6.3 The measurement problem
6.3.3 Decoherence as a non-unitary evolution
In the last four decades it has become clear that the theoretical description of a measurement, outlined above, cannot be considered complete. It neglects that the object-apparatus system is never totally isolated, but always interacts with its environment. This failure is removed by decoherence, which in recent years has been experimentally verified.13 The purpose of this section is to show that decoherence is
a non unitary process that has to be taken seriously, that is, it cannot completely be dismissed as having no implications for the measurement process. If this is true, then premise two of the strong measurement problem is false, since the state vector does not always evolve according to unitary dynamics. As a consequence, there is no proof of the insolubility of the measurement problem. Even though I want to stress that this does not constitute a solution to the measurement problem either.
To gain a better idea of how decoherence works, let us take a look at a simple model that can be found in Zurek (1982, 2003), Giulini et al. (1996, ch. 3.1.3) and Schlosshauer (2004, pp. 1276-1278). Suppose that we have a system | i that is surrounded by an environment |Ei. For simplicity, the object, which is measured, and the apparatus, which measures the object, are combined in the system | i. Writing them separately would not essentially change the model. Now, at time t = 0 the system is in a superposition of spin states
12Maudlin formulates this measurement problem in three statements that taken together are
inconsistent and therefore at least one of them has to be false. This is certainly a valid formulation; however, when we ask why the statements are inconsistent we have to come back to the insolubility proof that therefore lies at the heart of the problem.
13See the results described in Brune et al. (1996), Breuer & Petruccione (2002, ch. 4.5) and
| (t = 0)i = a |"i + b |#i .
Since the system is not isolated, it will soon become entangled with its environment and the state will read as
| (t > 0)i = a |"i |E"i + b |#i |E#i .
The effect of decoherence can be best seen when looking at the reduced density matrix of the system
⇢S(t > 0) = T rE[| (t > 0)i h (t > 0)|]
=|a2| |"i h"| + |b2| |#i h#| + z(t)ab⇤|"i h#| + z⇤(t)a⇤b|#i h"| .
Here the time dependence of the off-diagonal terms is given by the function z(t). It is the overlap of the states of the environment |En(t)i and |Em(t)i and generally
is dominated by an exponential function
| hEn(t)| |Em(t)i | = exp[ nm(t)], nm(t) 0.
As can be shown, “[t]he time dependence of the decoherence function nm(t)strongly
depends, in general, on the specific form of the system-[environment] coupling, on the various parameters of the underlying microscopic model, and also on the properties of the initial state.” (Breuer & Petruccione 2002, p. 222) Furthermore, in realistic models the decoherence function will lead to a fast decay of the off-diagonal terms of the system’s density matrix. So after some more time the latter is given as
⇢S(t 0)' |a2| |"i h"| + |b2| |#i h#| .
The density matrix has no interference terms and is therefore equal to the density matrix of a mixed state. This is one of the main consequences of decoherence, i.e., the cancelling of interference terms in the system’s reduced density matrix. Since, however, decoherence does not lead to a density matrix of a pure and non-superposed state, it is clear that the measurement problem cannot be solved by decoherence alone.14
Even though decoherence does not solve the measurement problem, it is significant that it involves a non-unitary evolution of the systems state. This can be seen for instance by looking at the expectation values for a suitable observable for the spin state of the system. If this observable is ˆS = ˆSS⌦ ˆIE, where ˆIE is the identity
operator for the environment, it is clear that
14There are two interesting points to note: (1) Decoherence is in principle reversible, because
the involved interaction Hamiltonians are, at least in some models, hermitian (cf. Zurek 1982, p. 1873, Giulini et al. 1996, p. 23, 45) and (2) the interference terms do not always vanish completely in finite time and could be measured in a suitable experiment, if we had the necessary technology (cf. Giulini et al. 1996, p. 54).
h (t > 0)| ˆS| (t > 0)i 6= h (t 0)| ˆS| (t 0)i or equivalently T rS n ˆ S⇢S(t > 0) o 6= T rS n ˆ S⇢S(t 0) o and therefore the criterion for non-unitarity is fulfilled.
In more general terms, that is, not in direct relation to the above simple model for decoherence, the non-unitarity of the system’s evolution can be seen from the following. If the density matrix of a system behaves unitarily, its time evolution is determined by the von Neumann equation
id⇢ dt =
h ˆ H, ⇢i.
This equation is equivalent to the Schrödinger equation (either one can be derived from the other), which determines the time evolution of the state’s wave function. In contrast to this, if decoherence is present, most models describe the time evolution of the system’s reduced density matrix by a master equation of the Lindblad form (cf. Giulini et al. 1996, pp. 111-112; Breuer & Petruccione 2002, pp. 119-222):
id⇢S dt = h ˆ H, ⇢S i N2 1 X i=1 ci ✓ ˆ Ai⇢SAˆ†i 1 2Aˆ † iAˆi⇢S 1 2⇢SAˆ † iAˆi ◆ .
The second term on the right hand side is the non-unitary part of the equation and implies a non-unitary evolution of the system’s wave function.15
It might be objected to the above point that the attribution of a state to the system alone is not possible, because it is entangled with its environment and the decisive feature of entanglement is non-separability. To answer this objection, we have to enter the ongoing discussion about entanglement measures and criteria for separability. This is already a large and fast growing field, making it impossible to give a complete survey here.16 As the situation is today, there is no single measure
for entanglement, applicable and useful in all situations, and no single criterion for separability, giving sufficient and necessary conditions for all possible states. I will therefore only discuss the most common measure, which gives the result a somewhat limited scope.
The oldest criterion for separability and at the same time measure for entanglement is given by the Bell inequalities. Though it has been shown that not all entangled states violate them, those that do not can possibly be transferred into states that do. Bell inequalities therefore represent a powerful tool to detect entanglement (cf. Munro et al. 2001). They are also intuitively very accessible, since they can
15The description of decoherence using a master equation has interestingly the same form as
statistical collapse models, like that of GRW, have (cf. Giulini et al. 1996, p. 68 f., 223 and ch. 8).
be tested in experiments.17 The connection to decoherence is now straightforward.
The strength of violation of the Bell inequalities is determined by the interference between the entangled states.18 In particular, the Bell inequalities are not violated,
if there is no interference. Since decoherence eliminates interference, it is clear that decohered states will satisfy the Bell inequalities and therefore are not entangled (cf. Bertlmann et al. 2002, ch. 11.6; Bertlmann et al. 2003). More precisely, since decoherence becomes stronger over time, a “time of disentanglement” (Blanchard et al. 2001) can be defined, after which a previously entangled state is separable. This influence of decoherence on entanglement has also been experimentally confirmed by de Riedmatten et al. (2006).19
However, it has to be stressed that decoherence does not erase all correlation between system and environment. This is not possible, since the evolution of ⇢S
clearly depends on that of the environment it is coupled to. Rather, decoherence “retains all ‘statistical’ correlations [...] while dropping all quantum correlations (entanglement).” (Zeh 2000, p. 22) The system is therefore often called an ‘open quantum system’. Again, it has to be stressed that the Bell inequalities are not sufficient and necessary to show the separability of fully decohered states in all cases. Additionally, the connections between different measures of entanglement and between different separability criteria are not entirely clear. Nevertheless, the satisfaction of the Bell inequalities alone already demonstrates that fully decohered states have no non-local features in experiments and thus cannot be genuinely entangled.
I now want to discuss the relevance of decoherence to the measurement problem. The theoretical description of decoherence, adumbrated above, rests on the trace operation, which is on the surface just an ad hoc manipulation of the theory; we are only interested in the degrees of freedom of the system and therefore we simply delete the degrees of freedom of the environment. However, this simplification of the actual physical situation is justified by the perspective of the local observer, who, if the Hilbert space of the total system is composed of the Hilbert spaces of the system and the environment, H = HS⌦ HE, can be fully described within HS. If we could
change from the local into a global perspective, then we would see that there is no decoherence and that the entanglement between system and environment remains intact. This is the reason why most, if not all, authors do not count decoherence as one of the fundamental processes in nature.
Should we therefore discard decoherence completely and not use it in any explan- ation of how measurement results come about? I think not. We have to be clear about what is doing the work in the explanation of phenomena and experimental
17Today, mostly the more refined method of “entanglement witnesses” is used to detect entanglement,
which can be considered, loosely speaking, to be a generalisation of the Bell inequalities (cf. Horodecki et al. 2009, ch. VI.5). Bell inequalities also not only detect, but also measure entanglement by their connection to the concurrence measure (cf. Emary & Beenakker 2004).
18Kiess et al. (1995) have performed an experiment in which they manipulated the interference
term to gain different degrees of violation of the Bell inequalities.
results involving decoherence. The explanatory work is not only done by the local perspective of the observer, but also by the underlying constitution of the world that makes the description using decoherence possible. The world could be different in a way that does not incorporate decoherence, hence the fact that we can use decoherence must be grounded in some feature of the world. This situation is analogous to other phenomena that involve dissipation. For example, the loss of speed of a tyre rolling over the floor can be explained by friction, even though it is clear that friction is just a simplified way to describe more fundamental forces. One might object, however, that the above analogy suggests that the non-unitarity of decoherence might be reduced to processes in nature that are in fact unitary.
At this point the distinction between local and global (or subjective and objective) solutions to the measurement problem becomes important. The measurement problem points towards the incapability of physics to explain definite measurement outcomes. However, in so far measurement outcomes are local events and always observed from a local perspective, a local solution to the measurement problem would be sufficient. For this reason we can make use of local phenomena like the non-unitarity of decoherence in explicating measurement outcomes.
Schlosshauer (2004, p. 1271) calls such a solution ‘subjective’, because it “does not make any claims about definiteness of the underlying physical reality”. I disagree here, because even though decoherence disappears globally, the fact that we locally observe decoherence clearly tells us something about the underlying physics. Unitary physics can have effects such that for the local observer they are non-unitary. The correlation of a system with a non-local environment has the effect that a certain property of the system, its coherence, is not observable locally anymore (cf. Hornberger 2009, p. 226). The actual justification for taking decoherence as a real process is of course the experimental results. Thus, decoherence is the description of a real process, the effects of which can only be observed locally. For these reasons I believe that decoherence and its non-unitarity can be relevant to explain definite measurement outcomes, despite being only a local phenomenon.
Even though the non-unitarity of decoherence is mentioned by several authors20it
has received little attention in the discussion about the measurement problem.21 The
reason is probably that decoherence alone does not solve the measurement problem, and to be clear, I will not change this situation. Nevertheless, it is important to investigate the relevance of decoherence a bit further. Let us recall premise two of the strong measurement problem given above: ‘The state vector always evolves according to linear and unitary dynamics.’ Given that my argument in favour of local solution is correct, this premise can be given a local form: ‘The state vector of a local system always evolves according to linear and unitary dynamics.’ Since this is a statement about the dynamics of a system and decoherence adds something new
20See for example Busch et al. (1996, p. 123), Giulini et al. (1996, p. 51, 249), Kupsch (2000, p.
128) and Breuer & Petruccione (2002, p. 109, 122).
21Much more attention to the non-unitarity has been paid in the field of quantum computation,
where it is particularly problematic, since a non-unitary evolution typically goes hand in hand with a loss of information (cf. Facchi et al. 2005).
to the dynamics, clearly, the non-unitary evolution of the decohered state has to be added to premise two. It is obvious that the non-unitarity cannot be added by a conjunction, because this would make premise two contradictory; the state vector cannot always evolve according to unitary dynamics and sometimes not. So we have to try a disjunction, and since unitary and non-unitary dynamics cannot happen at the same time, it has to be an exclusive disjunction. Premise two then reads: ‘Either the state vector of a local system evolves according to linear and unitary dynamics or according to non-unitary dynamics.’
The premise is fine now, but what about the conclusion? Does it still follow? Here we have to keep in mind that the strong measurement problem is just a vague translation of what really is a precise mathematical proof. What the proof sets out for is to show the insolubility of the measurement problem within standard QM. Since unitarity of the evolution is ubiquitous to establish the proof, it is clear that the conclusion does not follow from premise one with the new premise two. It might be argued that still at least sometimes the state vector evolves according to a standard Schrödinger equation and therefore at least in these cases the insolubility proof is correct. However, in all interesting cases, those that can count as measurement, decoherence and therefore non-unitary dynamics are effective and the conclusion does not follow.
The result of all this is that, since decoherence is part of every measurement from a local perspective, the insolubility of the measurement problem cannot be proven. The strong measurement problem is only retained if a global solution is demanded, but is false if a local solution suffices. However, it is most important to stress that decoherence does not solve the measurement problem, since it does not explain how a superposition can evolve into a definite state, given that the outcome of the decoherence process is a mixed state. The weak measurement problem is still unsolved. Nevertheless, the situation has changed considerably. Was it hopeless before to look for a solution to the measurement problem in ordinary physics, this is now a possibility. It cannot be strictly excluded that a state evolves according to non-unitary dynamics that could imply the collapse of a superposition into a definite state. Of course, from the fact that decoherence is non-unitary, it does not follow that there must be other (non-unitary) processes that actually solve the measurement problem. The point I wish to make by involving decoherence is merely to cut the chains in which the strong measurement problem has put the philosopher of physics, and thereby at least take the possibility of a physical solution into account. In the next section, I will look into QFT to see whether there are hints that could eventually lead to a solution of the measurement problem.