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In the famous EPR paper published in 1935, Einstein, Podolsky, and Rosen suggested a thought experiment, which later became known as the “EPR experiment”. Using the EPR experiment, they posited that quantum mechanics was incomplete. Einstein, however, was dissatisfied with the EPR paper and published a second work on the EPR experiment, in which he discussed the dilemma of choosing whether quantum mechanics was incomplete or nonlocal. Currently, most physicists choose the nonlocality of quantum mechanics over Einstein’s choice of the incompleteness of quantum mechanics. However, with an appropriate alternate hypothesis, both of these choices can be rejected. Herein, I demonstrate an approach to overcome the Einstein Dilemma by proposing a new interpretation invoked by a new formalism of quantum mechanics known as two-state vector formalism.

In 1935, Einstein, Podolsky, and Rosen published their famous EPR paper, which posited that the quantum mechanical wave function did not provide a complete description of physical reality ([

1) Quantum mechanics is incomplete, or

2) Quantum mechanics is nonlocal.

This dilemma is at the heart of the EPR experiment. In what follows, we call this dilemma the “Einstein Dilemma” after Redhead ( [

The total rejection of this principle [principle of action through a medium] would make impossible the idea of the existence of a (quasi-) closed system, and thereby also make impossible the establishment of empirically verifiable laws in our well-known sense ( [

Bohr published a paper with the same title as the EPR paper and rebutted the EPR argument [

There are many views on Bohr’s statement [

Redhead speculates that the locality in the Einstein Dilemma can be described with

In this work, I show that the existing interpretations of quantum mechanics, including that of Bohr, must be subject to the Einstein Dilemma (Section 2); further, I provide a new interpretation that makes it possible to circumvent the dilemma (Sections 3 and 4).

First, let us examine Redhead’s view that Bohr’s locality is equivalent to

Let us consider Bohm’s version of the EPR experiment ( [

If

We now examine other views of Bohr’s interpretation. Beller and Fine discuss that Bohr adopted positivism as his philosophy of science after the EPR paper was published [

Those familiar with this controversy might consider that Bohr’s interpretation could circumvent the dilemma if the view of Howard, Halvorson-Clifton, and Ozawa-Kitajima (HHCOK) of Bohr’s interpretation is correct [

According to the HHCOK version of Bohr’s view, before the measurement of electron I, the states of the electrons are the mixture states of the x-spin. It should be noted, however, that the mixture state of the x-spin is not the eigenstate of the x-spin, and thus, we cannot predict which value the x-spin would assume using only quantum mechanics. This assumption suggests that we have to choose one of the following options:

1) The x-spin has a sharp value before the measurement. It follows that quantum mechanics is incomplete because it cannot predict which value the x-spin would have before measurement (since the mixture state is not the eigenstate).

2) The x-spin has an unsharp value before the measurement. It follows that quantum mechanics violates LOC_{1} because the spin of electron II can have a sharp value only after the measurement of the spin of electron I.

Therefore, the HHCOK version of Bohr’s interpretation of quantum mechanics will also face the Einstein Dilemma. Accordingly, it does not matter which view of Bohr’s interpretation we consider, we cannot reject either part of the dilemma. Next, we examine the existing interpretations of quantum mechanics other than Bohr’s interpretation.

HHCOK speculate that Bohr’s interpretation does not accept the projection postulate. If we consider an interpretation that assumes the projection postulate, it follows that quantum mechanics is nonlocal (violation of

The many-worlds interpretation (MWI) postulates that quantum mechanics is nonlocal. Some readers might believe the MWI avoids the issue of nonlocality. Nevertheless, because the worlds split for a very short period of time, we cannot avoid nonlocality [

Next, we consider whether it is impossible to avoid the Einstein Dilemma. In the remainder of this paper, I suggest an interpretation that circumvents the dilemma. This interpretation is based on one of the formalisms of quantum mechanics, known as “two-state vector formalism (TSVF),” as proposed by Aharonov, Bergmann and Leibowitz (ABL) [

In this section, I briefly summarize TSVF. In conventional quantum mechanics (CQM), only the past state determines the probability that a physical quantity Q has a certain value

Assume that Q has eigenstates

Here,

where

At times, the probabilities calculated using the ABL rule differ from those calculated using the Born rule. It appears as if TSVF and CQM are different theories, although they are not. The ABL rule agrees with the Born rule when

There seems to be no method for verifying the prediction made using TSVF because we cannot know the intermediate state at t without destroying the state. Aharonov, Albert and Vaidman solved this problem by pro- posing a new measurement concept called ‘weak measurement’; measurements made according to this concept do not destroy the intermediate quantum state [

According to Aharonov, Albert and Vaidman [

Recently, physicists performed a weak measurement and confirmed that the theoretically predicted and mea- sured weak values showed good agreement [

In general, when the final state is an eigenstate,

Thus, the physical quantity has a sharp value even before the measurement. In this paper, I do not consider the case where the final state is not an eigenstate. Consider Bohm’s version of the EPR experiment. When the interaction between two electrons ends at

where

From the ABL rule (1), the probability that the state of the system at a given time

It follows from this result that we do not need to assume any type of nonlocality. Furthermore, quantum mechanics is not modified. Accordingly, we can simultaneously insist that quantum mechanics is complete and local. Here, we can avoid confronting the Einstein Dilemma.

Readers might suspect that using TSVF means that quantum mechanics is not complete because the final state can be considered a “hidden variable.” Actually, it depends on the definition of the hidden variable. For example, the de Broglie-Bohm interpretation needs hidden variables of both position and momentum because we can clearly interpret that quantum mechanics does not need both position and momentum.

However, it is not always clear whether the final state is a hidden variable, unlike the case with the de Broglie-Bohm interpretation. TSVF formalism does not modify quantum mechanics because TSVF considers the description of the state vector as complete, but we need two state vectors for completeness. TSVF does not require adding any other variables other than the state vector. Therefore, I insist that quantum mechanics is still complete if my interpretation using TSVF is correct.

Nevertheless, this interpretation still appears problematic in the following situation. Let us assume that we measure the z-spin of an electron at

According to the ABL rule, both the probability that the system state at

This contradicts the assumption that

If at t we measure the spin in the z-direction, we must find it up, because that’s how the particle was prepared at

Nonetheless, there is still concern that our interpretation contradicts the uncertainty relation (Kennard-Robertson inequality). Let us assume that an ensemble of electrons whose z-spin at

However, it is to be noted that the uncertainty relation considers only the past state. In our interpretation, we consider the future state as well, thus it is no problem to contradict with the uncertainty relation.

Here, we define new standard deviation of z-spin for our interpretation as

where

Likewise, we can obtain

In general, when we define

where

when the final state is an eigenstate of

Nevertheless, there remains the concern that the abovementioned interpretation may contradict no-go theorem such as the Kochen-Specker theorem [

In addition, Tollaksen discusses that the weak value of spin changes according the context of the experiment [

[

Upon his examination of quantum mechanics in the EPR paper, Einstein was presented with the dilemma that

1) Quantum mechanics is incomplete, or

2) Quantum mechanics is nonlocal.

Most physicists choose the nonlocality of quantum mechanics over Einstein’s choice of incompleteness. However, if possible, it is better to reject both choices.

According to TSVF, physical quantities can have sharp values before measurement. Therefore, we do not need to introduce the concept of nonlocality. Furthermore, TSVF does not assume any hidden variables; thereby ensuring that quantum mechanics is complete. In conclusion, we circumvented the Einstein Dilemma by con- sidering both the past state and the future state of a quantum mechanical system.

This work has been supported by Japan Society for the Promotion of Science (Grant No. 26370021).