EPR (Einstein, Podolsky, Rosen)

Paper that investigated the problem of entanglement.

Hidden variables

Suggestion that the underlying features of QM might be understandable by adding variables.  Imagine that you are witnessing a process but all you can see are the shadows of the objects involved. Objects could appear to pass through each other when only the shadows were viewed. Knowing that there is another dimension clarifies the observation.

eventuality

A value that an observable can have. Specifically a two valued outcome.

True/False : one way to characterize the outcome

QM: System can be in an unknown state č neither true not false.

indefinite

the eventuality is undetermined

potential

a state that has the potential to have some value

actualization of potentiality

to place the state in a definite state with a specific value for the potential outcome.

Bell’s inequality

Bell derived a way of distinguishing the outcomes predicted by quantum mechanics versus those predicted by a general hidden variable theory.

non-local

Events separated by some distance are correlated. The correlation is not necessarily based on some propagation of information.

http://en.wikipedia.org/wiki/Non-local#Nonlocality_in_quantum_mechanics

 

 

Following Abner Shimony in the New Physics, who creates a vocabulary that is distinct so that he can carefully  describe a quantum system. Shimony breaks QM into a set of basic postulates summarized here. He also discusses the problem of entanglement and measurement.

 

 

1

“Associated with every physical system is a complex linear vector space V, such that each vector of unit length represents a state of the system.”

2

“There is a one to one correspondence between the set of eventualities (observables) concerning the system and the set of subspaces of the vector space associate with the system, such that if e is an eventuality (observable) and E Is the subspace that corresponds to it, then e is true in a state |S> if and only if any vector that represents S belong to E; and is false in the state S if and only if any vector that represents S belongs to E(orthogonal) .”   A states described by a vector that has components in both subspaces represents a state with an unspecified value for this observable e.

3

“If |S> is a state and e is an eventuality (observable) corresponding to the subspace E, then the probability that e will turn out to be true if the initially the system is in state |S> and an operation is performed to actualize (measure) it.

v is a unit vector representing S … and PE is the projection.”

 

4

“If 1 and 2 are two physical systems, with which the vector spacesV1 and V2 are associated, then the composite system 1 + 2 consisting of 1 and 2 is associated with the tensor product V1 x V2.”

5

“If a system is in a nonreactive environment between 0 -> t, then there is a linear operator U(t) such that  U(t) |v> represents the state of the system at time t if |v> represents the state of the system at time 0. Furthermore, ||U(t) v||2 = || v||2 for all v in the vector space.”

 

 

 

The author (Shimony) is an expert on quantum measurement, EPR, and Schrodinger cat paradoxes.  He carefully explores the basic structure of quantum theories as a network of potentialities. Possible outcomes or values that a system can assume need to be interconnected in a way to reflect quantum nature.

 

Classically the state of a system determines all of its eventualities.

 

QM states in a maximal state of information do not specify all of their eventualities. There are aspects of the system which are indefinite.  Outcomes from measurements or actualizations of these potentialities are described by probability or chance.

 

The first point that needs to be established in order to treat the problem of quantum entanglement is to describe how two systems are described in QM. This is point 4 in the above table. We describe the combination of two systems as a product space. Consider

a system 1 that can have value for a two state system of T or F. Another system has an eventuality that similarly can be either T or F.

 

 

Separately the two states could be written as shown above.

 

The combined system might be

 

 

This a state that is NOT entangled and the outcomes for each state can be separated. Consider however the problem where the two systems are correlated by perhaps some conservation rule.

 

Notice that this state can not be separated. There is no product of vectors  that can produce this state. The oft quoted example is the decay of an atom into two back to back photons which must posses zero angular momentum. The example provided by Hong and Nu are position correlated photons.

 

Let us consider this experiment. There are correlated photons. The position of one photon will restrict the direction of the other photon.  A pair of photons is shown in red below. A pair consists of a signal (system 1) and an idler photon (system 2). The experimenter chooses to allow signal photons to interfere via a slit system (measure location on a screen y) . The experimenter can then choose to

  1. either measure the idler photons (T2, F2) or
  2. to allow them to go through slits and then be measured on a screen (y’). 

Idler measurement 1 can in principle be used to determine which slit the signal photon will hit and therefore eliminate the possibility of interference. While measurement 2 cannot distinguish the path for either signal or idler photon and so interference will occur.

 

Start with a correlated state

 

 

end with either:

 

Clearly experiment1:

 

 

and experiment 2

 

The cross terms for experiment 2 can result in interference.

 

The question of information transmission is typically addressed in these gendaken (thought) experiments.  Here again one can imagine removing the detectors and putting them back into the experiment.  The result of this procedure will be that the signal experiment will immediately show a transition from interference to non-interference. The first photon that reaches the new setup has a partner that is also at the signal slits. Thus it seems that one could change the apparatus instantly and the result would be detectable at the signal experiment. Have we built a system that allows super luminous transmission of information?

 

The photons must emerge so that either of the slits can be reached by the signal photons.  Now suppose that the idler system has one detector for the top slit but not a detector for the bottom slit.  We measure when the signal photon is tagged as going through the bottom slit but do not ascertain if it definitely went through the top slit.

 

Here the question arises is detecting a photon the same as putting a detector in a location and not detecting it. I believe I can make my detector 100% efficient so that you can say that the photon did not go through slit. [ If we take an extended wf and place many detectors into the area one and only detector will fire. The screen for the two slit problem is an example of this phenomena. The particle appears somewhere on the screen but jumps around.  Its wavefunction must extend across the screen in a classic interference pattern until at an instant an interaction occurs that gives away the position. Now the wf collapses.  At the instant before detection the photon is extended across some range.]  If the photon is not detected then I would say we are sure that the photon didn’t go through the slit however the photon could be anywhere else. The analysis of the wf gets complicated but covering the slit with a highly efficient detector could be considered the same as closing the slit when no photon is detected.   Now the correlation in terms of angular range must be more carefully analyzed. Can an idler photon miss the detector but be correlated to a signal photon that can pass through the slit?  Could a photon hit near the detector and be correlated with a signal photon that could go through the required slit for an interference.  Note as you try to reign in all the correlations the problem typically can be shown to produce an effect that is consistent with quantum and relativity.  A careful analysis of the two slit problem and the emission of a photon revisit this issue in a similar fashion. One must really take all possibilities into consideration.  The photon might have a wavelength that doesn’t localize the electron sufficiently or it might reach the opposite detector so that the ambiguity remains. Feynman considers these implications by imagining a light that illuminates the slits. Detecting a scattered photon might infer through which slit an electron went. However the wavelength of the photon is critical in determining how well position is resolved. The possibility of a misinterpretation due to the A photon scattering to the B-detector must also be analyzed. Students are reffered to this excellent discussion in Feynam’s intro physics books for further discussion.

 

To return to the double two slit problem. Let us assume that we cover one slit with a detector that is large enough to cover the area required to show that a photon could not have passed through the associated signal slit. If this detector has an area A then it subtends and angle A/d where d is the distance to the detector.  Now if we move further away the detector must increase in area to cover the same solid angle.  The distance across the detector will grow proportionally with the distance to the source. In order to control this experiment (i.e. turn off the detector or move it) one must send a signal that traverses the area and this can travel at maximum speed of c. So the further you are away from the source the longer it takes to change the experiment. While I have not carried the analysis at this point to a firm conclusion I hope it is clear that one needs to again be careful to consider features of the experiment design in order to address the questions of causality, information transmission and non-locality. 

 

Similarly, the problem of using a slit system at large distances means that we need subtend smaller and smaller angles until the back to back beams are almost parallel. This of course would eliminate any angular correlation between the idler and the signal photon.

 

To touch on this issue I will turn to the two photon experiment.  One can find numerous discussions of the problem and it illustrates the way these problems need to be addressed. Since this only a sideline discussion I will leave it here.

 

I welcome further discussion of the Hong experiment but without student input I will leave this problem as incomplete.

 

Atomic decay into two correlated photons is treated in detail in Shimony’s essay in the new physics.  Here the two experimenters are free to orient their polarizer at an arbitrary angle. If the direction of the photon is along the z-axis then the filters are perpendicular to z and can be rotated about the z-axis. If experimenter-1 sets his filter he can label the polarization of his photon (γ1).  The assumption before hand would be that the photon is in a definite but unknown state of polarization. You and imagine that it is a vector pointing somewhere in the x-y plane. The act of measuring the photon forces it to collapse to one of the two possible states T1, or F1.  The surprising result is that the second photon (γ2) must now assume a specific state. However the second experimenter doesn’t need to orient his apparatus so as to measure this polarization. We know that a beam can be prepared in a specific orientation of polarization and be measured wrt a different orientation. Thus a beam linearly polarized along the x-direction can impinge on a filter with a 30o orientation. There is some overlap or projection of the x-state onto the x’ state (cosθ). Experimenter 1 should observe 50% of his photons as x-polarized and 50% of his photons as y-polarized. So even though there is an explicit correlation for individual events experimenter 2 has no result that will imply which orientation is being used by experimenter 1. While some events are enhanced based on experiment-1 other events are reduced with the eventual outcome that experimenter 2 always measures a 50-50 split. However if all of the x-1 data are grouped by experimenter 2 a cosθ dependence will emerge and if the y-1 data are grouped a sinθ dependence will emerge.

 

x-y system

 

x’-y’ system

 

where

 

 

compound entagled state

or

 

using the above relationship I can cast the state into a mixture of two different orientations.

 

So it is possible to write the state wrt different filter orientations.

 

This is an example of how information transmission cannot be performed at super luminal speeds for an entangled states. Careful analysis of entanglement has of yet not revealed a mechanism that violates the principles of relativity.

 

Shimony discusses this as follows:

 

“… If the reduction of ψ is to be interpreted causally, then which of the events is the cause and which is the effect? ..”

For space-like separation the time ordering is undetermined and can be reversed based on a choice of a reference system.

“.. quantum mechanics presents us with a kind of causal connection which is different from anything that could be characterized classically…”

“.. This kind of causal connection has no classical analogue.. uantum mechanical potentiality has essentially broadened the concept of an event.”