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May 07, 2020

Lecture Notes PH 411/511 ECE 598 A. La Rosa Portland State University

INTRODUCTION TO QUANTUM MECHANICS _____________________________________________________________________________________

CHAPTER-12 QUANTUM ENTANGLEMENT The annihilation of the positronium This chapter is taken completely from the The Feynman Lectures, Vol III, Chapter 18 and

from the book Quantum Mechanics by D. J. Griffiths. The annihilation of the positronium process with the consequent generation of two entangled photons is described by Feynman in great detail, accounting for the conservation of energy, linear momentum, angular momentum and parity. Feynman argues there does not exists a paradox when stating that measurement on one side affect the result of measurement made at another far away location.

I. The QUANTUM THEORY and the BELL’s DISCOVERY I.1 The EPR paper on the (lack of) Completeness of the Quantum Theory I.2 The Bohm experimental version to settle the EPR paradox

II. ILLUSTRATION of an ENTANGLEMENT PROCESS: The ANNIHILATION of the POSITRONIUM

III. BELL’S THEOREM

IV. QUANTUM TELEPORTATION

V. ENTANGLEMENT from INDEPENDENT PARTICLE SOURCES

APPENDIX-1: Tensor Product of State-Spaces APPENDIX-2: THE EPR Paper on the (lack of) Completeness and Locality of the Quantum Theory.

Einstein, Podolsky, and Rosen, Phys. Rev. 47, 777 (1935).

| R ⟩

(2)

| x ⟩

| L ⟩ )

(2)

(2) (1)

(1)

(1)

| y ⟩

| R ⟩

| L ⟩

F

Polarizer

Polarizer

Polarizer

Polarizer

Polarizer Polarizer

Fig. 1 Positronium decay into a directional two-photon state |F ⟩. (All direction are

equally probable). A measurement along one direction (using polarization filters) makes the state |F⟩ to collapse into correlated-states determined by the polarizers.

I. The QUANTUM THEORY and the BELL’s DISCOVERY

Consider F~ to be an observable-operator associated to the observable f (quantity one measures experimentally), which has a complete orthonormal set of eigenfunctions { 1, 2,

… } and corresponding eigenvalues { f1, f2, … }. Let’s consider a system (an ensemble of

systems) is in the state described by,

= m

m cm = m

m ⟨m

What exactly does this wavefunction mean?

We have so far adopted the Born’s statistical interpretation, postulating that 2

jc gives the

probability P( fj ) of obtaining the value when making a measurement of f on a particular system of an ensemble.

Under this interpretation, the wave function does not uniquely determine the outcome of

a measurement; instead it provides a statistical distribution of possible results. Such an

interpretation has caused deep controversial discussions.

Suppose, for example, one measurement renders the value f4. What was the value f of the system before we made the measurement?

There are three main schools of thought for answering that question:

i) The realistic viewpoint: “The system had the value f4.”

That is, the physical system has the particular property being measured prior to the act of measurement. This view was advocated by Einstein.

Accordingly, quantum mechanics is an incomplete theory, for even when the system had the value f4, still quantum mechanics is unable to tell us so.

(The theory is silent about what is likely to be true in the absence of observation.) Einstein hoped for progress in physics to yield a more complete theory, and one where the

observer did not play a fundamental role.

Therefore, there is some other additional information (known as hidden variable), which together with the wave function is required for a complete description of the physical reality of the system. [But after 1927 Einstein regarded the hidden variables project — the project of developing a more complete theory by starting with the existing quantum theory and adding things, like trajectories or real states — an improbable route to that goal.]

In 1935 Einstein co-authored a celebrated paper supporting the realistic view point and questioning the completeness of the quantum theory.1 Fifteen years later Bhom proposed to analyze the EPR paper but thought an experiment involving the dissociation of a diatomic molecule where the two parts together should satisfy the conservation of angular momentum. Different EPR-Bohn type experimental setup have been suggested and implemented since.

ii) The orthodox viewpoint: “the system had no specific value of f.”

It is the act of measurement that forces the system to adopt a specific value.

A measurement forces a system to adopt a given value (corresponding to the the type of measurement being done). Or equivalently, a measurement makes the wavefunction to collapse into a given stationary state, thus “creating” an attribute on the system that was not there previously. “Measurements not only disturb what is to be measured, they produce it. We compel the system to assume a definite value of f.”2

For example, a two-electron system may be in the state 0S 2/1 [ 1)( 2)( - 1)( 2)( ] (where one electron is flying in the opposite direction of the other). Upon using a magnetic

field apparatus to measure the spin of the particles, one possible outcome is electron -1 in the state

1)( and electron-2 in the state 2)( . That is, the measurement has “created” these new states.

Furthermore, Bohr enunciated the principle of complementarity, which holds that objects have complementary properties that cannot all be observed or measured simultaneously.

iii) Agnostic response: duck the question on the grounds that it is “methaphysical”. There were so many direct applications of the (maybe incomplete) quantum mechanics

theory that many physicists left the conceptual foundation interpretations aside for the time being.

These three views on the interpretation of the wavefunction were subject of controversial discussions. But in 1964 John Bell astonished the physics community by showing that it makes an observable difference whether the particle had a precise (though unknown) value of f prior to the measurement, or not.3

Bell was able to lay down conditions that all deterministic local theories must satisfy.

It turns out those conditions are found to be violated by experiment.

System with hidden variables satisfy Bell’s inequalities Therefore,

No Bell’s inequalities fulfilled non-existence of hidden variables

Bell’s discovery effectively eliminated agnosticism as a viable option, and make it an experimental question whether i) or ii) is the correct choice.

Current experiments have decisively confirmed the orthodox interpretation. A system simply does not have a precise vale of f prior to measurement. It is the measurement process that insists on one particular number , and thereby in a sense creates the specific result, limited only by the statistical weighting imposed by the wavefunction.4

What if we made a second measurement, immediately after the first? Would we get f4 again? There is a consensus that the answer is yes. Evidently the first measurement radically

alters the wavefunction, so it is 4 right after the measurement. It is said that, upon

measurement, the wavefunction collapses to 4, and then the latter starts to evolve in accordance with the Schrodinger equation.

= m

m cm tmeasuremen

4 time withEvolution

Hence, if the second measurement is made quickly, then it will render the same value f4.

I.1 THE EPR Paper on the (lack of) Completeness and Locality of the Quantum Theory. Einstein, Podolsky, and Rosen, Phys. Rev. 47, 777 (1935).

This paper is available online http://journals.aps.org/pr/pdf/10.1103/PhysRev.47.777 Einstein, Podolsky, and Rosen questioned the completeness of the quantum mechanics theory. Here we highlight some of their statements:

It describes measurements of two non-commuting variables, position and momentum

The EPR paper emphasizes on the distinction between the objective reality (which should be independent of any theory), and the physical concepts with which a given theory operates

The concepts are intended to correspond with the objective reality, and by means of these concepts we picture this reality to ourselves.

The EPR text is concerned with the logical connections between two assertions. Quantum mechanics is incomplete. Incompatible quantities (those whose operators do not commute, like the x-

coordinate of position and linear momentum in direction x) cannot have simultaneous “reality” (i.e., simultaneously real values).

The authors assert that one or another of

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