Nowadays it is practically forgotten that for observables with degenerate spectra the original von Neumann projection postulate differs crucially from the version of the projection postulate which was later formalized by Lüders. The latter (and not that due to von Neumann) plays the crucial role in the basic constructions of quantum information theory. We start this paper with the presentation of the notions related to the projection postulate. Then we remind that the argument of Einstein-Podolsky-Rosen against completeness of QM was based on the version of the projection postulate which is nowadays called Lüders postulate. Then we recall that all basic measurements on composite systems are represented by observables with degenerate spectra. This implies that the difference in the formulation of the projection postulate (due to von Neumann and Lüders) should be taken into account seriously in the analysis of the basic constructions of quantum information theory. This paper is a review devoted to such an analysis.

We recall that for observables with nondegenerate spectra the two versions of the projection postulate, see von Neumann [

This paper is devoted to such an analysis. We start with a short recollection of the basic facts on the projection postulates and conditional probabilities in QM. Then we analyze the EPR argument against completeness of QM [

We analyze the quantum teleportation scheme. We will see again that it is fundamentally based on the use of the Lüders postulate. The formal application of the von Neumann postulate blocks the teleportation type schemes; see for more detail [

Finally, we remark that “one way quantum computing,” for example, [

The results of this analysis ought to be an alarm signal for people working in the quantum foundations. If one assumes that von Neumann was right, but Lüders as well as Einstein et al. were wrong, then all basic schemes of QI should be reanalysed. However, a deeper study of von Neumann’s considerations on the projection postulate [

The main conclusion of the present paper is that the study of the foundations of QM and QI is far from being completed; see also the recent monograph of Jaeger [

Everywhere below

where

For an observable

If we select only systems with the fixed measurement result

where

Lüders generalized this postulate to the case of operators having degenerate spectra. Let us consider the spectral decomposition for a self-adjoint operator

where

By Lüders’ postulate after a measurement of an observable

Thus for the corresponding density operator we have

If one does not make selections corresponding to the values

von Neumann had a completely different viewpoint on the postmeasurement state [

The result will not be a fixed pure state, in particular, not Lüders’ state

Following von Neumann, we choose an orthonormal basis

A measurement of the observable

von Neumann emphasized that the

Consider the

Consider any composite system

Any state

where

induces a projection of

In particular, for a state of the form

one of states

Thus by performing a measurement on the

is assigned to

However, the EPR considerations did not match von Neumann’s projection postulate, because the spectrum of

Finally, (after considering of operators with discrete spectra) Einstein et al. considered operators of position and momentum having continuous spectra. According to the von Neumann [

In Section

We will proceed across the quantum teleportation scheme, see, for example, [

The quantum teleportation scheme requires Alice and Bob to share a maximally entangled state before, for instance, one of the four Bell states:

Alice will then make a partial measurement in the Bell basis on the two qubits in her possession. To make the result of her measurement clear, we will rewrite the two qubits of Alice in the Bell basis via the following general identities (these can be easily verified):

If Alice indicates that her result is

If the message indicates

The main problem is that Alice’s measurement is represented by a degenerate operator in the 3-qubit space. It is nondegenerate with respect to her 2 quibits, but not in the total space. Thus the standard conclusion that by obtaining, for example,

In this section we try to formalize von Neumann’s considerations on the measurement of observables with degenerate spectra.

Consider an

We postulate (it is one of the steps in the formalization of von Neumann’s considerations).

Here

We would like to present the list of other properties of

Take an arbitrary

Performing the

The postmeasurement density operator

By (

Consider now the

For any pure state

Let

We performed a comparative analysis of two versions of the projection postulate—due to von Neumann and Lüders. We recalled that for observables with degenerate spectra these versions imply consequences which at least formally different. In the case of a composite system any measurement on a single subsystem is represented by an operator with degenerate spectrum. Such measurements play the fundamental role in quantum foundations and quantum information: from the original EPR argument to shemes of quantum teleportation and quantum computing. We formulated natural conditions reducing the von Neumann projection postulate to the Lüders projection postulate; see the theorem. This theorem closed mentioned loopholes in QI-schemes. However, conditions of this theorem are the subject of further analysis.