How the EPR Paradox Fits Between Entanglement and Nonlocality

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If we violate a Bell inequality, we know we have quantum entanglement, and it used to be thought that these two notions were one-and the same.

At least, this was commonly accepted until 1989, when R.F. Werner actually proved that this isn’t so. There really are some entangled states that are Bell-local — states that you can always describe with a Local Hidden Variable (LHV) Model, and which therefore will never violate any Bell inequality.

Since then, physicists have shown that Bell-nonlocality is formally stronger than entanglement as far as correlations go. If the correlations between positions [itex]x_A[/itex] and [itex]x_B[/itex] of particles [itex]A[/itex] and [itex]B[/itex] can be explained by an LHV model, then their joint probability distribution [itex]\rho(x_A,x_B)[/itex] factors when you condition on these hidden variables  [itex]\lambda[/itex] (as seen in the following equation).

[itex]\rho_{LHV}(x_A,x_B)=\int d\lambda \;\rho(\lambda)\; \rho(x_A|\lambda) \;\rho(x_B|\lambda).[/itex]

Bell inequalities are mathematical consequences of LHV models like this, which is why violating Bell inequalities rules out such LHV models.

If you add the extra assumptions that the probabilities [itex]\rho(x_A|\lambda)[/itex] and [itex]\rho(x_B|\lambda)[/itex] have to be described quantum-mechanically (i.e., as arising from some wavefunction or density matrix), then what you have is a model for a separable (i.e., not entangled) state:

[itex]\rho_{sep}(x_A,x_B)=\int d\lambda \rho(\lambda) \; Tr[\hat{\rho}_{\lambda}^{A}|x_{A}\rangle\langle x_{A}|] \; Tr[\hat{\rho}_{\lambda}^{B}|x_{B}\rangle\langle x_{B}|].[/itex]

So all separable states’ statistics can be described by some LHV model, but not all states whose statistics obey an LHV model are separable. This extra assumption of quantum mechanical origins of probabilities limits the possible probabilities in a big way because our statistics have to obey the uncertainty principle.

So here we have a relatively tidy picture of the difference between entanglement and Bell-nonlocality. The correlations in separable states obey a special kind of LHV model, where the statistics of both particles [itex]A[/itex] and [itex]B[/itex] also have to be quantum-mechanical in origin.

More recently though, people have been considering the intermediate case.
What if we say that only the statistics of particle [itex]B[/itex] have to be described quantum mechanically? i.e.,

[itex]\rho_{LHS-B}(x_A,x_B)=\int d\lambda \rho(\lambda) \;\rho(x_A|\lambda)\;Tr[\hat{\rho}_{\lambda}^{B}|x_{B}\rangle\langle x_{B}|].[/itex]

What we get by ruling out such intermediate models is a level of correlation that’s definitely stronger than entanglement, but definitely weaker than Bell-nonlocality. This level of correlation is called EPR-steering, and is synonymous with demonstrating the EPR-paradox. Inequalities derived from these sorts of LHV models (called Local Hidden State (LHS) models for [itex]B[/itex]) are called EPR-steering inequalities, which when violated, also demonstrate the EPR-paradox. Since here, the statistics of [itex]B[/itex] must be quantum mechanical in origin, all statistics of [itex]B[/itex] obey the uncertainty principle, even when conditioning on other variables.

Now, we have something of a hierarchy of correlation, where separable states, are a subset of non-steerable (we could all them EPR-local) states, which are in turn a subset of states obeying some sort of LHV model (we could call them Bell-Local). But what does EPR-steering have to do with the EPR paradox?

EPR-steering is a relatively new concept in quantum physics having only just been formalized in 2007 (and which doesn’t even have a Wikipedia article yet). That being said, the fundamental ideas behind EPR-steering date back to the days following the original EPR-paradox paper in 1935. In Schrodinger’s response to the EPR-paradox paper, he remarks that it is rather uncomfortable that the theory (of quantum mechanics) allows the state (of a particle) to be steered at the experimenter’s mercy in spite of their having no access to it. What this “steering” is, is best explained by looking back at a simplified version of the EPR-paradox.

Consider two particles [itex]A[/itex] and [itex]B[/itex], described by a single joint wavefunction highly entangled in position and momentum. We separate them by a long distance so that light cannot travel between them in the time we make our measurements. They are so highly correlated that measuring the position of [itex]A[/itex], allows you to predict the position of [itex]B[/itex] with near certainty, and the same goes for momentum. If we live in a local universe, then everything there is to know about the statistics of [itex]B[/itex] is within the past light cone of [itex]B[/itex] (i.e,. the set of all points in the past where light or slower things could conceivably affect [itex]B[/itex] at the time of measurement). Since particle [itex]A[/itex] is outside [itex]B[/itex]’s past light cone, its statistics couldn’t possibly tell you more about [itex]B[/itex] than knowing everything in [itex]B[/itex]’s past light cone. In addition, all this knowledge of the history of [itex]B[/itex] cannot reduce your measurement uncertainty by more than what the uncertainty principle allows. Because of this, if we condition our measurement results of [itex]B[/itex] on the results of [itex]A[/itex], we should not be able to get conditional uncertainties beyond the limits of the uncertainty principle. However, there’s no limit to how strong the correlations between [itex]A[/itex] and [itex]B[/itex] can actually be, and we can have both conditional position and conditional momentum uncertainties be arbitrarily small. The conditional uncertainty relations that can be violated by strong enough correlations are some of the first EPR-steering inequalities. The EPR “paradox” is that it seems that quantum physics can’t be a complete description of reality if the universe is also local. These are mutually exclusive.

As far as why it’s called “steering”, quantum mechanics tells us that in the EPR scenario if you measure the position of [itex]A[/itex], the quantum state of [itex]B[/itex] will be one of well-defined position. If you measure the momentum of [itex]A[/itex], then [itex]B[/itex] will be in a state of well-defined momentum. By choosing whether to measure the position or momentum of [itex]A[/itex], one has influence over the ensemble of states of [itex]B[/itex], though this influence can only be realized with the additional information about the outcome of the measurement. This phenomenon is known as quantum steering, since one can “steer” the ensemble of states of [itex]B[/itex] by manipulating [itex]A[/itex] (and this is the “steering” Schrodinger was talking about). However, quantum steering is not sufficient to enable faster than light signaling because the density matrix of [itex]B[/itex] is totally independent of manipulations of [itex]A[/itex] unless one also conditions on the measurement outcome of [itex]A[/itex], which would have to be communicated to [itex]B[/itex] by conventional means.

So how does the EPR paradox fit between entanglement and nonlocality? Right in the middle. The sets of Bell-Local, EPR-local, and separable states form concentric sets in a hierarchy of correlation, and research into applications of these correlations is a rapidly growing field with significant results coming out every month.



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