, x(τ) = eiHSτxe−iHSτ.
The curly
brackets { · , · } in the second line denote the anticom-
mutator {A, B} ≡ AB + BA. The functions

ν(τ) =
� ∞

0
dω J(ω) coth
� ω

2kT

�
cos (ωτ) ,
(43)

η(τ) =
� ∞

0
dω J(ω) sin (ωτ) ,
(44)

are known as the noise kernel and dissipation kernel, re-
spectively. The function J(ω), called the spectral density
of the environment, is given by

J(ω) ≡
�

i

c2
i

2miωi
δ(ω − ωi).
(45)

In general, spectral densities encapsulate the physi-
cal properties of the environment.
One frequently re-
places the collection of individual environmental oscilla-
tors by an (often phenomenologically motivated) contin-
uous function J(ω) of the environmental frequencies ω.
If we specialize to the important case of the system rep-
resented by a harmonic oscillator with self-Hamiltonian

HS =
1

2M p2 + 1

2MΩ2x2,
(46)

the resulting Born–Markov master equation is

d
dtρS(t) = −i
�
HS + 1

2M �Ω2x2, ρS(t)
�
−iγ
�
x,
�
p, ρS(t)
��

− D
�
x,
�
x, ρS(t)
��
− f
�
x,
�
p, ρS(t)
��
.
(47)


12

The coefficients �Ω2, γ, D, and f are defined as

�Ω2 ≡ − 2

M

� ∞

0
dτ η(τ) cos (Ωτ) ,
(48a)

γ ≡
1

MΩ

� ∞

0
dτ η(τ) sin (Ωτ) ,
(48b)

D ≡
� ∞

0
dτ ν(τ) cos (Ωτ) ,
(48c)

f ≡ − 1

MΩ

� ∞

0
dτ ν(τ) sin (Ωτ) .
(48d)

The first term on the right-hand side of Eq. (47) repre-
sents the unitary dynamics of a harmonic oscillator whose
natural frequency is shifted by �Ω. The second term de-
scribes momentum damping (dissipation) at a rate pro-
portional to γ, which depends only on the spectral den-
sity but not the temperature of the environment. The
third term is of the Lindblad double-commutator form
[see Eq. (31)] and describes decoherence of spatial coher-
ences over a distance ∆X at a rate D(∆X)2. Note that
D depends on both the spectral density J(ω) and the
temperature T of the environment. The fourth term also
represents decoherence, but its influence on the dynam-
ics of the system is usually negligible, especially at higher
temperatures. In the long-time limit γt ≫ 1, the master
equation (47) describes dispersion in position space given
by

∆X2(t) =
D

2m2γ2 t.
(49)

That is, the width ∆X(t) of the ensemble in position
space asymptotically scales as ∆X(t) ∝
√

t, just as in
classical Brownian motion; hence the term “quantum
Brownian motion.”
Figure 1 shows the time evolution of position-space and
momentum-space superpositions of two Gaussian wave
packets in the Wigner picture, as described by Eq. (47)
[28]. Interference between the two wave packets is rep-
resented by oscillations between the direct peaks. The
interaction with the environment damps these oscilla-
tions.
The damping occurs on different timescales for
the two initial conditions. While the momentum coordi-
nate is not directly monitored by the environment, the
intrinsic dynamics, through their creation of spatial su-
perpositions from superpositions of momentum, result
in decoherence in momentum space.
This interplay of
environmental monitoring and intrinsic dynamics leads
to the emergence of pointer states that are minimum-
uncertainty Gaussians (coherent states) well-localized in
both position and momentum, thus approximating clas-
sical points in phase space [5, 8, 16, 28, 44, 47, 48].
Let us consider the important case of an ohmic spectral
density J(ω) ∝ ω with a high-frequency cutoff Λ,

J(ω) = 2Mγ0

π
ω
Λ2

Λ2 + ω2 .
(50)

In the limit of a high-temperature environment (kT ≫ Ω
and kT ≫ Λ), we arrive at the Caldeira–Leggett master

x

p

x

p

FIG. 1. Evolution of superpositions of Gaussian wave packets
in quantum Brownian motion as studied in Ref. [28], visual-
ized in the Wigner representation. Time increases from top
to bottom. In the left column, the initial wave packets are
separated in position; in the right column, the separation is
in momentum.

equation [121],

d
dtρS(t) = −i
�
H′
S, ρS(t)
�
− iγ0
�
x,
�
p, ρS(t)
��

− 2Mγ0kT
�
x,
�
x, ρS(t)
��
,
(51)

where

H′
S = HS + 1

2M �Ω2x2 =
1

2M p2 + 1

2M
�
Ω2 − 2γ0Λ
�
x2

(52)
is the frequency-shifted Hamiltonian H′
S of the system.
This equation has been widely and successfully used to
model decoherence and dissipation processes, even in
cases where the assumptions were not strictly fulfilled
(for example, in quantum-optical settings, where often
kT ≲ Λ [122]).
In the position representation, the final term on the
right-hand side of Eq. (51) can be written as

− γ0

�x − x′

λdB

�2
ρS(x, x′, t),
(53)

where λdB = (2MkT)−1/2 is the thermal de Broglie wave-
length. This term describes spatial localization with a


13

decoherence rate τ −1
|x−x′| given by [18]

τ −1
|x−x′| = γ0

�x − x′

λdB

�2
.
(54)

This is Eq. (22), and as discussed there, given that λdB is
extremely small for macroscopic and even mesoscopic ob-
jects, we see that superpositions of macroscopically sepa-
rated center-of-mass positions will typically be decohered
on timescales many orders of magnitude shorter than the
dissipation (relaxation) timescale γ−1
0 . Over timescales
on the order of the decoherence time, we may therefore
often neglect the dissipative term in Eq. (51), leading to
the pure-decoherence master equation

d
dtρS(t) = −i
�
H′
S, ρS(t)
�
− 2Mγ0kT
�
x,
�
x, ρS(t)
��
. (55)

C.
Spin–boson models

In the spin–boson model, a qubit interacts with an
environment of harmonic oscillators. The seminal review
paper by Leggett et al. [29] discusses the dynamics of the
spin–boson model in great detail.
Let us first consider a simplified spin–boson model
where the self-Hamiltonian of the system is taken to be
HS = 1

2ω0σz, with eigenstates |0⟩ and |1⟩. In contrast
with the more general case discussed below, this Hamilto-
nian does not include a tunneling term − 1

2∆0σx, and thus
HS does not generate any nontrivial intrinsic dynamics.
We employ the familiar self-Hamiltonian, Eq. (41), for
an environment of harmonic oscillators, and choose the
bilinear interaction Hamiltonian Hint = σz ⊗�

i ciqi. Us-
ing the raising and lowering operators a† and a, we can
recast the total Hamiltonian as

H = 1

2ω0σz +
�

i
ωia†
iai + σz ⊗
�

i

�
gia†
i + g∗
i ai
�
. (56)

Note that since
�
H, σz
�
= 0, no transitions between |0⟩
and |1⟩ can be induced by H. There is no energy ex-
change between the system and the environment, and we
therefore deal with a model of decoherence without dis-
sipation. Such a model is a good representation of rapid
decoherence processes during which the amount of dissi-
pation is negligible, as is often the case in physical appli-
cations. The resulting evolution can be solved exactly [9].
For an ohmic spectral density with a high-frequency cut-
off, it is found that superpositions of the form α |0⟩+β |1⟩
are exponentially decohered on a timescale set by the
thermal correlation time (kT)−1 of the environment.
Inclusion of a tunneling term − 1

2∆0σx yields the gen-
eral spin–boson model defined by the Hamiltonian

H = 1

2ω0σz − 1

2∆0σx +
�

i

� 1

2mi
p2
i + 1

2miω2
i q2
i

�

+ σz ⊗
�

i
ciqi.
(57)

The rich non-Markovian dynamics of this model have
been analyzed in Refs. [29, 123]. The particular dynamics
strongly depend on the various parameters, such as the
temperature of the environment, the form of the spec-
tral density (subohmic, ohmic, or supraohmic), and the
system–environment coupling strength. For each param-
eter regime, a characteristic dynamical behavior emerges:
localization, exponential or incoherent relaxation, expo-
nential decay, and strongly or weakly damped coherent
oscillations [29].
In the weak-coupling limit, one can derive the Born–
Markov master equation in much the same way as in
the case of quantum Brownian motion (note the similar
structure of the Hamiltonians). The result is (see Ref. [9]
for details)

d
dtρS(t) = −i
�
H′
SρS(t) − ρS(t)H′†
S
�

− �D [σz, [σz, ρS(t)]] + ζσzρS(t)σy + ζ∗σyρS(t)σz.
(58)

The first term on the right-hand side of the master equa-
tion (58) represents the evolution under the environment-
shifted self-Hamiltonian H′
S, the second term corre-
sponds to decoherence in the σz eigenbasis of the system
at a rate given by �D, and the last two terms describe
the decay of the two-level system. H′
S is the renormal-
ized (and in general non-Hermitian) Hamiltonian of the
system. The coefficients ζ∗, �D, �f, and �γ are given by

ζ∗ = �f − i�γ,
(59a)

�D =
� ∞

0
dτ ν(τ) cos (∆0τ) ,
(59b)

�f =
� ∞

0
dτ ν(τ) sin (∆0τ) ,
(59c)

�γ =
� ∞

0
dτ η(τ) sin (∆0τ) ,
(59d)

with the noise and the dissipation kernels ν(τ) and η(τ)
taking the same form as in quantum Brownian motion
[see Eqs. (43) and (44)].

D.
Spin-environment models

A qubit linearly coupled to a collection of other
qubits—known also as a spin–spin model—is often a good
model of a single two-level system, such as a supercon-
ducting qubit, strongly coupled to a low-temperature en-
vironment [103, 104].
The model of a harmonic oscil-
lator interacting with a spin environment may be rele-
vant to the description of decoherence and dissipation in
quantum-nanomechanical systems and micron-scale ion
traps [124]. For details on the theory of spin-environment
models, see Refs. [104, 125–127].
A simple version of a spin–spin model is described by


14

the total Hamiltonian

H = HS + Hint = −1

2∆0σx + 1

2σz ⊗

N
�

i=1
giσ(i)
z

≡ −1

2∆0σx + 1

2σz ⊗ E.
(60)

Here, HS represents the intrinsic dynamics given by a
tunneling term, while Hint describes the environmental
monitoring of the observable σz.
The model can be solved exactly [128, 129], and
the resulting dynamics illustrate the dependence of the
preferred basis on the relative strengths of the self-
Hamiltonian of the system and the interaction Hamil-
tonian.
The preferred basis emerges as the local ba-
sis that is most robust under the total Hamiltonian.
When the interaction Hamiltonian dominates over the
self-Hamiltonian, the pointer states are found to be eigen-
states of the interaction Hamiltonian, in agreement with
the commutativity criterion, Eq. (18). Conversely, when
the modes of the environment are slow and the self-
Hamiltonian dominates the evolution of the system (the
quantum limit of decoherence [42]), the pointer states are
the eigenstates of the Hamiltonian of the system.
In the weak-coupling limit, spin environments can be
mapped onto oscillator environments [110, 130]. Specifi-
cally, the reduced dynamics of a system weakly coupled
to a spin environment can be described by the system
coupled to an equivalent oscillator environment described
by an explicitly temperature-dependent spectral density
of the form

Jeff(ω, T) ≡ J(ω) tanh
� ω

2kT

�
,
(61)

where J(ω) is the original spectral density of the spin
environment. (See Sec. 5.4.2 of Ref. [9] for details and
examples.)

V.
QUBIT DECOHERENCE, QUANTUM
ERROR CORRECTION, AND ERROR
AVOIDANCE

Quantum computation and quantum information pro-
cessing rely on coherent superpositions of mesoscopically
or macroscopically distinct states that are highly suscep-
tible to decoherence. Avoiding, controlling, and mitigat-
ing decoherence is therefore of paramount importance.
While the qubits need to be protected from detrimental
environmental interactions, we also need to be able to
control and measure them via a macroscopic apparatus.
The formidable challenge of designing a quantum com-
puter consists of meeting both demands in a balanced
way. Even so, decoherence induced by interactions with
the environment and the control apparatus, as well as
noise due to faulty gate operations, will likely be too
strong to allow for useful quantum computations to be
carried out [74, 131]. What is also needed is an active

mitigation of the effects of decoherence through active
quantum error correction [132–136].
We may distinguish two limiting cases for modeling
decoherence in qubits. The first limit is that of indepen-
dent qubit decoherence. Here, each qubit couples indepen-
dently to its own environment, without any interactions
between these environments. For example, this may be
the case if the qubits are spatially well-separated (rela-
tive to the typical coherence length of the environment)
and only couple to their immediate surroundings. Then
the error processes affecting the qubits will be completely
uncorrelated. Thus, if the probability of a particular er-
ror to affect one qubit is p, the probability of this error
to occur in K qubits will be pK. Many error-correcting
schemes are only efficient in correcting such single-qubit
errors, and thus the assumption of independent decoher-
ence frequently underlies these schemes. This assump-
tion, however, is unrealistic when the qubits are located
spatially close to each other. In this case, all qubits ap-
proximately feel the same environment, and it is likely
that errors will become correlated among multiple qubits.
The limiting case corresponding to this situation is that
of collective qubit decoherence, in which all qubits couple
to exactly the same environment.

A.
Correction of decoherence-induced quantum
errors

Consider a single qubit S, initially described by a pure
state |ψ⟩ and interacting with an environment E. One
can show that an arbitrary evolution of the combined
qubit–environment state can always be written in the
form

|ψ⟩ |e0⟩ −→ I |ψ⟩ |eI⟩ +
�

s=x,y,z
(σs |ψ⟩) |es⟩ ,
(62)

where the Pauli operators σs act on the Hilbert space of
S, and |eI⟩ and {|es⟩} are environmental states that are
not necessarily orthogonal or normalized. Thus, any in-
fluence of the environment on the qubit can be expressed
simply in terms of a weighted sum of the Pauli operators
and the identity operator acting on the original state of
the qubit. The effects of σx and σz on the qubit state are
often referred as a bit-flip error and phase-flip error, re-
spectively. If we restrict our attention to environmental
entanglement and the resulting decoherence effects, then
only phase-flip errors need to be taken into account.
For N qubits, Eq. (62) generalizes to

|ψ⟩ |e0⟩ −→
�

i
(Ei |ψ⟩) |ei⟩ .
(63)

Here |ψ⟩ is the initial N-qubit state, and the error op-
erators Ei are tensor products of N operators involv-
ing identity and Pauli operators. Equation (63) repre-
sents a worst-case scenario.
In many cases, simplified
versions can be used. One important case is that of par-
tial decoherence. Here, only a small number K < N of


15

qubits become entangled with the environment between
two successive applications of an error-correcting mech-
anism. Then it will be sufficient to restrict our attention
to the 2K possible error operators made up of at most K
operators σz and N − K identity operators. In the case
of independent qubit decoherence, we only need to con-
sider a collection of independent phase-flip errors acting
on single qubits, represented by error operators of the
form E = I ⊗ · · · ⊗ I ⊗ σz ⊗ I ⊗ · · · ⊗ I.
Given the entangled state on the right-hand side of
Eq. (63), the goal of quantum error correction is to re-
store the initial (unknown) state |ψ⟩. We let an ancilla,
described by an initial state |a0⟩, interact with the qubit
system such that

|a0⟩

��

i
(Ei |ψ⟩) |ei⟩

�

−→
�

i
|ai⟩ (Ei |ψ⟩) |ei⟩ .
(64)

Let us assume that the ancilla states |ai⟩ are at least
approximately mutually orthogonal, such that they can
be distinguished by measurement. We now measure the
observable OA = �

i ai|ai⟩⟨ai| on the ancilla, with ai ̸=
aj for i ̸= j. The projective measurement will yield a
particular outcome, say, ak, and lead to the reduction of
the entangled state,
�

i
|ai⟩ (Ei |ψ⟩) |ei⟩ −→ |ak⟩ (Ek |ψ⟩) |ek⟩ .
(65)

The outcome ak of the measurement tells us the counter-
transformation needed to restore the initial qubit state.
Applying E−1
k
= E†
k to the system gives

|ak⟩ (Ek |ψ⟩) |ek⟩
E−1
k
−−−→ |ak⟩ |ψ⟩ |ek⟩ .
(66)

Note that, as required in order to avoid introducing ad-
ditional decoherence in the computational basis of the
qubit system, we have obtained no information whatso-
ever about the state of the system.
This account of quantum error correction has been
highly idealized.
Let us mention three complications.
First, it is impossible to design an interaction between
the computational qubits and the ancilla that would al-
low us to distinguish, by measuring the ancilla, between
all possible errors. Second, in realistic settings the error
operators Ei may be very complex, and it remains to be
seen whether and how the corresponding countertrans-
formations can be applied without introducing signifi-
cant additional decoherence.
Third, the ancilla qubits
are physically similar to the computational qubits and
can therefore be expected to be equally prone to en-
vironmental interactions (and thus decoherence) as the
computational qubits themselves. Since the inclusion of
ancilla qubits increases the total number of qubits in the
quantum computer, and since decoherence rates typically
scale exponentially with the size of the system, it will re-
quire sophisticated experimental designs to ensure not
only that quantum error correction works in practice,
but also that it does not aggravate the problem of qubit
decoherence.

B.
Quantum computation on decoherence-free
subspaces

We introduced the concept of decoherence-free sub-
spaces (DFS) [49–58],
or pointer subspaces [3],
in
Sec. II C 2. DFS allow us to encode quantum informa-
tion in “quiet corners” of the Hilbert space to protect
it from environmental effects. In contrast with quantum
error correction, DFS prevent errors from happening in
the first place and thus represent a strategy for intrinsic
error avoidance.
The two limiting cases of independent qubit decoher-
ence and collective qubit decoherence delineate the lim-
its on the size of a DFS. To illustrate this relation-
ship, let us consider the case of collective decoherence
of an N-qubit system interacting with an oscillator bath
[49, 51, 53, 56, 137].
The interaction Hamiltonian for
this generalized spin–boson model is taken to be [com-
pare Eq. (56)]

Hint =

N
�

i=1
σ(i)
z
⊗
�

j

�
gija†
j + g∗
ijaj
�
≡

N
�

i=1
σ(i)
z
⊗ Ei.

(67)
The assumption of collective decoherence implies that
the couplings gij (and thus the environment operators
Ei) must be independent of the index i. Then Eq. (67)
becomes

Hint =

��

i
σ(i)
z

�

⊗ E ≡ Sz ⊗ E.
(68)

Recall that a DFS is spanned by a degenerate set of
eigenstates of the system operators Sα of the interaction
Hamiltonian [see Eq. (20)]. Thus, in our case the DFS
will be spanned by degenerate eigenstates of the collec-
tive spin operator Sz. Any N-qubit product state of the
computational basis states |0⟩ and |1⟩ (the eigenstates of
σz with eigenvalues +1 and −1, respectively) will be an
eigenstate of Sz. There are 2N +1 different possible inte-
ger eigenvalues m, ranging from m = −N (corresponding
to the basis state |1 · · · 1⟩) to m = +N (corresponding
to |0 · · · 0⟩). The largest number of mutually orthogonal
computational-basis states with the same eigenvalue m
of Sz is given by the set S0 of basis states with m = 0,
i.e., those with N/2 qubits in the state |0⟩. There are
n0 =
� N
N/2
�
such states in this set, spanning a DFS of di-
mension n0. For large values of N, we can approximate
the binomial coefficient using Stirling’s formula,

log2

� N
N/2

�
≈ N − 1

2 log2(πN/2)
N≫1
−−−→ N.
(69)

Therefore, in the limiting case of collective decoherence,
the dimension of our DFS approaches the dimension of
the original Hilbert space, and the encoding efficiency
approaches unity. For example, for N = 4 qubits, the set

S0 = { |0011⟩ , |0101⟩ , |0110⟩ , |1001⟩ , |1010⟩ , |1100⟩ }
(70)


16

spans a maximum-size DFS of dimension six, to be com-
pared with the dimension of the original Hilbert space,
which is 24 = 16. Thus, given the model for collective de-
coherence considered here, using four physical qubits we
can encode up to two logical qubits in a DFS (since en-
coding three logical qubits would already require a DFS
of dimension 23 = 8).
As mentioned in Sec. II C 2, the existence of a DFS
corresponds to a dynamical symmetry. Our model rep-
resents a case of perfect dynamical symmetry, since
the system–environment interaction, Eq. (68), is com-
pletely symmetric with respect to any permutations of
the qubits, thereby leading to a DFS of maximum size.
What happens if the symmetry is broken by additional
small independent coupling terms? It has been shown
[50, 138] that, to first order in the perturbation strength,
the storage of quantum information in DFS is stable to
such perturbations to all orders in time, but that the pro-
cessing of such quantum information encoded in DFS is
robust only to first order in time.
In the case of purely independent qubit decoherence,
the environment operators Ei appearing in Eq. (67) will
now differ from one another. To find a DFS, we follow
the usual strategy [see Eq. (20)] of determining a set of
orthonormal basis states {|si⟩} such that

�
I(1) ⊗ · · · ⊗ I(j−1) ⊗ σ(j)
z
⊗ I(j+1) ⊗ · · · ⊗ I(N)�
|si⟩

= λ(j) |si⟩
(71)

for all i and 1 ≤ j ≤ N. The only state fulfilling this
eigenvalue problem is |0 · · · 0⟩.
Since we need at least
a two-dimensional subspace to encode a single logical
qubit, the case of independent decoherence in the spin–
boson model does not allow for the existence of a DFS for
quantum computation. In the language of pointer sub-
spaces, there is only a single exact pointer state, and this
environment-superselected preferred state of the system
will be the ground state |0 · · · 0⟩.
In realistic settings, neither the assumption of purely
independent decoherence nor the limit of entirely collec-
tive decoherence will be entirely appropriate. We can,
however, use a DFS to protect the qubits from collective
decoherence effects, and we can recover from single-qubit
errors due to independent decoherence using active error-
correction methods. These two approaches can be con-
catenated [54] to enable universal fault-tolerant quantum
computation even when the restriction to single-qubit er-
rors is dropped [55, 139].

C.
Environment engineering and dynamical
decoupling

For reasonably large DFS to exist,
the system–
environment interaction must exhibit a sufficiently high
degree of symmetry.
Such symmetries are unlikely to
arise naturally in typical experimental settings.

One way of overcoming this limitation is based on envi-
ronment engineering. Here, one tries to generate certain
symmetries in the structure of the system–environment
interactions. For example, an appropriately engineered
symmetrization could make superposition states in Bose–
Einstein condensates correspond to (approximate) de-
generate eigenstates of the interaction Hamiltonian, in
which case such states would lie within a DFS, thereby
significantly enhancing their longevity [140].
In ion
traps, changing the parameters in the effective interac-
tion Hamiltonian for the trapped ion allows one to se-
lect different pointer subspaces and thereby control into
which DFS the trapped ion is driven [77, 79, 141, 142].
Another approach to the active creation of DFS is
known as dynamical decoupling [143–148].
Here time-
dependent modifications are introduced into the Hamil-
tonian of the system that counteract the influence of
the environment. These modifications take the form of
sequences of rapid projective measurements or strong
control-field pulses acting on the system (“quantum
bang-bang control” [143]).
Even if the structure of
the system–environment interaction Hamiltonian is not
known, decoherence can be suppressed arbitrarily well
in the limit of an infinitely fast rate of the decoupling
control field, thus dynamically creating a DFS (which
then represents a dynamically decoupled subspace). In
the realistic case of a finite control rate, sufficient (albeit
imperfect) protection from decoherence can be achieved
via this decoupling technique, provided the control rate
is larger than the fastest timescale set by the rate of for-
mation of environmental entanglement.

VI.
EXPERIMENTAL STUDIES OF
DECOHERENCE

Decoherence, of course, happens all around us, and
in this sense its consequences are readily observed. But
what we would like to do is to be able to experimen-
tally study the gradual and controlled action of deco-
herence. In this endeavor, several obstacles have to be
overcome. We need to prepare the system in a superpo-
sition of mesoscopically or even macroscopically distin-
guishable states with a sufficiently long decoherence time
such that the gradual action of decoherence can be re-
solved. We must be able to monitor decoherence without
introducing a significant amount of additional, unwanted
decoherence. We would also like to have sufficient con-
trol over the environment so we can tune the strength
and form of its interaction with the system.
Starting
in the mid-1990s, several such experiments have been
performed, for example, using cavity QED [12], meso-
scopic molecules [149], and superconducting systems such
as SQUIDs and Cooper-pair boxes [13]. Bose–Einstein
condensates [150] and quantum nanomechanical systems
[151, 152] are promising candidates for future experimen-
tal tests of decoherence.
These experiments are important for several reasons.


17

They are impressive demonstrations of the possibility
of generating nonclassical quantum states in mesoscopic
and macroscopic systems. They show that the quantum–
classical boundary is smooth and can be shifted by vary-
ing the relevant experimental parameters.
They allow
us to test and improve decoherence models, and they
help us design devices for quantum information process-
ing that are good at evading the detrimental influence
of the environment. Finally, such experiments may be
used to test quantum mechanics itself [13]. Such tests re-
quire sufficient shielding of the system from decoherence
so that an observed (full or partial) collapse of the wave-
function could be unambigously attributed to some novel
nonunitary mechanism in nature, such as those proposed
in dynamical reduction models [153–155]. This shielding,
however, is difficult to implement in practice, because
the large number of particles required for the reduction
mechanism to become effective will also lead to strong
decoherence [114, 156].
The superpositions realized in
current experiments are still not sufficiently macroscopic
to rule out collapse theories, although it has been demon-
strated [118] that matter-wave interferometry with large
molecular clusters (in the mass range between 106 and
108 amu) would be able to test the collapse theories pro-
posed in Refs. [154, 155]; such experiments may soon
become technologically feasible [11].

A.
Atoms in a cavity

In 1996 Brune et al. generated a superposition of ra-
diation fields with classically distinguishable phases in-
volving several photons [12, 150, 157]. This experiment
was the first to realize a mesoscopic Schr¨odinger-cat state
and allowed for the controlled observation and manipu-
lation of its decoherence. A rubidium atom is prepared
in a superposition of energy eigenstates |g⟩ and |e⟩ cor-
responding to two circular Rydberg states.
The atom
enters a cavity C containing a radiation field contain-
ing a few photons. If the atom is in the state |g⟩, the
field remains unchanged, whereas if it is in the state
|e⟩, the coherent state |α⟩ of the field undergoes a phase
shift φ, |α⟩ −→
��eiφα
�
; the experiment achieved φ ≈ π.
An initial superposition of the atom is therefore am-
plified into an entangled atom–field state of the form
1
√

2 (|g⟩ |α⟩ + |e⟩ |−α⟩).
The atom then passes through
an additional cavity, further transforming the superposi-
tion. Finally, the energy state of the atom is measured.
This disentangles the atom and the field and leaves the
latter in a superposition of the mesoscopically distinct
states |α⟩ and |−α⟩.
To monitor the decoherence of this superposition, a
second rubidium atom is sent through the apparatus. Af-
ter interacting with the field superposition state in the
cavity C, the atom will always be found in the same en-
ergy state as the first atom if the superposition has not
been decohered. This correlation rapidly decays with in-
creasing decoherence. Thus, by recording the measure-

ment correlation as a function of the wait time τ between
sending the first and second atom through the appara-
tus, the decoherence of the field state can be monitored.
Experimental results were in excellent agreement with
theoretical predictions [158, 159]. It was found that de-
coherence became faster as the phase shift φ and the
mean number ¯n = |α|2 of photons in the cavity C was
increased. Both results are expected, since an increase
in φ and ¯n means that the components in the superpo-
sition become more distinguishable. Recent experiments
have realized superposition states involving several tens
of photons [160] and have monitored the gradual deco-
herence of such states [161].

B.
Matter-wave interferometry

In these experiments (see Ref. [11] for a review), spatial
interference patterns are demonstrated for mesoscopic
molecules ranging from fullerenes [162] to molecular clus-
ters involving hundreds of atoms, with a total size of
up to 60 ˚A and masses of several thousand amu (see
Fig. 2) [163, 164].
Since the de Broglie wavelength of
such molecules is on the order of picometers, standard
double-slit interferometry is out of reach. Instead, the
experiments make use of the Talbot effect, an interfer-
ence phenomenon in which a plane wave incident on a
diffraction grating creates an image of the grating at mul-
tiples of a distance L behind the grating. In the experi-
ment, the molecular density (at a macroscopic distance L
from the grating) is scanned along the direction perpen-
dicular to the molecular beam.
An oscillatory density
pattern (corresponding to the image of the slits in the
grating) is observed, confirming the existence of coher-
ence and interference between the different paths of each
individual molecule passing through the grating. Recent
experiments have used an improved version of the origi-
nal Talbot–Lau setup [165], as well as optical ionization
gratings [166].
Decoherence is measured as a decrease of the visibil-
ity of this pattern (Fig. 2). The controlled decoherence
due to collisions with background gas particles [115, 116]
and due to emission of thermal radiation from heated
molecules [168] has been observed, showing a smooth de-
cay of visibility in agreement with theoretical predictions
[31, 117, 167]. These successes have led to speculations
that one could perform similar experiments using even
larger particles such as proteins and viruses [115, 169]
or carbonaceous aerosols [170]. Such experiments will be
limited by collisional and thermal decoherence and by
noise due to inertial forces and vibrations [115, 169, 170].

C.
Superconducting systems

Superconducting
quantum
interference
devices
(SQUIDs)
and
Cooper-pair
boxes
have
important
applications in quantum information processing.
A


18
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1263

NATURE COMMUNICATIONS | 2:263 | DOI: 10.1038/ncomms1263 | www.nature.com/naturecommunications

11 Macmillan Publishers Limited. All rights reserved.

ysics, single-particle 
regarded as a para-
feature of quantum 
objects of our mac-
rinciple has become 
ng feld of quantum 
ch in many labora-
nderstanding of the 
uantum systems and 
o the observation of 

m interference with 
r successful experi-
our study focuses on 
ion of the molecule 
ce. We do this with 
vide useful molecu-
1 compares the size 
8 and PFNS10, with 
traphenylporphyrin 
PF84 and TPPF152. 
molecules in a three-
apitza-Dirac-Talbot-

rated in a thermal 
ravitational free-fall 
meter itself consists 
amber at a pressure 
mbrane with 90-nm 
6 nm. Each slit of G1 
ecular position that, 
ads to a momentum 

delocalization and 
increasing distance 
ser light wave with a 
een the electric laser 
y creates a sinusoidal 
t matter waves. Te 
n such that quantum 
c molecular density 
structure is sampled 
cal to G1) across the 

of the transmitted 
MS).
added various tech-
to liquid samples, a 
tial to maintain the 
owed us to increase 
r and many optimi-
were needed to meet 
s with very massive 

tum interferograms 
re 3. In all cases the 
ude of the sinusoidal 
al, exceeds the maxi-
y a signifcant multi-
t shown for TPPF84 
ed interference con-
ith individual scans 
) and Vobs = 49% for 
n, we have observed 
10 and Vobs = 16 � 2% 

for TPPF152 (see Figure 3), in which our classical model predicts 
Vclass = 1%. Tis supports our claim of true quantum interference for 
all these complex molecules.

Te most massive molecules are also the slowest and therefore 

the most sensitive ones to external perturbations. In our particle 

Figure 1 | Gallery of molecules used in our interference study. (a) The 
fullerene C60 (m = 720 AMU, 60 atoms) serves as a size reference and 
for calibration purposes; (b) The perfluoroalkylated nanosphere PFNS8 
(C60[C12F25]8, m = 5,672 AMU, 356 atoms) is a carbon cage with eight 
perfluoroalkyl chains. (c) PFNS10 (C60[C12F25]10, m = 6,910 AMU, 430 
atoms) has ten side chains and is the most massive particle in the set. 
(d) A single tetraphenylporphyrin TPP (C44H30N4, m = 614 AMU, 78 
atoms) is the basis for the two derivatives (e) TPPF84 (C84H26F84N4S4, 
m = 2,814 AMU, 202 atoms) and (f) TPPF152 (C168H94F152O8N4S4, 
m = 5,310 AMU, 430 atoms). In its unfolded configuration, the latter is the 
largest molecule in the set. Measured by the number of atoms, TPPF152 
and PFNS10 are equally complex. All molecules are displayed to scale. The 
scale bar corresponds to 10 Å.

y

X

Detector

G1

G2

G3

S3

S2

S1

Oven

Lens

Laser

Z

Figure 2 | Layout of the Kapitza-Dirac-Talbot-Lau (KDTL) interference 
experiment. The effusive source emits molecules that are velocity-selected 
by the three delimiters S1, S2 and S3. The KDTL interferometer is composed 
of two SiNx gratings G1 and G3, as well as the standing light wave G2. The 
optical dipole force grating imprints a phase modulation �(x)��opt·P/(v·wy) 
onto the matter wave. Here �opt is the optical polarizability, P the laser 
power, v the molecular velocity and wy the laser beam waist perpendicular 
to the molecular beam. The molecules are detected using electron impact 
ionization and quadrupole mass spectrometry.

0
0.4
0.8
1.2
1.6
4

6

8
10

20

30

visibility (%)

pressure (in 10−6 mbar)

FIG. 2. Left: Molecular clusters used in recent interference
experiments, drawn to scale (the scale bar represents 10 ˚A).
Figure from Ref. [163].
(a) Fullerene C60 (m = 720 amu,
60 atoms). (b) Perfluoroalkylated nanosphere PFNS8 (m =
5672 amu, 356 atoms).
(c) PFNS10 (m = 6910 amu, 430
atoms). (d) Tetraphenylporphyrin TPP (m = 614 amu, 78
atoms).
(e) TPPF84 (m = 2814 amu, 202 atoms).
(f)
TPPF152 (m = 5310 amu, 430 atoms). Right: Visibility of
interference fringes of C70 fullerenes as a function of the pres-
sure of the background gas. Measured values (circles) agree
well with the theoretical prediction (solid line) [31, 117, 167]
describing an exponential decay of visibility with pressure.
Figure adapted from Ref. [115].

SQUID consists of a ring of superconducting material
interrupted by thin insulating barriers, the Josephson
junctions (Fig. 3a).
At sufficiently low temperatures,
electrons of opposite spin condense into bosonic Cooper
pairs.
Quantum-mechanical tunneling of Cooper pairs
through the junctions leads to the flow of a resistance-
free supercurrent around the loop (Josephson effect),
which creates a magnetic flux threading the loop. The
collective center-of-mass motion of a macroscopic num-
ber (∼ 109) of Cooper pairs can then be represented by
a wave function labeled by a single macroscopic variable,
namely, the total trapped flux Φ through the loop.
The two possible directions of the supercurrent define
a qubit with basis states {|⟳⟩ , |⟲⟩}.
By adjusting an
external magnetic field, the SQUID can be biased such

(a)

(a)

80%

60%

40%

5
0

probability for ⟳

(b)

Josephson junction

superconducting

ring

supercurrent

(b)

(a)

delay time τ (ns)

80%

60%

40%

5
10
15
20
25
30
35
0

probability for ⟳

(b)

Josephson junction

superconducting

ring

supercurrent

FIG. 3. (a) Schematic illustration of a SQUID. A supercon-
ducting ring is interrupted by Josephson junctions, leading
to a dissipationless supercurrent.
(b) Decoherence of a su-
perposition of clockwise and counterclockwise supercurrents
in a superconducting qubit. The damping of the oscillation
amplitude corresponds to the gradual loss of coherence from
the system. Figure adapted from Ref. [173].

that the two lowest-lying energy eigenstates |0⟩ and |1⟩
are equal-weight superpositions of the persistent-current
states |⟳⟩ and |⟲⟩.
Such superposition states involving µA currents were
first experimentally observed in 2000 using spectroscopic
measurements [171, 172]. Their decoherence was subse-
quently measured using Ramsey interferometry [173], as
follows. Two consecutive microwave pulses are applied to
the system. During the delay time τ between the pulses,
the system evolves freely. After application of the second
pulse, the system is left in a superposition of |⟳⟩ and |⟲⟩,
with the relative amplitudes exhibiting an oscillatory de-
pendence on τ. A series of measurements in the basis
{|⟳⟩ , |⟲⟩} over a range of delay times τ then allows one
to trace out an oscillation of the occupation probabilities
for |⟳⟩ and |⟲⟩ as a function of τ (Fig. 3b). The envelope
of the oscillation is damped as a consequence of decoher-
ence acting on the system during the free evolution of
duration τ. From the decay of the envelope we can infer
the decoherence timescale; the original experiment gave
20 ns [173], while subsequent experiments have achieved
decoherence times of several µs [174].
Superpositions states and their decoherence have also
been observed in superconducting devices whose key vari-
able is charge (or phase), instead of the flux variable used
in SQUIDs. Such Cooper-pair boxes consist of a small
superconducting island onto which Cooper pairs can tun-
nel from a reservoir through a Josephson junction. Two
different charge states of the island, differing by at least
one Cooper pair, define the basis states. Coherent os-
cillations between such charge states were first observed


19

in 1999 [175]. In 2002, Vion et al. [176] reported thou-
sands of coherent oscillations with a decoherence time
of 0.5 µs. Similar results have been obtained for phase
qubits [177, 178], demonstrating decoherence times of
several µs.

VII.
DECOHERENCE AND THE
FOUNDATIONS OF QUANTUM MECHANICS

Can decoherence address foundational problems? If so,
which ones, and how? Addressing these subtle questions
is beyond the scope of this review; a few brief remarks
must suffice here.
(See Refs. [6, 7, 9, 21] for in-depth
discussions.) Decoherence, at its heart, is a technical re-
sult concerning the dynamics and measurement statistics
of open quantum systems. From this view, decoherence
merely addresses a consistency problem, by explaining
how and when the quantum probability distributions ap-
proach the classically expected distributions. Since deco-
herence follows directly from an application of the quan-
tum formalism to interacting quantum systems, it is not
tied to any particular interpretation of quantum mechan-
ics, nor does it supply such an interpretation, nor does it
amount to a theory that could make predictions beyond
those of standard quantum mechanics.
The predictively relevant part of decoherence theory
relies on reduced density matrices, whose formalism and
interpretation presume the collapse postulate and Born’s

rule. If we understand the “quantum measurement prob-
lem” as the question of how to reconcile the linear, de-
terministic evolution described by the Schr¨odinger equa-
tion with the occurrence of random measurement out-
comes, then decoherence has not solved this problem
[6, 9].
Decoherence does, however, address an aspect
sometimes associated with the quantum measurement
problem, namely the preferred-basis problem (at least
in the sense described in Sec. II C). Further explorations
of the role of the environment, such as in quantum Dar-
winism (see Sec. II D), can help illuminate fundamental
questions concerning information transfer and amplifica-
tion in the quantum setting.
Decoherence has been used to identify internal con-
cistency issues in interpretations of quantum mechanics,
and the picture associated with the decoherence process
has sometimes been seen as suggestive of particular inter-
pretations of quantum mechanics [6, 7]. Indeed, histori-
cally decoherence theory arose in the context of Zeh’s [1]
independent formulation of an Everett-style interpreta-
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however, it seems that certain interpretations simply may
be more in need of decoherence than others for defin-
ing their structure; see Ref. [180] for the example of an
Everett-style interpretation [23]. At the end of the day,
any interpretation that does not involve entities, claims,
or structures in contradiction with the predictions of de-
coherence theory (which is to say, with the predictions of
quantum mechanics) will arguably remain viable.

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