CHSH inequality
CHSH inequality
In physics, the Clauser–Horne–Shimony–Holt (CHSH) inequality can be used in the proof of Bell's theorem, which states that certain consequences of entanglement in quantum mechanics cannot be reproduced by local hidden-variable theories. Experimental verification of the inequality being violated is seen as confirmation that nature cannot be described by such theories. CHSH stands for John Clauser, Michael Horne, Abner Shimony, and Richard Holt, who described it in a much-cited paper published in 1969. is a constraint—on the statistical occurrence of "coincidences" in a Bell test—which is necessarily true if an underlying local hidden-variable theory exists. In practice, the inequality is routinely violated by modern experiments in quantum mechanics.
Statement The usual form of the CHSH inequality is
where
<math>a</math> and <math>a'</math> are detector settings on side <math>A</math>, <math>b</math> and <math>b'</math> on side <math>B</math>, the four combinations being tested in separate subexperiments. The terms <math>E(a,b)</math> etc. are the quantum correlations of the particle pairs, where the quantum correlation is defined to be the expectation value of the product of the "outcomes" of the experiment, i.e. the statistical average of <math>A(a) \times B(b)</math>, where <math>A,B</math> are the separate outcomes, using the coding +1 for the '+' channel and −1 for the '−' channel. Clauser et al.'s 1969 derivation was oriented towards the use of "two-channel" detectors, and indeed it is for these that it is generally used, but under their method the only possible outcomes were +1 and −1. In order to adapt to real situations, which at the time meant the use of polarised light and single-channel polarisers, they had to interpret '−' as meaning "non-detection in the '+' channel", i.e. either '−' or nothing. They did not in the original article discuss how the two-channel inequality could be applied in real experiments with real imperfect detectors, though it was later proved that the inequality itself was equally valid. The occurrence of zero outcomes, though, means it is no longer so obvious how the values of E are to be estimated from the experimental data.
The mathematical formalism of quantum mechanics predicts that the value of <math>S</math> exceeds 2 for systems prepared in suitable entangled states and the appropriate choice of measurement settings (see below). The maximum violation predicted by quantum mechanics is <math>2 \sqrt{2}</math> (Tsirelson's bound) and can be obtained from a maximal entangled Bell state.
Experiments Many Bell tests conducted subsequent to Alain Aspect's second experiment in 1982 have used the CHSH inequality, estimating the terms using (3) and assuming fair sampling. Some dramatic violations of the inequality have been reported.
In practice most actual experiments have used light rather than the electrons that Bell originally had in mind. The property of interest is, in the best known experiments, the polarisation direction, though other properties can be used. The diagram shows a typical optical experiment. Coincidences (simultaneous detections) are recorded, the results being categorised as '++', '+−', '−+' or '−−' and corresponding counts accumulated.
Four separate subexperiments are conducted, corresponding to the four terms <math>E(a, b)</math> in the test statistic S (, above). The settings , , , and are generally in practice chosen—the "Bell test angles"—these being the ones for which the quantum mechanical formula gives the greatest violation of the inequality.
For each selected value of <math>a,b</math>, the numbers of coincidences in each category <math>\left\{ N_{++}, N_{--}, N_{+-}, N_{-+} \right\}</math> are recorded. The experimental estimate for <math>E(a, b)</math> is then calculated as: {{NumBlk||<math display="block"> E(a,b) = \frac {N_{++} - N_{+-} - N_{-+} + N_{--}} {N_{++} + N_{+-} + N_{-+}+ N_{--}}</math>|}}
Once all the 's have been estimated, an experimental estimate of S (Eq. ) can be found. If it is numerically greater than 2 it has infringed the CHSH inequality and the experiment is declared to have supported the quantum mechanics prediction and ruled out all local hidden-variable theories.
The CHSH paper lists many preconditions (or "reasonable and/or presumable assumptions") to derive the simplified theorem and formula. For example, for the method to be valid, it has to be assumed that the detected pairs are a fair sample of those emitted. In actual experiments, detectors are never 100% efficient, so that only a sample of the emitted pairs are detected. A subtle, related requirement is that the hidden variables do not influence or determine detection probability in a way that would lead to different samples at each arm of the experiment.
The CHSH inequality has been violated with photon pairs, beryllium ion pairs, ytterbium ion pairs, rubidium atom pairs, whole rubidium-atom cloud pairs, nitrogen vacancies in diamonds, and Josephson phase qubits.
Derivation The original 1969 derivation will not be given here since it is not easy to follow and involves the assumption that the outcomes are all +1 or −1, never zero. Bell's 1971 derivation is more general. He effectively assumes the "Objective Local Theory" later used by Clauser and Horne. Clauser and Horne show that the CHSH inequality can be derived from the CH74 one. As they tell us, in a two-channel experiment the CH74 single-channel test is still applicable and provides four sets of inequalities governing the probabilities p of coincidences.
Working from the inhomogeneous version of the inequality, we can write: <math display="block">- 1 \; \leq \; p_{jk}(a, b) - p_{jk}(a, b') + p_{jk}(a', b) + p_{jk}(a', b') - p_{jk}(a') - p_{jk}(b) \; \leq \; 0</math> where j and k are each '+' or '−', indicating which detectors are being considered.
To obtain the CHSH test statistic S (), all that is needed is to multiply the inequalities for which j is different from k by −1 and add these to the inequalities for which j and k are the same.
Optimal violation by a general quantum state In experimental practice, the two particles are not an ideal EPR pair. There is a necessary and sufficient condition for a two-qubit density matrix <math>\rho</math> to violate the CHSH inequality, expressed by the maximum attainable polynomial Smax defined in . This is important in entanglement-based quantum key distribution, where the secret key rate depends on the degree of measurement correlations.
Let us introduce a 3×3 real matrix <math>T_{\rho}</math> with elements <math>t_{ij} = \operatorname{Tr}[\rho\cdot(\sigma_i \otimes \sigma_j)]</math>, where <math>\sigma_1, \sigma_2, \sigma_3</math> are the Pauli matrices. Then we find the eigenvalues and eigenvectors of the real symmetric matrix <math>U_\rho = T_\rho^\text{T} T_\rho</math>, <math display=block> U_\rho \boldsymbol{e}_i = \lambda_i \boldsymbol{e}_i, \quad |\boldsymbol{e}_i| = 1, \quad i=1,2,3, </math> where the indices are sorted by <math>\lambda_1 \geq \lambda_2 \geq \lambda_3</math>. Then, the maximal CHSH polynomial is determined by the two greatest eigenvalues,
The projective measurement that yields either +1 or −1 for two orthogonal states <math>|\alpha\rangle, |\alpha^\perp\rangle</math> respectively, can be expressed by an operator <math>\Alpha = |\alpha\rangle\langle\alpha| - |\alpha^\perp\rangle\langle\alpha^\perp|</math>. The choice of this measurement basis can be parametrized by a real unit vector <math>\boldsymbol{a} \in \mathbb{R}^3, |\boldsymbol{a}|=1</math> and the Pauli vector <math>\boldsymbol{\sigma}</math> by expressing <math>\Alpha = \boldsymbol{a} \cdot \boldsymbol{\sigma}</math>. Then, the expected correlation in bases a, b is <math display=block> E(a,b) = \operatorname{Tr}[\rho(\boldsymbol{a} \cdot \boldsymbol{\sigma})\otimes(\boldsymbol{b} \cdot \boldsymbol{\sigma})] = \boldsymbol{a}^\text{T} T_\rho \boldsymbol{b}. </math> The numerical values of the basis vectors, when found, can be directly translated to the configuration of the projective measurements. states that for any quantum strategy <math>\mathcal{S}</math> for the CHSH game, the bias <math display="inline">\beta^_{\text{CHSH}}(\mathcal{S}) \leq \frac{1}{\sqrt{2}}</math>. Equivalently, it states that success probability <math display="block">\omega^_{\text{CHSH}}(\mathcal{S}) \leq \cos^2\left(\frac{\pi}{8}\right) = \frac{1}{2} + \frac{1}{2\sqrt{2}}</math> for any quantum strategy <math>\mathcal{S}</math> for the CHSH game. In particular, this implies the optimality of the quantum strategy described above for the CHSH game.
Tsirelson's inequality establishes that the maximum success probability of any quantum strategy is <math display="inline">\cos^2\left(\frac{\pi}{8}\right)</math>, and we saw that this maximum success probability is achieved by the quantum strategy described above. In fact, any quantum strategy that achieves this maximum success probability must be isomorphic (in a precise sense) to the canonical quantum strategy described above; this property is called the rigidity of the CHSH game, first attributed to Summers and Werner. More formally, we have the following result:
{{math theorem | name = Theorem (Exact CHSH rigidity) | math_statement = Let <math>\mathcal{S} = \left(|\psi\rangle, (A_0, A_1), (B_0, B_1)\right)</math> be a quantum strategy for the CHSH game where <math>|\psi\rangle \in \mathcal{A}\otimes\mathcal{B}</math> such that <math display="inline">\omega_{\text{CHSH}}(\mathcal{S}) = \cos^2\left(\frac{\pi}{8}\right)</math>. Then there exist isometries <math>V : \mathcal{A}\to\mathcal{A}_1\otimes\mathcal{A}_2</math> and <math>W : \mathcal{B}\to\mathcal{B}_1\otimes\mathcal{B}_2</math> where <math>\mathcal{A}_1,\mathcal{B}_1</math> are isomorphic to <math>\mathbb{C}^2</math> such that letting <math>|\theta\rangle = (V\otimes W)|\psi\rangle</math> we have <math display ="block"> |\theta\rangle = |\Phi\rangle_{\mathcal{A}_1, \mathcal{B}_1} \otimes |\phi\rangle_{\mathcal{A}_2,\mathcal{B}_2} </math> where <math display="inline">|\Phi\rangle = \frac{1}{\sqrt{2}}\left(|00\rangle + |11\rangle\right)</math> denotes the EPR pair and <math>|\phi\rangle_{\mathcal{A}_2,\mathcal{B}_2}</math> denotes some pure state, and <math display ="block">\begin{align} (V\otimes W)A_0|\psi\rangle = Z_{\mathcal{A}_1}|\theta\rangle, & \qquad (V\otimes W)B_0|\psi\rangle = Z_{\mathcal{B}_1}|\theta\rangle,\\ (V\otimes W)A_1|\psi\rangle = X_{\mathcal{A}_1}|\theta\rangle, & \qquad (V\otimes W)B_1|\psi\rangle = Z_{\mathcal{B}_1}|\theta\rangle. \end{align}</math>}}
Informally, the above theorem states that given an arbitrary optimal strategy for the CHSH game, there exists a local change-of-basis (given by the isometries <math>V, W</math>) for Alice and Bob such that their shared state <math>|\psi\rangle</math> factors into the tensor of an EPR pair <math>|\Phi\rangle</math> and an additional auxiliary state <math>|\phi\rangle</math>. Furthermore, Alice and Bob's observables <math>(A_0, A_1)</math> and <math>(B_0, B_1)</math> behave, up to unitary transformations, like the <math>Z</math> and <math>X</math> observables on their respective qubits from the EPR pair. An approximate or quantitative version of CHSH rigidity was obtained by McKague, et al. who proved that if you have a quantum strategy <math>\mathcal{S}</math> such that <math display="inline">\omega_{\text{CHSH}}(\mathcal{S}) = \cos^2\left(\frac{\pi}{8}\right) - \epsilon</math> for some <math>\epsilon > 0</math>, then there exist isometries under which the strategy <math>\mathcal{S}</math> is <math>O(\sqrt{\epsilon})</math>-close to the canonical quantum strategy. Representation-theoretic proofs of approximate rigidity are also known.