Quantum Zeno Effect in the Decoherent Histories
arXiv:0704.1551v1  [quant-ph]  12 Apr 2007
Quantum Zeno Effect in the Decoherent Histories
Petros Wallden∗
Abstract
The quantum Zeno effect arises due to frequent observation. That implies
the existence of some experimenter and its interaction with the system. In this
contribution, we examine what happens for a closed system if one considers
a quantum Zeno type of question, namely what is the probability of a system,
remaining always in a particular subspace. This has implications to the arrival
time problem that is also discussed. We employ the decoherent histories approach
to quantum theory, as this is the better developed formulation of closed system
quantum mechanics, and in particular, dealing with questions that involve time
in a non-trivial way. We get a very restrictive decoherence condition, that implies
that even if we do introduce an environment, there will be very few cases that
we can assign probabilities to these histories, but in those cases, the quantum
Zeno effect is still present.
1 Motivation
A remarkable property of quantum mechanics, is the so called quantum Zeno effect
[1]. This effect, is that frequent observation slow down the evolution of the state,
with the limit of continuous observation leading to “freezing” of the state1. This
has been experimentally verified. The intuitive explanation, is that the interaction of
the observer with the system leads to this apparent paradox. It would therefore be
interesting to see whether this effect persists if we consider a closed system. We would
try to see what is the probability of a closed system remaining in a particular subspace
of its Hilbert space with no external observer. This directly relates to the arrival time
problem as well (e.g. [2, 3]). Having said that, we should emphasize that in closed
systems, we cannot in general assign probabilities to histories, unless they decohere
and it is this property that resolves the apparent paradox that arises.
2 This paper
This contribution is largely based on Ref.[3]. In Section 3 we revise the quantum Zeno
effect and the decoherent histories, and introduce a new formula for the restricted
propagator that will be of use further. In Section 4.1 we see what probabilities we
would get if we had decoherence, that highlights the persistence of the quantum Zeno
effect. In Section 4.2 we get the decoherence condition that in Section 5 is stressed
how restrictive is by considering the arrival time problem. We conclude in Section 6.
∗Raman Research Institute, Theoretical Physics Group; Sadashivanagar, Bangalore - 560 080,
India; on leave from: Imperial College, Theoretical Physics Group; Blackett Laboratory, London
SW7 2BZ, UK; email: petros.wallden@gmail.com
1To be more precise, restriction to a subspace.
http://arxiv.org/abs/0704.1551v1
3 Introductory material
3.1 Quantum Zeno effect
In standard Copenhagen quantum mechanics, the measurement is represented by pro-
jecting the state to a subspace defined by the eigenstates that correspond to the range
of eigenvalues of the measured physical quantity. The latter is represented by a self-
adjoint operator. The state, otherwise evolves unitarily: Û(t) = exp(−iĤt), where
Ĥ is the Hamiltonian. It is then a mathematical fact, that frequent measurement, of
the same quantity (subspace) leads to slow down of the evolution, i.e. decreases the
probability that the state evolves outside the subspace in question. This resembles the
ancient Greek, Zeno paradox (Zήνων), and thus the name.
The continuum measurement limit, leads to zero probability of leaving the observed
subspace. The state continues to evolve (unitarily), but restricted in the subspace of
observation [4]. This implies that if we project to a one-dimensional subspace, the state
stops evolving. In most literature, the question is of a particle decaying or not, so the
last comment applies. In particular, the above phenomenon is still present for infinite
dimensional Hilbert spaces, but provided that the restricted Hamiltonian (Hr = PHP )
is self-adjoint, as we will see later.
3.2 Decoherent histories
Decoherent histories approach to quantum theory is an alternative formulation de-
signed to deal with closed systems and it was developed by Griffiths [5], Omnès [6],
and Gell-Mann and Hartle [7]. There is no external observer, no a-priori environment-
system split. The main mathematical aim of this approach, is to see when is it mean-
ingful to assign probabilities to a history of a closed quantum system and of course to
determine this probability.
Here we will revise the standard non-relativistic quantum mechanics in decoherent
histories formulation. To each history (α) corresponds a particular class operator Cα,
Cα = Pαne
−iH(tn−tn−1)Pαn−1 · · · e
−iH(t2−t1)Pα1 (1)
Where Pα1 etc are projection operators corresponding to some observable, H is the
Hamiltonian, and tn is the total time interval we consider. This class operator corre-
sponds to the history, the system is at the subspace spanned by Pα1 at time t1 at Pα2
at time t2 and so on. The probability for this history, provided we had some external
observer making the measurement at each time tk would be
p(α) = D(α, α) = Tr(CαρC
α) (2)
where ρ is the initial state. In the case of a closed system, Eq.(2) fails in general to be
probability due to interference2.
There are, however, certain cases where we can assign probabilities. This happens
if for a complete set of histories, they pairwise obey
D(α, β) = Tr(CαρC
) = 0 ∀ α 6= β (3)
In that case, the complete set of histories is called decoherent set of histories and we
can assign to each history of this set the probability of Eq.(2). In order to achieve a set
2The additivity of disjoint regions of the sample space is not satisfied by Eq.(2)
of histories that satisfy Eq.(3) in general we need to consider coarse grained histories,
or/and very specific initial state ρ3.
To sum up, in decoherent histories we need to first construct a class operators that
corresponds to the histories of interest4, and then confirm that these histories satisfy
Eq.(3). Only then we can give an answer.
3.3 The restricted propagator
A mathematical object that will be needed for computing the suitable class operators,
is the restricted propagator. This is the propagator restricted to some particular region
∆ (of the configuration space) that corresponds to a subspace of the total Hilbert space
denoted by H∆. The most common (but not the most general) is the path integral
definition:
gr(x, t | x0, t0) =
Dx exp(iS[x(t)]) = 〈x|gr(t, t0)|x0〉 (4)
The integration is done over paths that remain in the region ∆ during the time interval
[t, t0]. The S[x(t)] is as usual the action. The operator form of the above is given by
[8, 9]:
gr(t, t0) = lim
Pe−iH(tn−tn−1)P · · ·Pe−iH(t1−t0)P (5)
With tn = t, δt → 0 and n → ∞ simultaneously keeping δt × n = (t − t0). H is
the Hamiltonian operator. P is a projection operator on the restricted region ∆. We
therefore have
gr(x, t | x0, t0) = 〈x|gr(t, t0)|x0〉 (6)
Note here that the expression Eq.(5) is the defining one for cases that the restricted
region is not a region of the configuration space, but some other subspace of the total
Hilbert space H. The differential equation obeyed by the restricted propagator is:
−H)gr(t, t0) = [P,H ]gr(t, t0) (7)
Which is almost the Schrödinger equation, differing by the commutator of the projec-
tion to the restricted region with the Hamiltonian.
The most useful form, for our discussion was derived in Ref. [3]
gr(t, t0) = P exp (−i(t − t0)PHP )P (8)
Note that PHP is the Hamiltonian projected in the subspace H∆. To prove Eq.(8) we
multiply Eq. (7) with P we will then get
− PHP )gr(t, t0) = 0 (9)
using the fact that P [H,P ]P = 0 and that the propagator has a projection P at the
final time. This is Schrödinger equation with Hamiltonian PHP . It is evident that
3Note that the interaction of a system with an environment that brings decoherence, in the histories
vocabulary, is just a particular type of coarse graining where we ignore the environments degrees of
freedom.
4Note that the same classical question can be turned to quantum with several, possibly inequivalent
ways. Due to this property, the construction of the suitable class operator is important for questions
such as for example, the arrival time or reparametrization invariant questions.
this leads to the full propagator in H∆ provided that the operator PHP is self-adjoint
in this subspace [4]5.
4 Quantum Zeno histories
In this section we will examine the question what is the probability for a system to
remain in a particular subspace, during a time interval ∆t = t − t0. We will see the
probabilities and decoherence conditions for the general case, and then see what this
implies for the arrival time problem, which is just a particular example.
4.1 The class operator and probabilities
There are several ways of turning the above classical proposition to a quantum me-
chanical one. The most straight forward is the following. We consider a system being
in one subspace by projecting to that, and the history of always remaining in that
subspace corresponds to the limit of projecting to the region evolving unitarily but for
infinitesimal time and then projecting again, i.e. taking the δt between the proposi-
tions going to zero. The class operator for remaining always in that subspace follows
from Eq.(1) by taking each Pαk being the same (P ) and taking the limit of (tk − tk−1)
going to zero for each k. We then have
Cα(t, t0) = gr(t, t0) (10)
and the class operator for not remaining at this subspace during all the interval is
naturally
Cβ(t, t0) = g(t, t0)− gr(t, t0) (11)
with g(t, t0) = exp(−iH(t− t0)) the full propagator.
Let us, for the moment, assume that the initial state |ψ〉 is such, that we do
have decoherence. We will return later to see when this is the case. The (candidate)
probability is
p(α) = 〈ψ|g†r(t, t0)gr(t, t0)|ψ〉 (12)
Following Eq.(8) it is clear6 that
g†r(t, t0)gr(t, t0) = P (13)
which then implies
p(α) = 〈ψ|P |ψ〉 (14)
For an initial state that is in the subspace defined by P , the probability to remain in this
subspace is one. This is the usual account of the quantum Zeno effect. As it is stressed
in other literature, to have the quantum Zeno is crucial that the restricted Hamiltonian
Hr = PHP to be self-adjoint operator in the subspace. Note, that this only states
that the system remain in the subspace, but it does not “freeze” completely and in
particular follows unitary evolution in the subspace with Hamiltonian, the restricted
5A detailed proof from Eq.(5) can be found in [3].
6Provided PHP is self-adjoint in the subspace. This is true for finite dimensional Hilbert spaces
and has been shown to be true for regions of the configuration space in a Hamiltonian with at most
quadratic momenta [4].
one Hr. The form of Eq.(8) of the restricted propagator makes the latter comment
more transparent.
4.2 Decoherence condition
All this is well understood for open systems with external observers. To assign the
candidate probability of Eq.(12) as a proper probability of a closed system, we need
the system to obey the decoherence condition, i.e.
D(α, β) = 〈ψ|C
βCα|ψ〉 = 0 (15)
and this implies that
〈ψ|g†r(t, t0)g(t, t0)|ψ〉 = 〈ψ|P |ψ〉 (16)
which is a very restrictive condition and only very few states satisfy this, as we will
see in the arrival time example. The condition, essentially states that the overlap of
the time evolved state (g(t, t0)|ψ〉) with the state evolved in the subspace (gr(t, t0)|ψ〉)
should be the same at the times t0 and t. Given that the restricted Hamiltonian leads,
in general, to different evolution, the condition refers only to very special initial states
with symmetries, or for particular time intervals ∆t.
5 Arrival time problem
The arrival time problem is the following: What is the probability that the system
crosses a particular region ∆ of the configuration space, at any time during the time
interval ∆t = (t − t0). One can attempt to answer this, by considering what is the
probability that the system remains always in the complementary region ∆̄. So if Q
is the total configuration space, we have ∆ ∪ ∆̄ = Q and ∆ ∩ ∆̄ = ∅. Taking this
approach to the arrival time problem, the relation with the quantum Zeno histories is
apparent, since it is just the special case, where the subspace of projection is a region
of the configuration space (∆̄) and the Hamiltonian is quadratic in momenta, i.e.
|x〉〈x|dx
Ĥ = p̂2/2m+ V (x̂) (17)
This particular case is infinite dimensional, but as shown in Ref. [4] the restricted
Hamiltonian is indeed self-adjoint and the arguments of the previous section apply.
Before proceeding further, we should point out that one could construct different
class operators that would also correspond to the (classical) arrival time question. For
example, one could consider having POVM’s7 instead of projections at each moment
of time, or could have a finite (but frequent) number of projections (not taking the
limit where δt→ 0). These and other approaches are not discussed here.
Let us see now, what the quantum Zeno effect implies about the arrival time. It
states that a system initially localized outside ∆ will always remain outside ∆ (if it
decoheres) and therefore we can only get zero crossing probabilities. This is definitely
surprising, since for a wave packet that is initially localized in ∆̄ and its classical
trajectory crosses region ∆, we would expect to get crossing probability one. The
resolution comes due to the decoherence condition as will be argued later.
7Positive Operator Valued Measure
Returning to the decoherence condition Eq.(16) we see that there is the overlap of
the time evolved state with the restricted time evolved state. In the arrival time case,
the restricted Hamiltonian corresponds to the Hamiltonian in the restricted region (∆̄)
but with infinite potential walls on the boundary (i.e. perfectly reflecting). We then
get decoherence in the following four cases.
(a) The initial state |ψ〉 is in an energy eigenstate, and it also vanishes on the bound-
ary of the region.
(b) The restricted propagator can be expressed by the method of images8 and the
initial state shares the same symmetry.
(c) The full unitary evolution in the time interval ∆t remains in the region ∆̄.
(d) Recurrence: Due to the period of the Hamiltonian and the restricted Hamiltonian
their overlap happens to be the same after some time t as it was in time t0. This
depends sensitively on the time interval and it is thus of less physical significance.
It is now apparent that most initial states do not satisfy any of those conditions. In
particular, the wavepacket that classically would cross the region ∆, will not satisfy
any of these conditions, and we would not be able to assign the candidate probability
as a proper one, and thus we avoid the paradox. The introduction of an interacting
environment to our system, (that usually produces decoherence by coarse-graining the
environment) does not change the probabilities and contrary to the intuitive feeling, it
does not provide decoherence for the particular type of question we consider. This still
leave us with no answer for any of the cases that the system would classically cross the
region. The latter implies, that the straight forward coarse grainings we used, were
not general enough to answer fully the arrival time question 9.
As a final note, we should point out that the quantum Zeno effect in the decoherent
histories, has implications for the decoherent histories approach to the problem of time
(e.g. Refs. [9, 3]).
6 Conclusions
We examined the quantum Zeno type of histories of a closed system, using the deco-
herent histories approach. We show that the quantum Zeno effect is still present, but
only for the very few cases that we have decoherence. The situation does not change
with the introduction of interacting environment. We see that while in the open sys-
tem quantum Zeno, the delay of the evolution arises as interaction with the observer,
in the closed system we have the decoherence condition “replacing” the observer and
resolving the apparent paradox.
Acknowledgments: The author is very grateful to Jonathan J. Halliwell for many
useful discussions and suggestions, and would like to thank the organizers for giving
the opportunity to give this talk and hosting this very interesting and nice conference.
8Note that the restricted propagator can be expressed using the method of images, if and only if
there exist a set of energy eigenstates, vanishing on the boundary, that when projected on the region
∆̄ forms a dense subset of the subspace H
, i.e. span H
. This is equivalent with requiring that
the restricted energy spectrum (i.e. spectrum of the restricted Hamiltonian H
) is a subset of the
(unrestricted) energy spectrum, which is not in general the case.
9For more details, examples and discussion see Ref. [3].
References
[1] B. Misra and E.C.G. Sudarshan, J. Math. Phys. 18, 756 (1977).
[2] J.J.Halliwell and E.Zafiris, Phys.Rev. D57, 3351 (1998).
[3] P. Wallden, gr-qc/0607072.
[4] P. Facchi, S. Pascazio, A. Scardicchio, and L. S. Schulman, Phys. Rev. A 65,
012108 (2002).
[5] R.B. Griffiths, J. Stat. Phys., 36 219, (1984).
[6] R. Omnès. J. Stat. Phys. 53, 893 (1988).
[7] M. Gell-Mann and J. Hartle, in Complexity, Entropy and the Physics of Infor-
mation, SFI Studies in the Science of Complexity, Vol. VIII, edited by W. Zurek,
(Addison-Wesley, Reading, 1990).
[8] J. J. Halliwell, quant-ph/9506021.
[9] J. J. Halliwell and P. Wallden, Phys. Rev. D 73 024011 (2006), gr-qc/0509013.
