ERP and Entangled States: Before this tutorial, we recommend lecture coverage of EPR, entanglement, and Bell states (e.g., Ch 4.1 of McIntyre)

The experimental context for this module is a gedanken experiment (since realized in the lab) building on ideas from Einstein, and outlined by David Mermin in a 1985 Physics Today article: "Is the Moon There when Nobody Looks?"  It is described in the language of 2-particle entangled states of spin-1/2 objects. The setup is designed so two distant observers make simultaneous measurements on their particles (each part of an entangled pair), and the correlations of outcomes they measure would lead Einstein (or anyone believing that widely separated particles cannot influence each other) to argue that the measurable physical properties (outcomes) must already be there BEFORE a measurement is made.  Einstein, Podolsky, and Rosen called these "elements of reality." We now know that experiments disagree with this - and indeed, the non-classical outcomes seem to imply "spooky action at a distance." 

Prerequisites for students: 
The required formalism is largely described in the sections below, but to follow the notation, students need to have some prerequisite skills, including familiarity with Dirac notation for single-particle states, e.g. $\ket{+}$ for a spin-up state (in the Z-basis). (Through this section, we may casually switch notations e.g. to $\ket{\uparrow}$ or $\ket{0}$, if this makes a particular question easier to state or discuss.)  

In the downloadable lecture notes, we make use of a setup where measurement of a spin-1/2 object can be made along an arbitrary axis pointing in the $\hat{n}$ direction. We work with measurements constrained to the x-z plane.  Students are thus expected to know the following (and if not, you will need to teach this before this unit):

• A spin-1/2 object polarized in the $+\hat{n}$ direction (in the x-z plane, tilted by angle $\theta$ from the $z$-axis) can be written:
  $\ket{+}_n = \cos\frac{\theta}{2} \ket{+} + \sin\frac{\theta}{2} \ket{-}$
• The probability that a single spin-1/2 particle in any quantum state $\ket{\psi}$ will be measured as "up" in the $\hat{n}$ direction is thus given by  $| {}_n\bra{+} \ket{\psi}|^2$. 

Physicists continue to debate "interpretation" of quantum states, but this broader module is designed to introduce students to Bell tests and to present the experimental evidence that shows entangled states do not imply "local hidden variables" (or what Einstein called "elements of reality" in his EPR paper).

We thus begin our unit on entanglement (leading to an understanding of the EPR paradox) with a brief discussion of how one interprets a superposition state as philosophically distinct from a mixed state. We do not go into the mathematics of mixed states, as it is not needed for our discussion.

Both superposition and mixed states yield definite probabilities for measurement outcomes, but they are not the same: superposition states contain relative phase information which can lead to distinct experimental outcomes. (In a very simplistic way, one might say that a superposition is "this AND that" while a mixed state is "this OR that" with given probabilities. ) 

This section is also a good time to talk about the different language you are using to describe different states and what is interpretation vs experimental outcome.

To talk about entanglement, we need to introduce the notation for referring to two particles.  We use the tensor product ($\otimes$) to 'join' two states (and also to join two Hilbert spaces). 

The state $\ket{0} \otimes \ket{1} = \ket{01}$ (note that we often suppress the $\otimes$ symbol) is a two-particle state where the first particle is in the state $\ket{0}$ and the second particle is in the state $\ket{1}$.

Two-particle states that can be written as $\ket{\psi} \otimes \ket{\phi}$ are straightforward.  For example, we can have $\frac{1}{\sqrt{2}} ( \ket{0} - \ket{1}) \otimes  \frac{1}{\sqrt{3}}(\ket{0} + \sqrt{2} \ket{1})$ that can be expanded out as $\frac{1}{\sqrt{6}} ( \ket{00} + \sqrt{2}\ket{01} - \ket{10} - \sqrt{2} \ket{11})$. 

Entangled states are defined as states that CANNOT be written as $\ket{\psi} \otimes \ket{\phi}$.  

An example of an entangled state is $\frac{1}{\sqrt{2}} (\ket{00} + \ket{11})$. The example concept tests below help show the weirdness of entangled states.

Consider an experiment with a spin-0 source producing pairs of spin-1/2 particles in quantum state $\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{+-} - \ket{-+})$. This state (or equivalently, simply the fact that we start with a spin-0 system and must conserve total angular momentum) ensures that any measurement by observer #1 of the component of spin along any given direction will be perfectly anticorrelated with a measurement by observer #2 in the same direction.

How can we explain such perfect anticorrelation for well-separated observers?

A natural idea is that the particles must carry with them information about what outcome will be measured from the state. This is referred to as "local hidden variables". Einstein argued in his 1935 EPR paper that such information (or "elements of reality") must exist in order to explain perfect anticorrelations if the observers are very far apart. Otherwise, such anticorrelation would have to be an (absurd?) "spooky action at a distance".

In a highly simplified Gedanken experiment, as shown below, the observers agree to make only one of 3 choices for axis direction, axes $\hat{a}$, $\hat{b}$, or $\hat{c}$, oriented $120^{\circ}$ apart (all in the plane of the page.) The decision of what direction to measure along is made randomly, at the same time, immediately before the measurement.  This is done to ensure that observers #1 and #2 cannot communicate to each other what their measurement direction choices are. 

(Note: directions ($\hat{a}$, $\hat{b}$, $\hat{c}$) all lie in the plane of the page, it is easy to incorrectly misinterpret the diagram above as showing 3-D right handed coordinates) 

The existence of local hidden variables would mean that each particle carries an instruction set of what would be measured along any of these three direction choices. In this experiment, with $\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{+-} - \ket{-+})$, the two particles must have "opposite" instruction sets to ensure perfect anticorrelation. These hidden variables need to exist before any measurement is made and include instructions for measurements along all measurement directions.

The experiment repeats many times, and the outcome ($+$ or $-$) for each particle is recorded (but not the axis of measurement).  We compute the probability after many trials that the two observers record the SAME outcome (both $+$ or both $-$).  By enumerating all possibilities, we quickly establish that Prob(same) $\le$ 4/9, no matter what the distribution of "hidden instruction sets" might be. This is a Bell inequality; it arises simply from counting. It must hold for any local hidden-variable theory.

More details are found in the next section and our downloadable notes.

References: The original EPR paper is quite readable (Phys. Rev. 47, 777 (1935)), but we find the suggested experimental context of position and momentum harder for our students than the spin version. The setup we use follows Mermin's (excellent) paper "Is the moon there when nobody looks?" 

The previous section on Hidden Variables set up an example of a "Bell test." An initial 2-particle state $\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{+-} - \ket{-+})$ is created, and well-separated observers Alice and Bob each measure one of the two particles. They choose to measure spin along one of three randomly-selected axes, $120^{\circ}$ apart. Each individual measurement is recorded only as $+$ or $-$ (they do not record the axis measured). After many measurements, they compare outcomes and determine Probability(same outcome). 

A quantum calculation begins by computing Prob($++$) in the special case  that observer #1 measures in the z-direction, and observer #2 is in any of the (three) $\hat{n}$ directions:
$Prob(\#1 \ is\ +\hat{z}, \#2\ is +\hat{n}) = |\ \bra{+}\ \bra{+_n}\ \ket{\psi}\ |^2$

Using $\ket{+_n} = \cos{\theta/2}\ket{+} + \sin{\theta/2} \ket{-}$, we find this probability is $\frac{1}{2}\sin^2{\theta/2}$ (see lecture notes for more details)  Adding Prob(--) = Prob(++), we get (for this situation):
Prob(same) = $\sin^2{\theta/2}$ .
Averaging over the other possibilities for measurement combinations (e.g. that $\hat{n}$ can be oriented at $\theta = 0, \pm 120^{\circ}$ with equal likelihood), we conclude Prob(same) = 0.5 

This outcome has been well-tested experimentally. This violation of the "Bell inequality" (Prob(Same) $\le$ 4/9) experimentally validates quantum theory and is explicitly inconsistent with local hidden variable theories. 

Reference: 
"Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons ", Marissa Giustina et al., Phys. Rev. Lett. 115, 250401 (2015)