_{1}

Environment induced decoherence, and other quantum processes, have been proposed in the literature to explain the apparent spontaneous selection—out of the many mathematically eligible bases—of a privileged measurement basis that corresponds to what we actually observe. This paper describes such processes, and demonstrates that—contrary to common belief—no such process can actually lead to a preferred basis in general. The key observation is that environment induced decoherence implicitly assumes a prior independence of the observed system, the observer and the environment. However, such independence cannot be guaranteed, and we show that environment induced decoherence does not succeed in establishing a preferred measurement basis in general. We conclude that the existence of the preferred basis must be postulated in quantum mechanics, and that changing the basis for a measurement is, and must be, described as an actual physical process.

Assume that an observer O is performing a measurement on a quantum system S. For simplicity, suppose that O and S can both be described by Hilbert spaces of dimension

two, and let

The measurement is initiated with an interaction between the observer and the quantum system. Following [

One is tempted to interpret the state on the right hand side as describing a situation in which O measures S in the basis

The trouble is that the same resulting state can be written as well as [

where

bases for O and S respectively. Therefore the state

In practice, the actual observation is believed to be made in one basis and not in any other. If the experiment described is the Schrödinger cat experiment in which O is the experimenter and S is the cat, the cat is actually observed in the basis

never in the conjugate basis

However the laws of quantum mechanics do not explicitly tell us which basis is the preferred one in which actual observations are performed. This ambiguity is referred to as the preferred basis problem in quantum mechanics.

The preferred basis problem has been at the centre of many studies and much debate [

Definition 1 The combined system

for some nonnegative set of numbers

A classical mixture of states corresponds to a classical probabilistic sum of outcomes, in which the state for O and the state for S are jointly distributed over the separable orthonormal basis _{i}_{,j}. Property 2 in the Appendix shows an example of a family of density matrices that are not classical mixture of states.

The aim of this paper is to describe the theories such as environment induced decoherence, starting with a simple case of the environment induced decoherence proposed in [

We observe that all these theories make an implicit assumption on the initial state of the combined system, comprised of the observed system, the observer, and any third auxiliary system introduced by such theories. However, such assumption cannot be guaranteed to hold, as we have no prior knowledge of what the quantum state of any physical system is. And indeed it can be proved that these processes do not lead to the emergence of any preferred basis for many initial states that are possible for the combined system. This is at odds with the belief commonly shared so far, as for instance in [

We conclude that the existence of the preferred basis in quantum mechanics cannot be explained by quantum mechanics itself. The existence of the preferred basis must be a postulate, added to the existing laws of quantum mechanics.

The consequence of such a postulate is that any selection of the measurement basis― other than the preferred one―must be considered as an explicit, actual physical process. As any actual physical processes, the selection of this basis cannot be independent of the rest of the universe, depending on the initial state of the universe, state which is not known. The choice of the measurement basis cannot be proven to be independent of the system being observed, and this fact is made explicit by postulating the existence of the preferred basis in the laws of quantum mechanics.

Environment induced decoherence [

It is argued that such interaction reduces in a short frame of time any quantum state describing

Coming back to our first example, following [

where

As in the previous example, there is a first interaction between O and S leading to the state

At this point, the environment induced decoherence states that there is an unavoidable interaction between O and its environment E. Suppose that such an interaction can be written as

For i = 0, 1, where

In environment induced decoherence, one claims that any information in the environment is lost or ignored. The density matrix for the subsystem_{OS}, is obtained by tracing over the degree of freedom associated with E, namely:

Mathematically, the density matrix ρ_{OS} above is a classical mixture of states and can be interpreted as describing a classical situation in which O observes

The proponents of the decoherence theory assert that this is indeed what actually happens physically―the form of the interaction with the environment has selected a preferred basis, in which the density matrix of the composite system

Our first remark is that even if the state of _{OS} in a basis different from the basis in which ρ_{OS} is diagonal.

This remark being set aside (although in the author’s opinion, this is sufficient to call for the need to postulate the existence of the preferred basis), another fundamental issue

with the decoherence theory is that the mechanism above ignores the possibility that the environment could have been entangled with the system or the observer previous to the experiment. The initial state of the whole setup

As an example, suppose that the observed system and the environment were entangled, such that the initial state of

Indeed, we cannot rule out an interaction between S and E prior to the experiment. For instance, starting with the state

Applying the same interactions above to this initial state

and tracing over E leads now to

where

mixture of states, as proved in Property 2 in the Appendix.

This example shows that the environment induced decoherence does not in general lead to a classical mixture state in a preferred basis: the initial state of the system is unknown and we cannot rule out entanglement between the observed system, the observer and the environment, which can lead to a final state in which we do not obtain a classical mixture of states.

We have shown in the previous section that the environment induced decoherence as described in [

Is any other process, under the constraints of quantum mechanical laws, capable of producing classical mixture of states for the observed system and the observer, regardless of the initial state for the overall setup comprising the observed system, the observer, and any auxiliary third physical system?

However complex such a process may be, it can be described as follows: the overall setup is comprised of the three components, the observed system, S, the observer, O and the auxiliary system, E. The process makes these three components interact, possibly in a most complex manner. After this interaction, represented by an unitary operator U, we interest ourselves with the state describing

Let _{OS}, the initial state described by the density matrix:

lead to the final state_{OS} for the final state of

This in particular holds for any density matrix

We have therefore shown:

Property 1 In any measurement process involving an observer, an observed system and any auxiliary system, no physical setup which is obeying quantum mechanical laws can guarantee to produce a classical mixture of states for the observer and the observed system.

The result in the previous section shows that quantum mechanical processes cannot by themselves explain the existence of preferred bases in quantum mechanics. Current formulation of the quantum mechanical laws is not sufficient to imply the existence of a preferred basis. To be rigorously complete, quantum mechanical laws must be supplemented by an explicit assumption on what the preferred basis is, at least for the space describing the observer.

Postulate 1 The Hilbert space describing the observer is endowed with a preferred basis _{O} for O, the probability that O observes the result i is given by

Note that we need to assume the preferred basis for the observer only, as the existence of the preferred basis is experienced only at the observer’s level. Changing the basis used for S or E in the calculation above has no impact on the outcomes and the associated probabilities experienced at the observer O’s level.

As quantum mechanics cannot explain the existence of preferred basis, we need to accept it as a postulate. There exists a privileged basis for the Hilbert space describing the observer, and as a consequence there is no theoretical freedom in the selection of the basis in which measurements are performed. Observation of quantum states can be done in different bases, but there must be an actual physical process corresponding to this basis change: for instance, in a photon polarisation measurement, the measurement basis is changed by actually acting on a phase shifter. In the Stern-Gerlach experiment the measurement basis is changed by rotating the magnets creating the magnetic field. Our point is that one does not change the measurement basis only by thought, and that an actual physical change must occur to act on a measurement basis.

The fact that a measurement basis change is not a theoretical concept but is an actual physical process has a fundamental consequence. So far, it has been common to assume (for instance, in the thought experiment leading to the Bell inequality [

By making the measurement basis selection an explicit physical process that also obeys to the laws of quantum mechanics, we have no longer the theoretical independence between an observed system and the measurement basis in which the system is observed. Indeed, the observed system and the setup selecting the measurement basis are both quantum systems and are described jointly by a composite quantum system, for which the initial state is unknown. As proved in [

As such, Postulate 1 of this paper is not a mere mathematical axiom that is only required for a formal completeness of the laws of quantum mechanics. It implies a theoretical dependence between the choice of the measurement basis and the observed physical system, i.e. a potential à-priori dependence between the observer and the observed.

Inamori, H. (2016) No Quantum Process Can Explain the Existence of the Preferred Basis: Decoherence Is Not Universal. Journal of Quantum Information Science, 6, 214-222. http://dx.doi.org/10.4236/jqis.2016.63014

The following property gives an example of a family of density matrices that are not classical mixture of states, for the Hilbert spaces for O and S:

Property 2 Let

with _{1}, j_{1}) and (i_{2}, j_{2}), with i_{1} ≠ i_{2} and j_{1} ≠ j_{2}. Then the density matrix

Proof 1 The proof uses standard mathematical techniques as described for instance in [

Let’s consider

On the other hand, if we assume that ρ is a classical mixture of states, then by definition, there exists an orthonormal basis

with

Taking the trace of its square, we get _{0}, j_{0}) such that

Take now the trace over S. We have, on one hand,

This density matrix has a rank strictly greater than 1as α_{i}_{,j} is non-zero at least for two couples (i_{1}, j_{1}) and (i_{2}, j_{2}), with i_{1} ≠ i_{2} and j_{1} ≠ j_{2}.

On the other hand, if we assume that ρ is a classical mixture of states, then using Equation (16), we get