Einstein-Podolsky-Rosen (EPR) argument argued that quantum mechanics could not be a complete theory because such entanglement implies either faster-than-light influences or incomplete physical descriptions.[^1]
EPR argued that any reasonable physical theory should meet the criteria of local realism, where:
- “local” refers objects that are influenced only by their immediate surroundings
- “real” refers to objects that have definite properties independent of measurement. According to EPR, quantum entanglement is incompatible with local realism, since it implies that particles do not have definite values independent of measurement and can be influenced across space arbitrarily quick[^3].
[explain hidden variables proposed by Einstein]
Theory
Photonic quantum states
Polarization is the direction of electric field oscillation with respect to a plane of reference. Quantum states can be defined in terms of polarization as a two-level quantum system. They can be represented and measured in various orthonormal bases. The most simple one is the H/V (horizontal/vertical) basis, which can be represented using column vectors:
A photon that is oriented horizontally can pass through a horizontally aligned polarizer, but cannot do so in a vertically aligned polarizer.
We can also define superposition bases, such as the D/A (diagonal/anti-diagonal) basis, representing a / rotation:
\ket{A} = \frac{\ket{H} - \ket{V}}{\sqrt{2}}$$ A diagonally aligned photon is an equal (50\%) superposition of the horizontal and vertical states. Another commonly used superposition basis is the R/L (right/left circular) basis, representing a complex rotation: $$\ket{R} = \frac{\ket{H} + i\ket{V}}{\sqrt{2}},\ \ket{L} = \frac{\ket{H} - i\ket{V}}{\sqrt{2}}$$ ### Measurement basis transformation To measure in polarization bases, optical wave-plates are used to transform the polarization state. Wave-plates introducing a phase difference between orthogonal polarization components by manipulating the birefringent material. Birefringence is an optical property that has a refractive index dependent on polarization and direction of light. This experiment utilizes two types of waveplates, each with a precision of $2 /degree$: 1. **Half wave-plate (HWP)**: rotates the polarization direction by an angle $\theta$ according to $$\text{HWP}[\theta] = \begin{pmatrix} \cos(2\theta) & \sin(2\theta) \\ \sin(2\theta) & -\cos(2\theta) \end{pmatrix}$$ 2. **Quarter wave-plate (QWP):** rotates the polarization direction by an angle of $\theta$ according to $$ \text{QWP}[\theta] = \frac{1}{\sqrt{2}}\begin{pmatrix} 1+i\cos(2\theta) & i\sin(2\theta) \\ i\sin(2\theta) & 1-i\cos(2\theta) \end{pmatrix}$$ For the H/V basis, all waveplate angles are set to $\theta=0^o$ given that the pump photons from the source is $\ket{H}$. In order to measure polarization correlations in different measurement bases, the photon states must be converted into the respective basis using the waveplates, given that the initial measurements were performed in the H/V basis. - **converting to the D/A basis from the H/V basis**: set the QWP angle to $45^o$ and the HWP to $22.5^o$ - **converting to the R/L basis from the H/V basis**: set the QWP angle to $-45^o$ and the HWP to $0^o$ *Derivation*: On the Bloch sphere, different polarization bases occupy different locations. Given that the initial measurement is along the vertical axis (H/V direction), wave-plates are used to rotate the sphere so that our desired basis aligns with the vertical measurement axis. For the D/A basis measurement, the HWP at $22.5^o$ rotates polarization by $2 \times 22.5 = 45^o$ needed to map $\ket{D} \to \ket{H}$ and the QWP at $45^o$ ensures the proper form For the R/L basis measurement, ### Quantum entanglement and Bell states **Quantum entanglement** is the idea that particles (in this case, photons) can be correlated in a quantum mechanical sense, so that changing the state of one particle changes the state of the other when they "reconnect" with one another. Bell states are maximally entangled quantum states. There are four possible Bell states for polarization-entangled photons:\begin{align} \ket{\phi^+} = \frac{\ket{HH} + \ket{VV}}{\sqrt{2}},\ \ket{\phi^-} = \frac{\ket{HH} - \ket{VV}}{\sqrt{2}},\ \ket{\psi^+} = \frac{\ket{HV} + \ket{VH}}{\sqrt{2}},\ \ket{\psi^-} = \frac{\ket{HV} - \ket{VH}}{\sqrt{2}} \end{align}
John S. Bell proposed a new thought experiment that could test whether the behavior of quantum entangled particles was consistent with local realism. Consider two detectors that can perform measurements whose outcomes could only ever take one of two values; the measurement outcomes can be represented as $A = \pm 1$ and $B= \pm 1$. Also, detector A can choose to take measurements with input settings $a$ and $a'$; likewise detector B can choose to take measurements with input settings $b$ and $b'$. Locality is imposed by assuming that the output from the two detectors depends only on the input setting (a, b) and other properties prepared at the source that could affect the measurement (these properties are called "hidden variable", proposed by Einstein, represented by $\lambda$). Realism is imposed by assuming the inputs (a , a' , b , b') result in a predetermined output ($\pm 1$). The expectation value $E(a,b)$ for a given experimental run: $$E(a,b) = \int \partial{\lambda} \ A(a, \lambda) B(b, \lambda)$$ The value $\lambda$ is considered to vary for each experimental run and follows a probability distribution $\int \partial \lambda f(\lambda)$, where both particles A and B would share the same value of $\lambda$ on any given run. This is demonstrated by table below. | $\lambda$ | a | a' | b | b' | | ------------- | ----- | ----- | ----- | ----- | | $\lambda_{1}$ | 1 | -1 | 1 | 1 | | $\lambda_2$ | 1 | 1 | -1 | -1 | | . . . | . . . | . . . | . . . | . . . | The combination of expectation values, resulting from the different detector input settings, gives the parameter $S$ known as the Clauser-Horne-Shimony-Holt (CHSH) parameter. $$ S = E(a,b) + E(a' , b) - E(a, b') - E(a' , b')$$ As can be seen from table 1, for any particular $\lambda_i$, there are two possibilities: - Either $B(b , \lambda) + B(b', \lambda) = 0$ and $B(b, \lambda) - B(b' , \lambda) = \pm 2$ - or $B(b , \lambda) + B(b', \lambda) = \pm 2$ and $B(b, \lambda) - B(b' , \lambda) = 0$ This leads to: $$A(a, \lambda) [B(b , \lambda) + B(b', \lambda)] + A(a' , \lambda)[B(b , \lambda) - B(b', \lambda)] = \pm 2$$ $$ \int \partial \lambda f(\lambda) \ A(a, \lambda) [B(b , \lambda) + B(b', \lambda)] + A(a' , \lambda)[B(b , \lambda) - B(b', \lambda)] \leq 2$$ And so the CSHS inequality becomes $$|S| = |E(a,b) + E(a' , b) - E(a, b') - E(a' , b')| \leq 2$$ **spontaneous parametric downconversion (SPDC)** = a non-linear optical process where a photon spontaneously splits into two other photons of lower energies[^2] [^1]: Fine, Arthur, (2020) "The Einstein-Podolsky-Rosen Argument in Quantum Theory", _The Stanford Encyclopedia of Philosophy_, Edward N. Zalta (Summer 2020 ed.), https://plato.stanford.edu/entries/qt-epr/ [^2]: Couteau, C. (2018). Spontaneous parametric down-conversion. _Contemporary Physics_, _59_(3), 291–304. https://doi.org/10.1080/00107514.2018.1488463 [^3]: Kaiser, D. (n.d.). _Lecture Notes for 8.225 / STS.042, “Physics in the 20th Century”: Bell’s Inequality and Quantum Entanglement_.