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101-455: The usual form of the CHSH inequality is where a {\displaystyle a} and a ′ {\displaystyle a'} are detector settings on side A {\displaystyle A} , b {\displaystyle b} and b ′ {\displaystyle b'} on side B {\displaystyle B} ,

202-518: A 0 ⟩ {\displaystyle |0\rangle ,|a_{0}\rangle } respectively, so precisely | ( ⟨ 0 | ⊗ ⟨ a 0 | ) | Φ ⟩ | 2 = 1 2 cos 2 ⁡ ( π 8 ) {\textstyle |(\langle 0|\otimes \langle a_{0}|)|\Phi \rangle |^{2}={\frac {1}{2}}\cos ^{2}\left({\frac {\pi }{8}}\right)} . Similarly,

303-1233: A 0 ⟩ = cos ⁡ θ | 0 ⟩ + sin ⁡ θ | 1 ⟩ , | a 1 ⟩ = − sin ⁡ θ | 0 ⟩ + cos ⁡ θ | 1 ⟩ , | b 0 ⟩ = cos ⁡ θ | 0 ⟩ − sin ⁡ θ | 1 ⟩ , | b 1 ⟩ = sin ⁡ θ | 0 ⟩ + cos ⁡ θ | 1 ⟩ , {\displaystyle {\begin{aligned}&|a_{0}\rangle =\cos \theta \,|0\rangle +\sin \theta \,|1\rangle ,\qquad |a_{1}\rangle =-\sin \theta \,|0\rangle +\cos \theta \,|1\rangle ,\\&|b_{0}\rangle =\cos \theta \,|0\rangle -\sin \theta \,|1\rangle ,\qquad |b_{1}\rangle =\sin \theta \,|0\rangle +\cos \theta \,|1\rangle ,\end{aligned}}} with θ = π / 8 {\displaystyle \theta =\pi /8} . The following table shows how

404-501: A ′ , b ′ ) + E ( a ′ , b ) ] {\displaystyle 2+\left[E\left(a',b'\right)+E\left(a',b\right)\right]} and 2 − [ E ( a ′ , b ′ ) + E ( a ′ , b ) ] {\displaystyle 2-\left[E\left(a',b'\right)+E\left(a',b\right)\right]} . That is: | E (

505-726: A ′ , λ ) B _ ( b ′ , λ ) ] ρ ( λ ) d λ + ∫ [ 1 ± A _ ( a ′ , λ ) B _ ( b , λ ) ] ρ ( λ ) d λ {\displaystyle \int \left[1\pm {\underline {A}}\left(a',\lambda \right){\underline {B}}\left(b',\lambda \right)\right]\rho (\lambda )d\lambda +\int \left[1\pm {\underline {A}}\left(a',\lambda \right){\underline {B}}(b,\lambda )\right]\rho (\lambda )d\lambda } which, using

606-403: A ′ , λ ) B _ ( b , λ ) ] ρ ( λ ) {\displaystyle \left[1\pm {\underline {A}}\left(a',\lambda \right){\underline {B}}(b,\lambda )\right]\rho (\lambda )} are both non-negative to rewrite the right-hand side of this as ∫ | A _ (

707-420: A ′ , b ) | {\displaystyle 2\;\geq \;\left|E(a,b)-E\left(a,b'\right)\right|+\left|E\left(a',b'\right)+E\left(a',b\right)\right|\;\geq \;\left|E(a,b)-E\left(a,b'\right)+E\left(a',b'\right)+E\left(a',b\right)\right|} (by the triangle inequality again), which is the CHSH inequality. In their 1974 paper, Clauser and Horne show that the CHSH inequality can be derived from

808-593: A ⊕ b = x ∧ y {\displaystyle a\oplus b=x\land y} , where ∧ denotes a logical AND operation and ⊕ denotes a logical XOR operation. If this equality holds, then Alice and Bob win, and if not then they lose. It is also required that Alice and Bob's responses can only depend on the bits they see: so Alice's response a {\displaystyle a} depends only on x {\displaystyle x} , and similarly for Bob. This means that Alice and Bob are forbidden from directly communicating with each other about

909-467: A , b ′ ) | ≤ 2 ± [ E ( a ′ , b ′ ) + E ( a ′ , b ) ] {\displaystyle \left|E(a,b)-E\left(a,b'\right)\right|\;\leq 2\;\pm \left[E\left(a',b'\right)+E\left(a',b\right)\right]} which means that the left-hand side is less than or equal to both 2 + [ E (

1010-557: A , b ′ ) + p j k ( a ′ , b ) + p j k ( a ′ , b ′ ) − p j k ( a ′ ) − p j k ( b ) ≤ 0 {\displaystyle -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} where j and k are each '+' or '−', indicating which detectors are being considered. To obtain

1111-927: A , λ ) B _ ( b ′ , λ ) | | [ 1 ± A _ ( a ′ , λ ) B _ ( b , λ ) ] ρ ( λ ) d λ | {\displaystyle \int \left|{\underline {A}}(a,\lambda ){\underline {B}}(b,\lambda )\right|\left|\left[1\pm {\underline {A}}\left(a',\lambda \right){\underline {B}}\left(b',\lambda \right)\right]\rho (\lambda )d\lambda \right|+\int \left|{\underline {A}}(a,\lambda ){\underline {B}}(b',\lambda )\right|\left|\left[1\pm {\underline {A}}\left(a',\lambda \right){\underline {B}}(b,\lambda )\right]\rho (\lambda )d\lambda \right|} By ( 4 ), this must be less than or equal to ∫ [ 1 ± A _ (

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1212-424: A , λ ) B _ ( b , λ ) | | [ 1 ± A _ ( a ′ , λ ) B _ ( b ′ , λ ) ] ρ ( λ ) d λ | + ∫ | A _ (

1313-400: A , λ ) B _ ( b , λ ) ρ ( λ ) d λ {\displaystyle E(a,b)=\int {\underline {A}}(a,\lambda ){\underline {B}}(b,\lambda )\rho (\lambda )d\lambda } where A and B are the outcomes. Since the possible values of A and B are −1, 0 and +1, it follows that: Then, if a ,

1414-631: A , b {\displaystyle a,b} with the settings x , y {\displaystyle x,y} . The tensor product Tsirelson bound is then the supremum of the value attained in this Bell expression by making measurements A x a : H A → H A {\displaystyle A_{x}^{a}:{\mathcal {H}}_{A}\to {\mathcal {H}}_{A}} and B y b : H B → H B {\displaystyle B_{y}^{b}:{\mathcal {H}}_{B}\to {\mathcal {H}}_{B}} on

1515-483: A , b ) {\displaystyle E(a,b)} is then calculated as: Once all the E 's have been estimated, an experimental estimate of S (Eq. 2 ) 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

1616-475: A , b ) − E ( a , b ′ ) | ≤ 2 − | E ( a ′ , b ′ ) + E ( a ′ , b ) | {\displaystyle \left|E(a,b)-E\left(a,b'\right)\right|\;\leq \;2-\left|E\left(a',b'\right)+E\left(a',b\right)\right|} from which we obtain 2 ≥ | E (

1717-459: A , b ) − E ( a , b ′ ) | + | E ( a ′ , b ′ ) + E ( a ′ , b ) | ≥ | E ( a , b ) − E ( a , b ′ ) + E ( a ′ , b ′ ) + E (

1818-642: A , b , x , y ; [ A x a , B y b ] = 0 {\displaystyle \forall a,b,x,y;[A_{x}^{a},B_{y}^{b}]=0} on a quantum state | ψ ⟩ ∈ H {\displaystyle |\psi \rangle \in {\mathcal {H}}} : Since tensor product algebras in particular commute, T t ≤ T c {\displaystyle T_{t}\leq T_{c}} . In finite dimensions commuting algebras are always isomorphic to (direct sums of) tensor product algebras, so only for infinite dimensions it

1919-670: A = 0° , a ′ = 45° , b = 22.5° , and b ′ = 67.5° 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 a , b {\displaystyle a,b} , the numbers of coincidences in each category { N + + , N − − , N + − , N − + } {\displaystyle \left\{N_{++},N_{--},N_{+-},N_{-+}\right\}} are recorded. The experimental estimate for E (

2020-715: A ′, b and b ′ are alternative settings for the detectors, Taking absolute values of both sides, and applying the triangle inequality to the right-hand side, we obtain We use the fact that [ 1 ± A _ ( a ′ , λ ) B _ ( b ′ , λ ) ] ρ ( λ ) {\displaystyle \left[1\pm {\underline {A}}\left(a',\lambda \right){\underline {B}}\left(b',\lambda \right)\right]\rho (\lambda )} and [ 1 ± A _ (

2121-941: A different transliteration ), the author of the article in which the first one was derived. The first Tsirelson bound was derived as an upper bound on the correlations measured in the CHSH inequality . It states that if we have four ( Hermitian ) dichotomic observables A 0 {\displaystyle A_{0}} , A 1 {\displaystyle A_{1}} , B 0 {\displaystyle B_{0}} , B 1 {\displaystyle B_{1}} (i.e., two observables for Alice and two for Bob ) with outcomes + 1 , − 1 {\displaystyle +1,-1} such that [ A i , B j ] = 0 {\displaystyle [A_{i},B_{j}]=0} for all i , j {\displaystyle i,j} , then For comparison, in

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2222-429: A distribution ρ(λ). The quantum correlation is the key statistic in the CHSH inequality and some of the other Bell inequalities, tests that open the way for experimental discrimination between quantum mechanics and local realism or local hidden-variable theory . Quantum correlations give rise to various phenomena, including interference of particles separated in time. This quantum mechanics -related article

2323-430: A pair of functions f A , f B : { 0 , 1 } ↦ { 0 , 1 } {\displaystyle f_{A},f_{B}:\{0,1\}\mapsto \{0,1\}} , where f A {\displaystyle f_{A}} is a function determining Alice's response as a function of the message she receives from Charlie, and f B {\displaystyle f_{B}}

2424-491: A proof by Antonio Acín in 2006, he realized that the one he had in mind didn't work, and issued the question as an open problem. Together with Miguel Navascués and Stefano Pironio , Antonio Acín had developed an hierarchy of semidefinite programs, the NPA hierarchy, that converged to the commuting Tsirelson bound T c {\displaystyle T_{c}} from above, and wanted to know whether it also converged to

2525-690: A quantum state | ψ ⟩ ∈ H A ⊗ H B {\displaystyle |\psi \rangle \in {\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}} : The commuting Tsirelson bound is the supremum of the value attained in this Bell expression by making measurements A x a : H → H {\displaystyle A_{x}^{a}:{\mathcal {H}}\to {\mathcal {H}}} and B y b : H → H {\displaystyle B_{y}^{b}:{\mathcal {H}}\to {\mathcal {H}}} such that ∀

2626-2066: A quantum strategy for the CHSH game where | ψ ⟩ ∈ A ⊗ B {\displaystyle |\psi \rangle \in {\mathcal {A}}\otimes {\mathcal {B}}} such that ω CHSH ( S ) = cos 2 ⁡ ( π 8 ) {\textstyle \omega _{\text{CHSH}}({\mathcal {S}})=\cos ^{2}\left({\frac {\pi }{8}}\right)} . Then there exist isometries V : A → A 1 ⊗ A 2 {\displaystyle V:{\mathcal {A}}\to {\mathcal {A}}_{1}\otimes {\mathcal {A}}_{2}} and W : B → B 1 ⊗ B 2 {\displaystyle W:{\mathcal {B}}\to {\mathcal {B}}_{1}\otimes {\mathcal {B}}_{2}} where A 1 , B 1 {\displaystyle {\mathcal {A}}_{1},{\mathcal {B}}_{1}} are isomorphic to C 2 {\displaystyle \mathbb {C} ^{2}} such that letting | θ ⟩ = ( V ⊗ W ) | ψ ⟩ {\displaystyle |\theta \rangle =(V\otimes W)|\psi \rangle } we have | θ ⟩ = | Φ ⟩ A 1 , B 1 ⊗ | ϕ ⟩ A 2 , B 2 {\displaystyle |\theta \rangle =|\Phi \rangle _{{\mathcal {A}}_{1},{\mathcal {B}}_{1}}\otimes |\phi \rangle _{{\mathcal {A}}_{2},{\mathcal {B}}_{2}}} where | Φ ⟩ = 1 2 ( | 00 ⟩ + | 11 ⟩ ) {\textstyle |\Phi \rangle ={\frac {1}{\sqrt {2}}}\left(|00\rangle +|11\rangle \right)} denotes

2727-526: A real unit vector a ∈ R 3 , | a | = 1 {\displaystyle {\boldsymbol {a}}\in \mathbb {R} ^{3},|{\boldsymbol {a}}|=1} and the Pauli vector σ {\displaystyle {\boldsymbol {\sigma }}} by expressing A = a ⋅ σ {\displaystyle \mathrm {A} ={\boldsymbol {a}}\cdot {\boldsymbol {\sigma }}} . Then,

2828-399: A single entangled qubit pair, then there exists a strategy which allows Alice and Bob to succeed with a probability of ~85%. We first establish that any deterministic classical strategy has success probability at most 75% (where the probability is taken over Charlie's uniformly random choice of x , y {\displaystyle x,y} ). By a deterministic strategy, we mean

2929-837: A strategy can be viewed as a probability distribution over deterministic strategies, and thus its success probability is a weighted sum over the success probabilities of the deterministic strategies. But since every deterministic strategy has a success probability of at most 75%, this weighted sum cannot exceed 75% either. Now, imagine that Alice and Bob share the two-qubit entangled state: | Φ ⟩ = 1 2 ( | 00 ⟩ + | 11 ⟩ ) {\textstyle |\Phi \rangle ={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )} , commonly referred to as an EPR pair . Alice and Bob will use this entangled pair in their strategy as described below. The optimality of this strategy then follows from Tsirelson's bound . Upon receiving

3030-411: 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 E ( a , b ) {\displaystyle E(a,b)} in the test statistic S ( 2 , above). The settings

3131-468: Is a stub . You can help Misplaced Pages by expanding it . Tsirelson%27s bound A Tsirelson bound is an upper limit to quantum mechanical correlations between distant events. Given that quantum mechanics violates Bell inequalities (i.e., it cannot be described by a local hidden-variable theory ), a natural question to ask is how large can the violation be. The answer is precisely the Tsirelson bound for

CHSH inequality - Misplaced Pages Continue

3232-517: Is a free parameter. We also calculate the acute angle θ = arctan ⁡ λ 2 + λ 1 tan 2 ⁡ φ λ 1 + λ 2 tan 2 ⁡ φ {\displaystyle \theta =\operatorname {arctan} {\sqrt {\frac {\lambda _{2}+\lambda _{1}\tan ^{2}{\varphi }}{\lambda _{1}+\lambda _{2}\tan ^{2}{\varphi }}}}} to obtain

3333-471: Is a function determining Bob's response based on what he receives. To prove that any deterministic strategy fails at least 25% of the time, we can simply consider all possible pairs of strategies for Alice and Bob, of which there are at most 8 (for each party, there are 4 functions { 0 , 1 } ↦ { 0 , 1 } {\displaystyle \{0,1\}\mapsto \{0,1\}} ). It can be verified that for each of those 8 strategies there

3434-719: 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: Theorem (Exact CHSH rigidity)  —  Let S = ( | ψ ⟩ , ( A 0 , A 1 ) , ( B 0 , B 1 ) ) {\displaystyle {\mathcal {S}}=\left(|\psi \rangle ,(A_{0},A_{1}),(B_{0},B_{1})\right)} be

3535-467: Is always at least one out of the four possible input pairs ( x , y ) ∈ { 0 , 1 } 2 {\displaystyle (x,y)\in \{0,1\}^{2}} which makes the strategy fail. For example, in the strategy where both players always answer 0, we have that Alice and Bob win in all cases except for when x = y = 1 {\displaystyle x=y=1} , so using this strategy their win probability

3636-458: Is another measurement basis used to communicate the secret key ( a 0 {\displaystyle {\boldsymbol {a}}_{0}} assuming Alice uses the side A). The bases a 0 , b {\displaystyle {\boldsymbol {a}}_{0},{\boldsymbol {b}}} then need to minimize the quantum bit error rate Q , which is the probability of obtaining different measurement outcomes (+1 on one particle and −1 on

3737-685: Is based on the Khalfin–Tsirelson–Landau identity. If we define an observable and A i 2 = B j 2 = I {\displaystyle A_{i}^{2}=B_{j}^{2}=\mathbb {I} } , i.e., if the observables' outcomes are + 1 , − 1 {\displaystyle +1,-1} , then If [ A 0 , A 1 ] = 0 {\displaystyle [A_{0},A_{1}]=0} or [ B 0 , B 1 ] = 0 {\displaystyle [B_{0},B_{1}]=0} , which can be regarded as

3838-476: Is customary to divide by the total number of observed coincidences. The legitimacy of this method relies on the assumption that the observed coincidences constitute a fair sample of the emitted pairs. Following local realist assumptions as in Bell's paper, the estimated quantum correlation converges after a sufficient number of trials to where a and b are detector settings and λ is the hidden variable , drawn from

3939-402: Is equal to 2 ± [ E ( a ′ , b ′ ) + E ( a ′ , b ) ] {\displaystyle 2\pm \left[E\left(a',b'\right)+E\left(a',b\right)\right]} . Putting this together with the left-hand side, we have: | E ( a , b ) − E (

4040-415: Is exactly 75%. Now, consider the case of randomized classical strategies, where Alice and Bob have access to correlated random numbers. They can be produced by jointly flipping a coin several times before the game has started and Alice and Bob are still allowed to communicate. The output they give at each round is then a function of both Charlie's message and the outcome of the corresponding coin flip. Such

4141-551: Is more general. He effectively assumes the "Objective Local Theory" later used by Clauser and Horne. It is assumed that any hidden variables associated with the detectors themselves are independent on the two sides and can be averaged out from the start. Another derivation of interest is given in Clauser and Horne's 1974 paper, in which they start from the CH74 inequality. The following is based on page 37 of Bell's Speakable and Unspeakable ,

CHSH inequality - Misplaced Pages Continue

4242-558: Is not even known to be decidable. The best known computational method for upperbounding it is a convergent hierarchy of semidefinite programs , the NPA hierarchy, that in general does not halt. The exact values are known for a few more Bell inequalities: For the Braunstein–Caves inequalities we have that For the WWŻB inequalities the Tsirelson bound is For the I 3322 {\displaystyle I_{3322}} inequality

4343-505: Is possible that T t ≠ T c {\displaystyle T_{t}\neq T_{c}} . Tsirelson's problem is the question of whether for all Bell expressions T t = T c {\displaystyle T_{t}=T_{c}} . This question was first considered by Boris Tsirelson in 1993, where he asserted without proof that T t = T c {\displaystyle T_{t}=T_{c}} . Upon being asked for

4444-726: Is strictly better than what was possible in the classical case. An arbitrary quantum strategy for the CHSH game can be modeled as a triple S = ( | ψ ⟩ , ( A 0 , A 1 ) , ( B 0 , B 1 ) ) {\displaystyle {\mathcal {S}}=\left(|\psi \rangle ,(A_{0},A_{1}),(B_{0},B_{1})\right)} where The optimal quantum strategy described above can be recast in this notation as follows: | ψ ⟩ ∈ C 2 ⊗ C 2 {\displaystyle |\psi \rangle \in \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}}

4545-456: Is the Grothendieck constant of order d . Note that since K G R ( 2 ) = 2 {\displaystyle K_{G}^{\mathbb {R} }(2)={\sqrt {2}}} , this bound implies the above result about the CHSH inequality. In general, obtaining a Tsirelson bound for a given Bell inequality is a hard problem that has to be solved on a case-by-case basis. It

4646-460: Is the expected value of the product of the alternative outcomes. In other words, it is the expected change in physical characteristics as one quantum system passes through an interaction site. In John Bell's 1964 paper that inspired the Bell test , it was assumed that the outcomes A and B could each only take one of two values, -1 or +1. It followed that the product, too, could only be -1 or +1, so that

4747-567: Is the EPR pair | ψ ⟩ = 1 2 ( | 00 ⟩ + | 11 ⟩ ) {\textstyle |\psi \rangle ={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )} , the observable A 0 = Z {\displaystyle A_{0}=Z} (corresponding to Alice measuring in the { | 0 ⟩ , | 1 ⟩ } {\displaystyle \{|0\rangle ,|1\rangle \}} basis),

4848-718: Is the difference between the winning and losing probabilities of S {\displaystyle {\mathcal {S}}} . In particular, we have β CHSH ∗ ( S ) = 1 4 ∑ x , y ∈ { 0 , 1 } ( − 1 ) x ∧ y ⋅ ⟨ ψ | A x ⊗ B y | ψ ⟩ . {\displaystyle \beta _{\text{CHSH}}^{*}({\mathcal {S}})={\frac {1}{4}}\sum _{x,y\in \{0,1\}}(-1)^{x\wedge y}\cdot \langle \psi |A_{x}\otimes B_{y}|\psi \rangle .} The bias of

4949-783: The Z {\displaystyle Z} and X {\displaystyle X} 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 S {\displaystyle {\mathcal {S}}} such that ω CHSH ( S ) = cos 2 ⁡ ( π 8 ) − ϵ {\textstyle \omega _{\text{CHSH}}({\mathcal {S}})=\cos ^{2}\left({\frac {\pi }{8}}\right)-\epsilon } for some ϵ > 0 {\displaystyle \epsilon >0} , then there exist isometries under which

5050-631: The Pauli matrices . Then we find the eigenvalues and eigenvectors of the real symmetric matrix U ρ = T ρ T T ρ {\displaystyle U_{\rho }=T_{\rho }^{\text{T}}T_{\rho }} , U ρ e i = λ i e i , | e i | = 1 , i = 1 , 2 , 3 , {\displaystyle U_{\rho }{\boldsymbol {e}}_{i}=\lambda _{i}{\boldsymbol {e}}_{i},\quad |{\boldsymbol {e}}_{i}|=1,\quad i=1,2,3,} where

5151-412: The "hidden variable" λ. λ is assumed to be drawn from a fixed distribution of possible states of the source, the probability of the source being in the state λ for any particular trial being given by the density function ρ(λ), the integral of which over the complete hidden variable space is 1. We thus assume we can write: E ( a , b ) = ∫ A _ (

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5252-477: 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

5353-401: 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: − 1 ≤ p j k ( a , b ) − p j k (

5454-418: 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 cos 2 ⁡ ( π 8 ) {\textstyle \cos ^{2}\left({\frac {\pi }{8}}\right)} , and we saw that this maximum success probability

5555-400: 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

5656-857: The CHSH inequality, expressed by the maximum attainable polynomial S max defined in Eq. 2 . 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 T ρ {\displaystyle T_{\rho }} with elements t i j = Tr ⁡ [ ρ ⋅ ( σ i ⊗ σ j ) ] {\displaystyle t_{ij}=\operatorname {Tr} [\rho \cdot (\sigma _{i}\otimes \sigma _{j})]} , where σ 1 , σ 2 , σ 3 {\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}} are

5757-428: The CHSH test statistic S ( 2 ), 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. 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 ρ {\displaystyle \rho } to violate

5858-1357: The EPR pair and | ϕ ⟩ A 2 , B 2 {\displaystyle |\phi \rangle _{{\mathcal {A}}_{2},{\mathcal {B}}_{2}}} denotes some pure state, and ( V ⊗ W ) A 0 | ψ ⟩ = Z A 1 | θ ⟩ , ( V ⊗ W ) B 0 | ψ ⟩ = Z B 1 | θ ⟩ , ( V ⊗ W ) A 1 | ψ ⟩ = X A 1 | θ ⟩ , ( V ⊗ W ) B 1 | ψ ⟩ = Z B 1 | θ ⟩ . {\displaystyle {\begin{aligned}(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{aligned}}} Informally,

5959-520: The NPA hierarchy to produce a halting algorithm to compute the Tsirelson bound, making it a computable number (note that in isolation neither procedure halts in general). Conversely, if T t {\displaystyle T_{t}} is not computable, then T t ≠ T c {\displaystyle T_{t}\neq T_{c}} . In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen claimed to have proven that T t {\displaystyle T_{t}}

6060-921: The Tsirelson bound ⟨ B ⟩ ≤ 2 2 {\displaystyle \langle {\mathcal {B}}\rangle \leq 2{\sqrt {2}}} follows. Tsirelson also showed that for any bipartite full-correlation Bell inequality with m inputs for Alice and n inputs for Bob, the ratio between the Tsirelson bound and the local bound is at most K G R ( ⌊ r ⌋ ) , {\displaystyle K_{G}^{\mathbb {R} }(\lfloor r\rfloor ),} where r = min { m , n , − 1 2 + 1 4 + 2 ( m + n ) } , {\displaystyle r=\min \left\{m,n,-{\frac {1}{2}}+{\sqrt {{\frac {1}{4}}+2(m+n)}}\right\},} and K G R ( d ) {\displaystyle K_{G}^{\mathbb {R} }(d)}

6161-435: The Tsirelson bound and nothing more. Three such principles have been found: no-advantage for non-local computation, information causality and macroscopic locality. That is to say, if one could achieve a CHSH correlation exceeding Tsirelson's bound, all such principles would be violated. Tsirelson's bound also follows if the Bell experiment admits a strongly positive quantal measure. There are two different ways of defining

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6262-413: The Tsirelson bound is not known exactly, but concrete realisations give a lower bound of 0.250 875 384 514 , and the NPA hierarchy gives an upper bound of 0.250 875 384 513 9766 . It is conjectured that only infinite-dimensional quantum states can reach the Tsirelson bound. Significant research has been dedicated to finding a physical principle that explains why quantum correlations go only up to

6363-431: The Tsirelson bound of a Bell expression. One by demanding that the measurements are in a tensor product structure, and another by demanding only that they commute. Tsirelson's problem is the question of whether these two definitions are equivalent. More formally, let be a Bell expression, where p ( a b | x y ) {\displaystyle p(ab|xy)} is the probability of obtaining outcomes

6464-890: 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 V , W {\displaystyle V,W} ) for Alice and Bob such that their shared state | ψ ⟩ {\displaystyle |\psi \rangle } factors into the tensor of an EPR pair | Φ ⟩ {\displaystyle |\Phi \rangle } and an additional auxiliary state | ϕ ⟩ {\displaystyle |\phi \rangle } . Furthermore, Alice and Bob's observables ( A 0 , A 1 ) {\displaystyle (A_{0},A_{1})} and ( B 0 , B 1 ) {\displaystyle (B_{0},B_{1})} behave, up to unitary transformations, like

6565-675: The auxiliary unit vectors c = e 1 cos ⁡ φ + e 2 sin ⁡ φ , c ′ = e 1 sin ⁡ φ − e 2 cos ⁡ φ , {\displaystyle {\begin{aligned}{\boldsymbol {c}}&={\boldsymbol {e}}_{1}\cos \varphi +{\boldsymbol {e}}_{2}\sin \varphi ,\\{\boldsymbol {c}}'&={\boldsymbol {e}}_{1}\sin \varphi -{\boldsymbol {e}}_{2}\cos \varphi ,\end{aligned}}} where φ {\displaystyle \varphi }

6666-451: The average value of the product would be where, for example, N ++ is the number of simultaneous instances ("coincidences") of the outcome +1 on both sides of the experiment. However, in actual experiments, detectors are not perfect and produce many null outcomes. The correlation can still be estimated using the sum of coincidences, since clearly zeros do not contribute to the average, but in practice, instead of dividing by N total , it

6767-413: The average win probability for a randomly chosen input is cos 2 ⁡ ( π 8 ) {\textstyle \cos ^{2}\left({\frac {\pi }{8}}\right)} . Since cos 2 ⁡ ( π 8 ) ≈ 85 % {\textstyle \cos ^{2}\left({\frac {\pi }{8}}\right)\approx 85\%} , this

6868-418: The average. We analyze the case where x = y = 0 {\displaystyle x=y=0} here: In this case the winning response pairs are a = b = 0 {\displaystyle a=b=0} and a = b = 1 {\displaystyle a=b=1} . On input x = y = 0 {\displaystyle x=y=0} , we know that Alice will measure in

6969-462: The bases above creates the constraint φ = π / 4 {\displaystyle \varphi =\pi /4} . The CHSH game is a thought experiment involving two parties separated at a great distance (far enough to preclude classical communication at the speed of light), each of whom has access to one half of an entangled two-qubit pair. Analysis of this game shows that no classical local hidden-variable theory can explain

7070-1121: The bases that maximize Eq. 2 , a = T ρ c ′ / | T ρ c ′ | , a ′ = T ρ c / | T ρ c | , b = c cos ⁡ θ + c ′ sin ⁡ θ , b ′ = c cos ⁡ θ − c ′ sin ⁡ θ . {\displaystyle {\begin{aligned}{\boldsymbol {a}}&=T_{\rho }{\boldsymbol {c}}'/|T_{\rho }{\boldsymbol {c}}'|,\\{\boldsymbol {a}}'&=T_{\rho }{\boldsymbol {c}}/|T_{\rho }{\boldsymbol {c}}|,\\{\boldsymbol {b}}&={\boldsymbol {c}}\cos \theta +{\boldsymbol {c}}'\sin \theta ,\\{\boldsymbol {b}}'&={\boldsymbol {c}}\cos \theta -{\boldsymbol {c}}'\sin \theta .\end{aligned}}} In entanglement-based quantum key distribution , there

7171-472: The basis | 0 ⟩ , | 1 ⟩ {\displaystyle |0\rangle ,|1\rangle } , and Bob will measure in the basis | a 0 ⟩ , | a 1 ⟩ {\displaystyle |a_{0}\rangle ,|a_{1}\rangle } . Then the probability that they both output 0 is the same as the probability that their measurements yield | 0 ⟩ , |

7272-473: The basis { | a 0 ⟩ , | a 1 ⟩ } {\displaystyle \{|a_{0}\rangle ,|a_{1}\rangle \}} , while if y = 1 {\displaystyle y=1} he measures in the basis { | b 0 ⟩ , | b 1 ⟩ } {\displaystyle \{|b_{0}\rangle ,|b_{1}\rangle \}} , where |

7373-568: The basis vectors, when found, can be directly translated to the configuration of the projective measurements. The optimal set of bases for the state ρ {\displaystyle \rho } is found by taking the two greatest eigenvalues λ 1 , 2 {\displaystyle \lambda _{1,2}} and the corresponding eigenvectors e 1 , 2 {\displaystyle {\boldsymbol {e}}_{1,2}} of U ρ {\displaystyle U_{\rho }} , and finding

7474-581: The bit x {\displaystyle x} from Charlie, Alice will measure her qubit in the basis { | 0 ⟩ , | 1 ⟩ } {\displaystyle \{|0\rangle ,|1\rangle \}} or in the basis { | + ⟩ , | − ⟩ } {\displaystyle \{|+\rangle ,|-\rangle \}} , conditionally on whether x = 0 {\displaystyle x=0} or x = 1 {\displaystyle x=1} , respectively. She will then label

7575-416: The classical case (or local realistic case) the upper bound is 2, whereas if any arbitrary assignment of + 1 , − 1 {\displaystyle +1,-1} is allowed, it is 4. The Tsirelson bound is attained already if Alice and Bob each makes measurements on a qubit , the simplest non-trivial quantum system. Several proofs of this bound exist, but perhaps the most enlightening one

7676-544: The classical case, it already follows that ⟨ B ⟩ ≤ 2 {\displaystyle \langle {\mathcal {B}}\rangle \leq 2} . In the quantum case, we need only notice that ‖ [ A 0 , A 1 ] ‖ ≤ 2 ‖ A 0 ‖ ‖ A 1 ‖ ≤ 2 {\displaystyle {\big \|}[A_{0},A_{1}]{\big \|}\leq 2\|A_{0}\|\|A_{1}\|\leq 2} , and

7777-463: The correlations that can result from entanglement. Since this game is indeed physically realizable, this gives strong evidence that classical physics is fundamentally incapable of explaining certain quantum phenomena, at least in a "local" fashion. In the CHSH game, there are two cooperating players, Alice and Bob, and a referee, Charlie. These agents will be abbreviated A , B , C {\displaystyle A,B,C} respectively. At

7878-554: The expected correlation in bases a, b is E ( a , b ) = Tr ⁡ [ ρ ( a ⋅ σ ) ⊗ ( b ⋅ σ ) ] = a T T ρ b . {\displaystyle 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}}.} The numerical values of

7979-437: 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 . 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

8080-811: The fact that the integral of ρ ( λ ) is 1, is equal to 2 ± [ ∫ A _ ( a ′ , λ ) B _ ( b ′ , λ ) ρ ( λ ) d λ + ∫ A _ ( a ′ , λ ) B _ ( b , λ ) ρ ( λ ) d λ ] {\displaystyle 2\pm \left[\int {\underline {A}}\left(a',\lambda \right){\underline {B}}\left(b',\lambda \right)\rho (\lambda )d\lambda +\int {\underline {A}}\left(a',\lambda \right){\underline {B}}(b,\lambda )\rho (\lambda )d\lambda \right]} which

8181-537: The four combinations being tested in separate subexperiments. The terms E ( a , b ) {\displaystyle E(a,b)} 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 A ( a ) × B ( b ) {\displaystyle A(a)\times B(b)} , where A , B {\displaystyle A,B} are

8282-479: The game is played. The states are arranged in the order that puts each state between the two most similar. They could correspond, for example, to photons polarized at angles of 0°, 22.5°, 45°, ... 180° (with 180° and 0° being the same state). To analyze the success probability, it suffices to analyze the probability that they output a winning value pair on each of the four possible inputs ( x , y ) {\displaystyle (x,y)} , and then take

8383-561: The indices are sorted by λ 1 ≥ λ 2 ≥ λ 3 {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \lambda _{3}} . Then, the maximal CHSH polynomial is determined by the two greatest eigenvalues, S max ( ρ ) = 2 λ 1 + λ 2 . {\displaystyle S_{\text{max}}(\rho )=2{\sqrt {\lambda _{1}+\lambda _{2}}}.} There exists an optimal configuration of

8484-407: The main change being to use the symbol ‘ E ’ instead of ‘ P ’ for the expected value of the quantum correlation. This avoids any suggestion that the quantum correlation is itself a probability. We start with the standard assumption of independence of the two sides, enabling us to obtain the joint probabilities of pairs of outcomes by multiplying the separate probabilities, for any selected value of

8585-876: The measurement bases a, a', b, b' for a given ρ {\displaystyle \rho } that yields S max with at least one free parameter. The projective measurement that yields either +1 or −1 for two orthogonal states | α ⟩ , | α ⊥ ⟩ {\displaystyle |\alpha \rangle ,|\alpha ^{\perp }\rangle } respectively, can be expressed by an operator A = | α ⟩ ⟨ α | − | α ⊥ ⟩ ⟨ α ⊥ | {\displaystyle \mathrm {A} =|\alpha \rangle \langle \alpha |-|\alpha ^{\perp }\rangle \langle \alpha ^{\perp }|} . The choice of this measurement basis can be parametrized by

8686-792: The observable A 1 = X {\displaystyle A_{1}=X} (corresponding to Alice measuring in the { | + ⟩ , | − ⟩ } {\displaystyle \{|+\rangle ,|-\rangle \}} basis), where X {\displaystyle X} and Z {\displaystyle Z} are Pauli matrices . The observables B 0 = 1 2 ( X + Z ) {\textstyle B_{0}={\frac {1}{\sqrt {2}}}(X+Z)} and B 1 = 1 2 ( Z − X ) {\textstyle B_{1}={\frac {1}{\sqrt {2}}}(Z-X)} (corresponding to each of Bob's choice of basis to measure in). We will denote

8787-557: The other). The corresponding bases are a 0 = T ρ e 1 / | T ρ e 1 | , b = e 1 . {\displaystyle {\begin{aligned}{\boldsymbol {a}}_{0}&=T_{\rho }{\boldsymbol {e}}_{1}/|T_{\rho }{\boldsymbol {e}}_{1}|,\\{\boldsymbol {b}}&={\boldsymbol {e}}_{1}.\end{aligned}}} The CHSH polynomial S needs to be maximized as well, which together with

8888-413: The particular Bell inequality in question. In general, this bound is lower than the bound that would be obtained if more general theories, only constrained by "no-signalling" (i.e., that they do not permit communication faster than light), were considered, and much research has been dedicated to the question of why this is the case. The Tsirelson bounds are named after Boris S. Tsirelson (or Cirel'son, in

8989-526: The probability that either of these successful outcomes happens is cos 2 ⁡ ( π 8 ) {\textstyle \cos ^{2}\left({\frac {\pi }{8}}\right)} . In the case of the 3 other possible input pairs, essentially identical analysis shows that Alice and Bob will have the same win probability of cos 2 ⁡ ( π 8 ) {\displaystyle \cos ^{2}\left({\frac {\pi }{8}}\right)} , so overall

9090-451: The probability that they both output 1 is exactly | ( ⟨ 1 | ⊗ ⟨ a 1 | ) | Φ ⟩ | 2 = 1 2 cos 2 ⁡ ( π 8 ) {\textstyle |(\langle 1|\otimes \langle a_{1}|)|\Phi \rangle |^{2}={\frac {1}{2}}\cos ^{2}\left({\frac {\pi }{8}}\right)} . So

9191-1057: The quantum strategy described above is 1 2 {\textstyle {\frac {1}{\sqrt {2}}}} . Tsirelson's inequality, discovered by Boris Tsirelson in 1980, states that for any quantum strategy S {\displaystyle {\mathcal {S}}} for the CHSH game, the bias β CHSH ∗ ( S ) ≤ 1 2 {\textstyle \beta _{\text{CHSH}}^{*}({\mathcal {S}})\leq {\frac {1}{\sqrt {2}}}} . Equivalently, it states that success probability ω CHSH ∗ ( S ) ≤ cos 2 ⁡ ( π 8 ) = 1 2 + 1 2 2 {\displaystyle \omega _{\text{CHSH}}^{*}({\mathcal {S}})\leq \cos ^{2}\left({\frac {\pi }{8}}\right)={\frac {1}{2}}+{\frac {1}{2{\sqrt {2}}}}} for any quantum strategy S {\displaystyle {\mathcal {S}}} for

9292-464: The rigidity of the CHSH game lets us test for a specific entanglement as well as specific quantum measurements. This in turn can be leveraged to test or even verify entire quantum computations—in particular, the rigidity of CHSH games has been harnessed to construct protocols for verifiable quantum delegation, certifiable randomness expansion, and device-independent cryptography. Quantum correlation In quantum mechanics , quantum correlation

9393-500: 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

9494-445: 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

9595-557: The start of the game, Charlie chooses bits x , y ∈ { 0 , 1 } {\displaystyle x,y\in \{0,1\}} uniformly at random, and then sends x {\displaystyle x} to Alice and y {\displaystyle y} to Bob. Alice and Bob must then each respond to Charlie with bits a , b ∈ { 0 , 1 } {\displaystyle a,b\in \{0,1\}} respectively. Now, once Alice and Bob send their responses back to Charlie, Charlie tests if

9696-405: The strategy S {\displaystyle {\mathcal {S}}} is O ( ϵ ) {\displaystyle O({\sqrt {\epsilon }})} -close to the canonical quantum strategy. Representation-theoretic proofs of approximate rigidity are also known. Note that the CHSH game can be viewed as a test for quantum entanglement and quantum measurements, and that

9797-686: The success probability of a strategy S {\displaystyle {\mathcal {S}}} in the CHSH game by ω CHSH ∗ ( S ) {\displaystyle \omega _{\text{CHSH}}^{*}({\mathcal {S}})} , and we define the bias of the strategy S {\displaystyle {\mathcal {S}}} as β CHSH ∗ ( S ) := 2 ω CHSH ∗ ( S ) − 1 {\displaystyle \beta _{\text{CHSH}}^{*}({\mathcal {S}}):=2\omega _{\text{CHSH}}^{*}({\mathcal {S}})-1} , which

9898-470: The tensor product Tsirelson bound T t {\displaystyle T_{t}} , the most physically relevant one. Since one can produce a converging sequencing of approximations to T t {\displaystyle T_{t}} from below by considering finite-dimensional states and observables, if T t = T c {\displaystyle T_{t}=T_{c}} , then this procedure can be combined with

9999-462: The two possible outputs resulting from each measurement choice as a = 0 {\displaystyle a=0} if the first state in the measurement basis is observed, and a = 1 {\displaystyle a=1} otherwise. Bob also uses the bit y {\displaystyle y} received from Charlie to decide which measurement to perform: if y = 0 {\displaystyle y=0} he measures in

10100-534: The value of S {\displaystyle S} 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 2 2 {\displaystyle 2{\sqrt {2}}} ( Tsirelson's bound ) and can be obtained from a maximal entangled Bell state . Many Bell tests conducted subsequent to Alain Aspect 's second experiment in 1982 have used

10201-425: The values of the bits sent to them by Charlie. However, Alice and Bob are allowed to decide on a common strategy before the game begins. In the following sections, it is shown that if Alice and Bob use only classical strategies involving their local information (and potentially some random coin tosses), it is impossible for them to win with a probability higher than 75%. However, if Alice and Bob are allowed to share

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