1. INTRODUCTION

In this paper we attempt a little revisionist history. In particular, we show how a very simple argument establishing the impossibility of a local realistic description of quantum theory—Bell’s theorem—was lingering on the edge of Schrödinger’s and Einstein’s consciousness in 1935–36.

2. EINSTEIN’S LESS FAMOUS ARGUMENT FOR INCOMPLETENESS OF QUANTUM MECHANICS

In June of 1935 Einstein wrote to Schrödinger [1] bemoaning that the Einstein–Podolky–Rosen (EPR) paper [2] “buried in the erudition” the simplicity of the point he was trying to make [3,4]. In this letter he defines completeness of a state description as

and a separation hypothesis between systems enclosed in different boxes as

For our purposes it is important only that the separation hypothesis, together with the assumption of “real states of real systems” implies local realism, although it potentially encompasses more.

Einstein goes on to consider entangled particles $A$ and $B$, and to point out that, depending on the choice of kind of measurement on $A$ (the type of observable, not its outcome), we ascribe different state functions ${\mathrm{\Psi }}_B$, ${\underline{\mathrm{\Psi }}}_B$ to system $B$.

Einstein’s description does not, in fact, carefully distinguish the ensemble of quantum states that is obtained on $B$ for a single fixed measurement on $A$ and the different ensembles of quantum states that correspond to distinct choices of measurement on $A$ [5,6]. In a moment we will emphasize why the choice of different (and incompatible) measurements on $A$ was a necessary part of his argument (and amusingly he points out, contra-EPR, that he “doesn’t give a damn” whether the states ${\mathrm{\Psi }}_B$, ${\underline{\mathrm{\Psi }}}_B$ are eigenfunctions of observables on $B$!), but first let us note that this description of all the possible ensembles achievable in such experiments was subsequently carefully characterized by Schrödinger who, within a year, proved the quantum steering theorem [7 –9]:

Theorem 1. Given an entangled state $|{\psi }_{AB}〉$ of two systems $A$, $B$, a measurement on system $A$ can collapse system $B$ to the ensemble of states $\{|{\varphi }_i〉\}$ with associated probabilities $p_i$, if and only if

The reason two (incompatible) measurements are necessary for Einstein’s argument for incompleteness is that if one considers only a single measurement on $A$ it is trivially possible to maintain a one-to-one correspondence between a real state $\lambda$ of system $B$ and the quantum state: in this setting, the steering statistics for an entangled state on $AB$ is indistinguishable from those of a mixture of quantum/real states $\{|{\varphi }_i〉\leftrightarrow {\lambda }_i\}$ for $B$, arranged such that the measurement on $A$ needs reveal only which member of the ensemble pertains. By choosing to steer to one of two different ensembles of quantum states $\{|{\varphi }_i〉\}$, $\{|{\varphi }_i^{\prime }〉\}$ with at least some elements distinct, this is no longer possible.

Einstein concluded that, assuming local realism, many different quantum states must be associated with any given real state of $B$. Note, however, that since these different quantum states for $B$ are operationally distinct, it clearly cannot be the case that those different quantum states are all only ever associated with that one single real state of $B$. They must somehow differ in the ensemble of real states they correspond to. Such a difference can be reflected either in terms of the members of the ensemble (i.e., sometimes being associated with completely different real states) or in terms of the frequencies (probabilities) over the ensemble, or both.

In the EPR, paper the initial state of $AB$ used is maximally entangled, and the ensembles steered to are those of orthogonal quantum states (position or momentum eigenstates). For this steering scenario it is well known (see, e.g., [11]) that the Wigner function provides a local (but as per Einstein’s argument, necessarily incomplete) description of reality. We begin by showing that steering between two ensembles of orthogonal states for a qubit also can be explained within such a local realistic theory.

3. STEERING BETWEEN TWO ENSEMBLES OF ORTHOGONAL STATES

Let us formalize Einstein’s conclusion and its implications, simplifying to the easiest case possible: two different measurements on $A$ that steer the quantum state of a qubit $B$ to ensembles $\{|x〉,|X〉\}$, $\{|y〉,|Y〉\}$, where $|x〉,|X〉,|y〉,|Y〉$ are all different, $〈x|X〉=0=〈y|Y〉$, and the members of each ensemble are equally likely. As Schrödinger had proven, this is possible for any entangled state $|{\psi }_{AB}〉$ for which

In a realistic description, every quantum state corresponds to a probability distribution over a set of real states $\lambda$. When an entangled quantum state is prepared on $AB$, let $\nu (\lambda )$ denote the ensemble of real states for $B$. It must be the case that $\nu (\lambda )$ can be resolved into the steering ensembles as

Now, Einstein (explicitly) and Schrödinger (at least for the sake of argument) assumed that a complete description of reality is possible, and that in such a theory the quantum state would therefore be incomplete in the precise sense Einstein defined [12]. Even for the simple case of steering between two ensembles captured by the generic decomposition of Eq. (2) this yields some extra consistency conditions that need to be satisfied. For example, it must be possible to find a probability density $\nu (\lambda )$ over some space of real states that can be decomposed into probability densities $x(\lambda )$, $X(\lambda )$ that are disjoint, because $|x〉$ and $|X〉$ are orthogonal. Denoting by $S_{\mu }$ the support of the probability density $\mu (\lambda )$ we have that

In the original EPR argument, the scenario considered involves steering of $B$ between the ensembles of position and momentum, and then analysis of the conclusions that can be drawn if a subsequent position/momentum measurement is performed on $B$. Similarly, here we analyze the restrictions that the incomplete description of reality must obey if measurements of the projectors onto the ensemble—i.e., $\{|x〉〈x|,|X〉〈X|\}$ or $\{|y〉〈y|,|Y〉〈Y|\}$—are performed. Such consideration shows that we must also obey consistency conditions of the form

It is useful to identify four disjoint regions of the space of real states: $S_1\equiv S_x\cap S_y$, $S_2\equiv S_x\cap S_Y$, $S_3\equiv S_X\cap S_y$, and $S_4\equiv S_X\cap S_Y$, and to use the notation

Since ${|〈x|y〉|}^2={|〈X|Y〉|}^2=\alpha$ and ${|〈x|Y〉|}^2={|〈X|y〉|}^2=1-\alpha$, from equations of the form of Eq. (4) we must have

By integrating Eq. (2) over the appropriate regions we identify a final set of consistency conditions:

These are satisfied by taking ${\nu }_1={\nu }_4=\alpha /2$, while ${\nu }_2={\nu }_3=(1-\alpha )/2$.

So far, then, all of these essentially trivial consistency conditions—which any local incomplete description of reality must obey—are easily complied with.

In Section 5 we will show that, if we add the possibility of an extra measurement on A being used to steer to a third ensemble of orthogonal states we find a contradiction, indicating any realistic theory explaining quantum theory must be nonlocal—Bell’s theorem. However, we now turn to a proof that yields the same conclusion, but uses steering between only two ensembles, one of which contains a pair of nonorthogonal states.

4. STEERING BETWEEN TWO ENSEMBLES, ONE OF WHICH CONTAINS NONORTHOGONAL STATES, IMPLIES THE UNTENABILITY OF LOCAL REALISM

To show that incompleteness cannot save local realism, we now consider the possibility of steering a qubit between two ensembles, where one of the ensembles contains nonorthogonal states. Most probably Einstein, but certainly Schrödinger, knew that this was possible—it is consistent with Einstein’s calculation summarized in Endnote [6] and is mentioned explicitly in Schrödinger’s proof of the steering theorem (he limits only to ensembles wherein the states are linearly independent).

Consider then steering the state $\rho$ depicted in Fig. 1(a) either to its eigen-ensemble $\{|x〉,|X〉\}$ or to the ensemble $\{|a〉,|b〉\}$ that is an equal mixture of nonorthogonal states $|a〉=\cos \frac{\theta }{2}|x〉+\sin \frac{\theta }{2}|X〉$ and $|b〉=\cos \frac{\theta }{2}|x〉-\sin \frac{\theta }{2}|X〉$, where $\theta \in (0,\frac{\pi }{2})$. In the incomplete theory these quantum states correspond to preparation of real physical states according to probability distributions $\mu (\lambda )$, $\mu =x,X,a,b$, which have support on sets labeled $S_{\mu }$. Local realism implies $S_x\cup S_X=S_a\cup S_b$; moreover, because $|x〉$ and $|X〉$ are orthogonal, $S_x\cap S_X=\varnothing$, while $S_a$ and $S_b$ can overlap. This is depicted in Fig. 1(b) where a convenient labeling for various regions of support is shown.

 

Fig. 1. (a) Two ensembles of $\rho$. (b) The space of real states, with disjoint regions of support labeled 1–6 such that $S_x=S_1\cup S_2\cup S_3$, $S_X=S_4\cup S_5\cup S_6$, $S_a=S_1\cup S_2\cup S_4\cup S_5$, and $S_b=S_2\cup S_3\cup S_5\cup S_6$.

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Once again, using the notation in Eq. (5), we have some simple consistency conditions; for example, normalization imposes

Consider the case that the system has been steered to the quantum state $|X〉$, and we wish to compute the probability of obtaining a measurement outcome $|a〉〈a|$ or alternatively of $|b〉〈b|$. A quick glance at Fig. 1 leads naturally to the constraints

A. The Subtlety of Deficiency

The steering-based proof of Bell’s theorem we have just presented has actually made a subtle assumption. Let us reconsider the case wherein the system has been steered to the quantum state $|X〉$ and we wish to compute the probability of obtaining a measurement outcome $|a〉〈a|$. Above we assumed this imposes $X_4+X_5={|〈a|X〉|}^2={\sin }^2\theta /2$. However, while it is certainly the case that all real states $\lambda$ in $S_a$ must deterministically yield the outcome $|a〉〈a|$, any real states in $S_b$ can potentially also yield this outcome, because $|a〉$ and $|b〉$ are not orthogonal. Within regions 3 and 6 they need not even do so deterministically, because $0<|〈a|b〉|<1$. That is, the incomplete realistic theory may have a property defined in [14] as deficiency. In fact, for the case of three- and higher-dimensional quantum systems, it follows [14] from the Kochen–Specker theorem that any realistic theory (local or otherwise) must be deficient, that is, it must be the case that the set of real states that a system, prepared in quantum state $|a〉$, may actually be in, is necessarily strictly smaller than the set of real states that would reveal the measurement outcome $|a〉〈a|$ with some finite probability.

To see that allowing the local realistic theory to be deficient does not save it from Bell’s theorem, we formally capture the possibility of deficiency by defining a response function or indicator function ${\xi }_a(\lambda )$, which for every $\lambda$ is simply the probability that a particular real state yields the $|a〉〈a|$ outcome. Since $0\le {\xi }_a(\lambda )\le 1$, we must have

Once again, Eqs. (7) and (10) cannot be simultaneously satisfied. For example, substituting Eq. (7) into Eq. (10) gives

5. STEERING BETWEEN THREE ENSEMBLES OF ORTHOGONAL STATES

We now return to the steering of Section 3, and show that steering among three ensembles of orthogonal states can sometimes violate the consistency conditions. For simplicity presume for the moment that the third ensemble of orthogonal states $\{|z〉,|Z〉\}$ is such that the state $|z〉=(|x〉+|y〉)/\sqrt{2(1+\alpha )}$, i.e., $|z〉$ “bisects” the states $|x〉$, $|y〉$. Then

Clearly $z_2=z_3$ and $Z_2=Z_3$. We must also have

There is no way to satisfy all these equations, subject to the requirement $z_j,Z_j\ge 0$. For example, an independent set of the above equations is

From these we obtain

Once again, the failure to keep the incomplete realistic models consistent indicates the initial assumption of local realism is unviable.

It is interesting to note that, if the states $\{|x〉,|y〉,|z〉\}$ had been chosen to be the eigenstates of the Pauli operators ${\sigma }_x$, ${\sigma }_y$, ${\sigma }_z$, then $\alpha =\beta =1/2$ and the consistency conditions would be satisfiable. Contrast this with the fact that the inconsistency obtained when $|z〉$ bisects $|x〉$, $|y〉$ holds regardless of how close these latter two (distinct) states are. As such, we see that “how far apart” the triples of states are does not capture the difficulty or otherwise of reproducing their steering properties in an incomplete model of physical reality.

To investigate this further we have analyzed the possible triples of overlaps

Conjecture 1. The triple of overlaps $\alpha ,\beta ,\gamma$ demonstrate a violation of local realism if and only if the point $[\alpha ,\beta ,\gamma ]$ does not lie in the convex hull of the four points $[\mathrm{1,0},0]$, $[\mathrm{0,1},0]$, $[\mathrm{0,0},1]$, $[\mathrm{1,1},1]$.

We have also performed some preliminary forays into the question of how steering among four ensembles may differ. One thing we noticed in this regard is that, if we look at steering performed on a Werner state (mixture of maximally entangled and maximally mixed state), then the lowest probability of the maximally entangled state for which violation of local realism can still be demonstrated is 4/5 when steering among three ensembles and $1/\sqrt{2}$ when steering among four ensembles.

6. OUTLOOK

In deriving Bell’s theorem from steering, a number of observations crop up that merit further investigation.

One intriguing feature is that we never make use of the assumption $\mu (\lambda )\ge 0$; rather, positivity was required only for integrals of the distributions over certain regions within the space of real states. Thus the proof rules out certain options for quasi-representations of the quantum state, as well.

A second observation concerns the deficiency property introduced in Section 4. It has been shown [14] that measurement-outcome contextuality [13] manifests itself in incomplete models of reality via deficiency. This in turn makes it strictly impossible for such a model to obey conditions of the form in Eq. (4)—i.e., all the nonorthogonality of quantum states cannot be attributed to classical nonorthogonality of their associated probability distributions. Perhaps combining this observation with steering of entangled systems of dimension three or higher can yield different steering-type proofs that local realism is untenable.

Finally, the proof in Section 4 did not require an equation of the form

7. CONCLUSIONS

Before concluding, let us mention some relevant work. The proof in Section 4 is readily extended to show nonlocality for all nonmaximally entangled pure states, reproducing the conclusions of Gisin, Popescu, and Rohrlich [16, 17]. Although our proofs are algebraic and thus reminiscent of Greenberger–Horne–Zeilinger (GHZ) [18] and Hardy [19] type arguments against local realism, the proof in Section 5 would seem closest to Mermin’s exposition of Bell inequalities in [20]. Harrigan and Spekkens [21] perform a more careful and thorough exposition of Einstein’s argument above for incompleteness and the relationship to locality. In [22], Werner presents an alternative route that could have led Einstein to Bell’s argument.

Finally, one may wonder whether the quantum state can still be argued to be incomplete in Einstein’s sense above even when separability is given up. While it is in fact possible to obey all consistency conditions generalizing those above for such an incomplete realistic theory [24], it turns out that an assumption of separability for product quantum states leads to the exact opposite conclusion, namely, that the quantum state must be complete [25].

In conclusion, if Einstein and Schrödinger had probed only a little further into whether an incomplete description of physical reality can actually fully explain the gedanken experiment that they had used to rule out completeness of quantum theory, perhaps the tension among locality, realism, and quantum theory would have been brought to the fore significantly earlier.