A succinct executive summary of the proof of QNL given in Section 3.1 contains two assertions: 1) QM predicts quantum correlations and consequent violation of the CH/CHSH inequality, implying QNL, and 2) the experiments verify that the inequality is violated in nature. It sounds impressive and convincing, but what if contrary to popular belief QM does not in fact predict quantum correlations and thus does not predict inequality violation for an EPRB experiment? The proof falls apart because then an observed violation would prove only that the experiments are incorrectly designed, implemented, or analyzed, and we would be motivated to identify the experimental problem(s). Indeed, I argue that both of these assertions are false and that the proof of QNL is invalid.

In this section, I develop the argument against the commonly accepted view that QM predicts quantum correlations and consequent violation of the CH/CHSH inequality. In Section 4 I develop the argument that the experiments conducted to date are invalid. Before proceeding it is important to give an overview of the projection postulate (also known as ‘collapse of the wavefunction’) to ensure full appreciation of my argument.

The projection postulate was first proposed by Dirac [14]. He claimed that when a measurement yields an eigenvalue of an operator, the system is left in the corresponding eigenstate. This idea was endorsed by von Neuman [15], and in modern textbooks of quantum mechanics it has since taken on the status of a postulate, the so-called Dirac-von Neumann projection postulate. For example, Rae [16] gives as his postulate 4.2: “Every dynamical variable may be represented by a Hermitian operator whose eigenvalues represent the possible results of carrying out a measurement of the value of the dynamical variable. Immediately after such a measurement, the wavefunction of the system will be identical to the eigenfunction corresponding to the eigenvalue obtained as a result of the measurement.”

When the eigenvalues are degenerate, the Dirac-von Neumann projection postulate gives the resulting state as an evenly-wieghted mixture of the eigenfunctions singled out by the measurement. Formally, Dirac-von Neumann projection tranforms the pre-measurement density matrix to the post-measurement density matrix as follows:

$$\begin{array}{}{\displaystyle {\rho}_{post}=A/Tr(A)}\end{array}$$(5)where A is the density operator corresponding to the measurement. The resulting state contains no trace of the original state and is therefore considered to be maximally disturbing.

Lüders extended the Dirac-von Neumann projection postulate to revise the treatment of degeneracy [17], proposing the so-called Lüders rule, which has become generally accepted as the correct projection postulate. Formally, Lüders’ rule tranforms the pre-measurement density matrix to the post-measurement density matrix as follows:

$$\begin{array}{}{\displaystyle {\rho}_{post}=(A{\rho}_{pre}A)/Tr(A{\rho}_{pre})}\end{array}$$(6)The resulting state preserves the weightings in the original state and is therefore considered to be minimally disturbing (although I argue later that the truly minimally disturbing measurement would no projection at all). Note that in the absence of degeneracy the Lüders rule reduces to Dirac-von Neumann projection.

Supporters of QNL sometimes argue that projection is irrelevant to the QNL debate because (they claim) quantum correlations can be derived without appealing to the projection postulate. This is incorrect, as I show elsewhere [18]. It is true that the quantum correlations can be derived assuming a *joint measurement*, consisting of a single sampling that includes the orientation settings at both sides. However, in EPRB experiments, there are two separated independent samplings at the two sides, each of which can include only its orientation setting, and one side must project the other side in conformance with Lüders’ rule if the quantum correlations are to be derived. It is very important to properly distinguish joint sampling from the separated sampling that occurs in a real EPRB experiment. I explain this very clearly in the above-cited paper.

The projection postulate appears to be satisfying in that it seems reasonable to assume that if we measure a given eigenvalue then the system must be in the corresponding eigenstate (*e.g*., Polkinghorne writes that the two "must obviously correspond" [19]), and in that its use leads to repeated measurements on the same system always giving the same result. Nevertheless, the projection postulate has been strongly challenged from its first introduction and it remains highly controversial [20–27]. I now briefly list some of the arguments that have been directed against the general idea of projection after which I challenge in more detail the application of Lüders’ rule to EPRB. Refer to the cited references for further details on the following arguments.

Projection is an indeterministic violation of unitary dynamic evolution, *i.e*., Schrodinger’s equation. QM provides no guidance about what constitutes a measurement and when projection is to be applied. QM also provides no guidance about how symmetry is to be broken for simultaneous measurements. For example, if the two sides in an EPRB experiment measure simultaneously, which one projects the other?

Several interpretations of QM dispense entirely with the projection postulate, *e.g*., the many-worlds and modal interpretations.

Projection is arguably not actually needed or used in any significant quantum calculations. Projection is not present in quantum field theory.

Projection is simply false for many real physical scenarios. Some measurements do not give the same result upon repetition, *e.g*., because the system is annihilated (photons disappear upon detection), or because the measurement involves an unpredictable violent disturbance.

In the Copenhagen interpretation a state refers to an ensemble, so it is incongruent that a measurement on a single member of an ensemble can define the distribution for the entire ensemble. In other words, QM is statistical but projection is deterministic.

Postulating projection in effect simply postulates QNL, obviating any need for experiments and rendering the proof of QNL tautological.

While these considerations are enough to throw the projection postulate into serious doubt, an objection more directly germane to EPRB can be given: Lüders projection cannot be valid for EPRB because it requires superluminal transmission of information. Without Lüders projection one can fall back to von Neumann projection (the projected state is a mixture) or to null projection (no projection at all), both of which fail to predict quantum correlations. The argument goes as follows (refer to [18] for full details).

Consider a spin 1/2 singlet state. The input singlet state is rotationally invariant, so it can be expressed in the *a* measurement basis as follows:

$$\begin{array}{}{\displaystyle {\psi}_{singlet}=\frac{1}{\sqrt{2}}(|-{1}_{a},{1}_{a}\u3009-|{1}_{a},-{1}_{a}\u3009)}\end{array}$$(7)The *a* subscripts denote the *a* measurement basis. If the measurement at A produces a value of −1_{a}, then Lüders’ rule gives the renormalized projected state as:

$$\begin{array}{}{\displaystyle {\psi}_{L}=|-{1}_{a},{1}_{a}\u3009}\end{array}$$(8)This is separable so the projected B state is |1_{a}〉, and the corresponding 2*x*2 density matrix is |1_{a}〉〈1_{a}|. Therefore, the projected 2*x*2 density matrix in the Z basis for the case of *O*_{A} = −1 is given by:

$$\begin{array}{}{\displaystyle {O}_{A}}& =-1:\phantom{\rule{thinmathspace}{0ex}}{\rho}_{B}^{2x2}=|{1}_{a}\u3009\u3008{1}_{a}|\\ & =\left[\begin{array}{cc}{\mathrm{cos}}^{2}(a/2)& \mathrm{sin}(a/2)\mathrm{cos}(a/2)\\ \mathrm{sin}(a/2)\mathrm{cos}(a/2)& {\mathrm{sin}}^{2}(a/2)\end{array}\right]\end{array}$$(9)Similarly, the projected density matrix in the Z basis for the case of *O*_{A} = 1 is given by:

$$\begin{array}{}{\displaystyle {O}_{A}}& =1:\phantom{\rule{thinmathspace}{0ex}}{\rho}_{B}^{2x2}=|-{1}_{a}\u3009\u3008-{1}_{a}|\\ & =\left[\begin{array}{cc}{\mathrm{sin}}^{2}(a/2)& -\mathrm{sin}(a/2)\mathrm{cos}(a/2)\\ -\mathrm{sin}(a/2)\mathrm{cos}(a/2)& {\mathrm{cos}}^{2}(a/2)\end{array}\right]\end{array}$$(10)These projections result from straightforward application of orthodox Lüders projection and are fully derived in [18]. Calculation of the expectation value 〈*AB*〉 now proceeds straighforwardly using the projected state and yields quantum correlation. However, all is not well because it is easily shown that this projection requires superluminal transmission of information. While a debate rages over whether superluminal influences actually violate special relativity and how, for my purposes it is sufficient to note that no satisfactory covariant account of state reduction has been proposed to date, despite intense efforts to find one [28].

Orthodox quantum theory employing Lüders projection, as shown, entails that the measurement at side A produces a projected state at side B. The projected state is one of the two states given in equations (9) and (10). The projected state must be locally present at side B, to be available for the local measurement at side B. It is obvious from perusal of equations (9) and (10) that the projected state contains information about both the measurement angle *a* at side A (*a* appears directly in the density matrices) and information about the outcome (the outcome determines which of the two states is projected). It is clear, not only from the demonstration here, but from other analyses [13, 18, 30], that both parameter and outcome independence must be violated to account for the EPR correlations.

Transmission of information is not problematic as long as the transmission is not required to occur at superluminal speeds, because special relativity would be violated, and that is something that theorists must eschew to maintain a consistent axiom set for physics. However, it is easy to see that the quantum separated solution using Lüders’ rule requires superluminal transmission in the paradigmatic EPRB experiment with a large separation between the measurement sides. The experiment can be set up such that the measurement at side B occurs before enough time elapses to allow subluminal transmission of information. If the EPR correlations are to be obtained in such a scenario, then parameter and outcome information become available in the projected state localized to side B within the arbitrarily small inter-measurement time, which is much shorter than the time required to physically transmit the information. Therefore, the general separated solution using Lüders’ rule requires superluminal transmission of information, violating special relativity.

One could argue that, although the information is indeed present at side B superluminally, there is no way for side B to extract it (per the no-signaling theorem), and so there is a ‘peaceful coexistence’ with special relativity. This notion supposes that information that cannot be extracted is not really information, and that superluminal transfer of such “information” therefore does not violate special relativity. However, this view is easily refuted, and I emphasize that the information must have been present at side B superluminally.

Consider two separated stations A and B. Station A possesses a real variable *a*, and a second randomly selected real variable *r*. Station A generates *b* = *a* + *r* and sends *b* to station B. Station B cannot extract *a* from *b*, however, *b* nevertheless contains information about *a*. This is simply proven because station B can pass *b* to a third station C, which also receives *r* from station A. Station C can access *a* by subtracting *r* from *b*. It is clear that information about *a* existed at station B and that the information was passed through station B to station C, despite station B’s inability to extract that information. A more intuitive everyday case showing the irrelevance of inability to access the information runs as follows. I write a message and lock it in a box. I send the key to my friend Charlie. I give the box to my friend Bob. Bob possesses the information of the message but cannot access it. We know the information is there from common sense but also because Bob can give the box to Charlie who can open the box and access the information.

Special relativity therefore prevents Lüders projection from validly applying to EPRB. Alternative calculation with von Neumann projection or null projection does not yield quantum correlation [18]. Although von Neumann projection does not yield quantum correlation, it does still require superluminal transmission of at least one bit of information to signal projection of the singlet state to a mixture at side B, as required. Theoretically, however, even one bit of information transferred superluminally forces us to reject von Neumann projection for EPRB.

The correct form of projection must be chosen based on the specific arrangement of an experiment. Hegerfeldt and Sala Mayato [31] correctly argue that different forms of projection “may appear naturally, depending on the realization of a particular measurement apparatus” and “Their applicability depends on the circumstances, *i.e*., the details of the measurement apparatus.” The only correct solution for EPRB must exclude all information transfer, that is, it must exclude projection completely. In the absence of projection, quantum correlation is not possible. QNL is a mistake, and the misapplication of the Lüders projection postulate is the source of apparent nonlocality. I agree with Isham when he states that Lüders projection “is best regarded as a definition of what is meant by an ‘ideal measurement’ in the case of a degenerate eigenvalue” [29], rather than an obligatory physical process.

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