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Give Alice the first qubit $A$ and Bob the second $B$. Compare the Bell state $|{\Phi }^{+}⟩$ with a preparation that tosses a fair classical bit and gives both parties its value. Their density operators, written in the computational basis ordered $00,01,10,11$, areThey have identical diagonal quantum state populations. The off-diagonal entries of the Bell state retain quantum coherence between $00$ and $11$; the classical mixture has none in this basis.

If both parties perform a Pauli measurement of $Z$, the Born rule gives $P(00)=P(11)=1/2$ and $P(01)=P(10)=0$ for both preparations. Perfect agreement here therefore cannot establish entanglement: an ordinary shared random bit already explains it.

Now both measure $X$, whose normalized eigenstates are $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$ and $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$. Changing basis givesThe Bell state again has matching outcomes, each with probability $1/2$. For either product state $|00⟩$ or $|11⟩$, each of the four $X$ outcome pairs has probability $1/4$, so their classical mixture does too. Equivalently, the Pauli correlator $⟨X\otimes X⟩$ is one for the Bell state and zero for this mixture, while $⟨Z\otimes Z⟩$ is one for both.

Taking a partial trace gives ${\rho }_A={\rho }_B=I_2/2$ for both preparations. Thus every measurement confined to Alice's system has the same statistics in the two cases, and so does every measurement confined to Bob's. They reveal the difference by comparing records through classical communication. Conditional on Alice's result, Bob can predict a matching $X$ result for the Bell state; without that result, his outcomes remain evenly split. An uncommunicated local quantum channel, including a measurement with its outcome discarded, leaves his reduced density matrix unchanged. This is quantum no-signalling, even when conditional states change. ^[ref]

The Von Neumann entropy makes the joint-versus-local distinction numerical. With base-two logarithms, the Bell state has eigenvalues $1,0,0,0$, giving $S(AB)=0$, whereas the mixture has $1/2,1/2,0,0$, giving $S(AB)=1$. Both have $S(A)=S(B)=1$. Their quantum mutual information $I(A:B)=S(A)+S(B)-S(AB)$ is consequently two and one bits, respectively. The value two is not two readable classical bits in either local qubit: the matching $Z$ records have ordinary mutual information one bit in both cases. Random records can come from a globally pure state. ^[ref]

These two matching-axis measurements distinguish the specified preparations; they alone are not a test of a Bell inequality. A local hidden-variable theory could assign shared random answers separately to these two axes. Bell theorem concerns incompatible correlations across a suitable larger collection of measurement settings. Bell's 1964 paper established the obstruction for local hidden-variable descriptions of the predicted correlations. ^[ref]

A setting-independent hidden variable fixes responses for every measurement choice, and each response depends only on the local setting. Stochastic local hidden-variable theories can be represented as mixtures of these assignments by including local random seeds in the hidden variable. Deterministic counterfactual assignments are the starting point of the chained modular Bell inequality.

The certainty of a prediction made without disturbing the system is proposed as sufficient evidence that the predicted quantity corresponds to an element of physical reality. It motivates the discussion of completeness and locality in foundations of quantum mechanics, but is distinct from the conditional factorization assumed by local hidden-variable theory.

For $|G_N^{-}⟩=(|0{⟩}^{\otimes N}-|1{⟩}^{\otimes N})/\sqrt{2}$, let $O_j$ have one Pauli X gate at site $j$ and Pauli Y gates elsewhere, and let $X_N=X^{\otimes N}$. The relations $Y|0⟩=i|1⟩$, $Y|1⟩=-i|0⟩$ show that $O_j|G_N^{-}⟩$ is proportional to $|G_N^{-}⟩$ exactly when $N$ is odd. In that case its eigenvalue is ${\lambda }_N=(-1{)}^{(N+1)/2}$, while $X_N$ always has eigenvalue $-1$.

Under the EPR criterion of reality and locality, each site's $X$ and $Y$ outcomes have predetermined values $x_j,y_j\in \{\pm 1\}$. Multiplying the $N$ certain equations $x_j{\prod }_{k\ne j}{y}_k={\lambda }_N$ gives ${\prod }_j{x}_j={\lambda }_N$ when $N$ is odd, since every $y_k$ appears an even number of times. This contradicts the certain quantum value $-1$ precisely when $N\equiv 3(\mathrm{mod}4)$. Thus the displayed GHZ state gives an infinite family of perfect-correlation contradictions with local hidden-variable theory, without an inequality.

The Schmidt decomposition and local unitary operations put any entangled two-qubit pure state intoBoth Schmidt coefficients are nonzero. Alice performs a two-outcome generalized measurement with Kraus operatorsTheir effects sum to the identity, so this is a legitimate local measurement in quantum mechanics. On success,The normalized output is the Bell state ${\Phi }_{+}$. Bob's unitary matrix $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ maps it to the spin singlet state. Alice communicates her success flag; no shared quantum operation is needed. At $\lambda =1/2$ the filter is the identity and success is certain. This is the singlet target case of optimal stochastic conversion of a two-qubit pure state.

For dichotomic local outcomes $A_x,B_y\in \{-1,1\}$, the CHSH inequality isIt assumes a local hidden-variable theory with freely chosen settings independent of the hidden-variable distribution. For the spin singlet state, direct evaluation of its Pauli matrices correlations givesChoose $a_0=e_z$, $a_1=e_x$, $b_0=(e_z+e_x)/\sqrt{2}$, and $b_1=(e_z-e_x)/\sqrt{2}$. The four correlations are $-1/\sqrt{2},-1/\sqrt{2},-1/\sqrt{2},+1/\sqrt{2}$, respectively, giving

The success flag must be recorded and communicated before the subsequent independent choice of Bell-test settings. This conditioning is legitimate because setting-independent heralding preserves Bell locality: a putative local hidden-variable theory for the entire sequential experiment would, after successful filtering, still have local response functions with a setting-independent conditional hidden-variable distribution. More explicitly, if the hidden variable is $\xi$ and $h$ is the already determined success flag, conditioning changes its distribution to $\rho (\xi \mid h=1)$ but leavesClassical messages used in preparation can be included in $\xi$; none are exchanged during the spacelike Bell measurements. Such a conditioned model still obeys the CHSH inequality, contradicting the displayed singlet correlations. Therefore no local hidden-variable theory reproduces all sequential measurement correlations of any entangled two-qubit pure state. This argument does not discard outcomes on the basis of the later measurement settings.

For each hidden variable $\lambda$, a deterministic local hidden-variable model assigns values $a_i(\lambda ),b_j(\lambda )\in \{0,\dots ,d-1\}$ to all settings. The responses depend only on the local setting; their averaging distribution is setting independent, as required by measurement independence.

Let $T(\lambda )$ be the sum of the nonnegative modular representatives in the chained modular Bell inequality. Before reduction modulo $d$, the chain telescopes:Reduction changes this integer only by multiples of $d$. Since all reduced terms are nonnegative,The smallest possible nonnegative value in this residue class is $d-1$. Averaging over the hidden variable provesAdding local random seeds to $\lambda$ gives the same result for stochastic local hidden-variable theories.

For the coplanar qubit realization of the chained Bell inequality, shareAt angle $\theta$ in the $xz$ plane, use the projective measurement of the observableHere ${\sigma }_x,{\sigma }_z$ are Pauli matrices, and the classical bit $a$ records eigenvalue $(-1{)}^a$. Since $⟨{\sigma }_z\otimes {\sigma }_z⟩=⟨{\sigma }_x\otimes {\sigma }_x⟩=1$ and both crossed expected values vanish in this Bell state,The one-party expected values vanish, so the complete joint probabilities areFor binary outcomes, $[a-b]$ is one exactly when $a\ne b$, while $[b-a-1]$ is one exactly when $a=b$. Consequently

For $N\ge 2$, set $\delta =\pi /(2N)$ and choose Alice's angle ${\theta }_i=2(i-1)\delta$ and Bob's angle ${\varphi }_i=(2i-1)\delta$. Every adjacent pair $A_i,B_i$ or $B_i,A_{i+1}$ differs by $\delta$, giving expectation ${\sin }^2(\delta /2)$. The closing pair has angle difference $(2N-1)\delta =\pi -\delta$; its agreement probability is also ${\sin }^2(\delta /2)$. Thus every one of the $2N$ terms contributes equally:Indeed $\sin x<x$ for $x>0$, so this value is below ${\pi }^2/(8N)\le {\pi }^2/16<1$. For an explicit two-setting instance, Alice measures ${\sigma }_z,{\sigma }_x$ and Bob measures $({\sigma }_z+{\sigma }_x)/\sqrt{2},(-{\sigma }_z+{\sigma }_x)/\sqrt{2}$. The four relevant expected values sum toviolating the local bound one. At $N=1$ there is no violation; the two terms always sum to one. As $N\to \infty$, the displayed quantum value is asymptotic to ${\pi }^2/(8N)$ and approaches the algebraic lower bound zero.

The violation rules out a local hidden-variable theory with setting-independent preassigned local responses reproducing these quantum correlations. It does not permit faster-than-light communication: the joint formula gives each local outcome probability $1/2$, independent of the remote setting, as required by the no-signalling theorem. The contradiction concerns the joint correlations and the Bell assumptions, not the ability to send a signal.

Consider two spacelike-separated measurement regions $R_A$ and $R_B$, with locally chosen settings $a,b\in \{0,1\}$ and recorded outcomes $A,B\in \{-1,1\}$. A relevant past region supplies a sufficient specification $\lambda$ of the common preparation and other hidden physical information needed to screen the outcomes. This is not necessarily just a quantum state written at the source, nor a conditioning on arbitrary records which already contain the future choices. The spacetime shielding and sufficiency requirements are substantive causal assumptions.

Use Bell local causality: once the relevant past is sufficiently specified, learning facts in the remote measurement region changes no local outcome probability. In particular, requirewhere conditional probabilities are defined. The probability chain rule givesThis factorization combines parameter independence with outcome independence. Neither probability is required to be zero or one: the argument includes stochastic local hidden-variable theories.

Also assume measurement independence, $\rho (\lambda \mid a,b)=\rho (\lambda )$, so all four setting pairs average over one normalized nonnegative past-variable distribution. This is a statistical condition on the setting procedure and relevant hidden information, not a proof of a metaphysical doctrine of free will. Finally the recorded outcomes must represent the same ensemble of trials. One can include nondetections in a specified bounded-outcome rule; if instead one discards events depending on setting or outcome, a further sampling assumption is required. Setting-dependent postselection is not covered silently by the argument.

Let ${\alpha }_a(\lambda )={\sum }_{A}AP(A\mid a,\lambda )$ and ${\beta }_b(\lambda )={\sum }_{B}BP(B\mid b,\lambda )$. They lie in $[-1,1]$, and the factorization givesFor each $\lambda$,Averaging with the same $\rho$ proves the CHSH inequalityNo separate assumption of simultaneous predetermined values for incompatible quantum observables was needed. The common probability model and its screening relations did the work.

A Reichenbach common cause principle can motivate the same calculation: a complete classical common cause in the past screens the two outcomes, while each setting affects only its own wing. Together with setting-independent sampling this gives the displayed factorization. It is not enough to provide a different screening variable for each pair of settings: those constructions need not form one compatible common-cause model for the four correlations.

A stochastic Einstein locality formulation instead constrains objective chances by facts in the appropriate causal past. To use it as a sufficient premise here, specify both local dependence on settings and a joint screening condition for the two outcomes. Merely requiring local marginal chances not to depend on a remote setting gives parameter independence, which is weaker than factorization. Thus the precise stochastic locality version matters; one should not infer a Bell inequality from every condition loosely described as no superluminal influence.

Let $R=|\psi ⟩⟨\psi |$ and choose an orthonormal basis $\{|e_{\mu }⟩\}$ of ${\mathcal{H}}_2$. The reduced density matrix on $S_1$ is the partial traceHere $I_1$ is the identity on ${\mathcal{H}}_1$, and the bras and kets in the sum contract only the second tensor factor. Equivalently, if $|\psi ⟩={\sum }_{a,\mu }{c}_{a\mu }|f_a⟩\otimes |e_{\mu }⟩$, then $({\rho }_1{)}_{ab}={\sum }_{\mu }{c}_{a\mu }{c}_{b\mu }^{\ast }$. The definition is independent of the chosen basis: it is characterized by $\mathrm{Tr}(M{\rho }_1)=\mathrm{Tr}[(M\otimes I_2)R]$ for every operator $M$ on ${\mathcal{H}}_1$.

The generalized measurement postulate takes $A_i$ to be operators on ${\mathcal{H}}_2$ satisfying ${\sum }_i{A}_i^{†}{A}_i=I_2$. The Born rule and the conditional state update giveIf the outcome is not supplied to $S_1$, the appropriate post-measurement density matrix is $R^{\prime }={\sum }_i(I_1\otimes A_i)R(I_1\otimes A_i^{†})$. For any local operator $M$, cyclicity of the full trace givesSince this holds for every $M$, $\boxed{{\mathrm{Tr}}_2{R}^{\prime }={\rho }_1}$. This is the no-communication theorem.

The invariance concerns the nonselective measurement in quantum mechanics. A conditional reduced density matrix ${\rho }_{1|i}={\mathrm{Tr}}_2|{\psi }_i⟩⟨{\psi }_i|$ can change, although ${\sum }_i{p}_i{\rho }_{1|i}={\rho }_1$. For example, measuring one half of a Bell pair in the computational basis changes the other half conditionally to $|0⟩$ or $|1⟩$; before learning the outcome, its density matrix remains $I/2$. Thus distant measurement in quantum measurements cannot transmit a controllable signal without classical communication. This quantum no-signalling is compatible with special relativity, even though some entangled states have correlations incompatible with a local hidden-variable theory.

The EPR criterion of reality says that if a physical quantity can be predicted with certainty without disturbing the system, then it represents an element of physical reality. For separated particles, the EPR locality assumption regards choices of measurements on the other particles as nondisturbing. The perfect correlations below therefore imply predetermined local values for both $X={\sigma }_x$ and $Y={\sigma }_y$, independent of which distant measurements are chosen.

Put $|0⟩=|\uparrow ⟩$ and $|1⟩=|\downarrow ⟩$. The Pauli matrices satisfyFor three particles, each operator with one $X$ and two $Y$ takes $|000⟩$ to $-|111⟩$ and $|111⟩$ to $-|000⟩$, whereas $X\otimes X\otimes X$ interchanges the two strings without a minus sign. Thus the GHZ state in the question has the certain outcomesEach local $X$ or $Y$ value can be predicted from measurements at the other two sites in one of these contexts. If the EPR criterion of reality and locality assign values $x_j,y_j\in \{\pm 1\}$, the first three equalities requireMultiplying these ordinary numbers and using $y_j^2=1$ yields $x_1{x}_2{x}_3=1$, contradicting the certain quantum outcome $-1$. This derives the GHZ theorem explicitly; the multiplication is of hypothetical local values, not of mutually incompatible local observables.

For general $N$, let $O_j=X_j{⨂}_{k\ne j}{Y}_k$, with the tensor factors ordered by their site labels, and let $X_N=X^{\otimes N}$. Directly,For the difference of the two strings to be an eigenstate, these phases must agree, which happens exactly when $N-1$ is even. For odd $N$ the common phase is $(-1{)}^{(N-1)/2}$, and interchanging the strings negates their difference. ConsequentlyFor even $N$, $X_N$ still has eigenvalue $-1$, but every $O_j$ takes the difference state to a phase times the sum state, which is orthogonal to it. Thus it is not an eigenstate of any $O_j$; those operators each have outcomes $\pm 1$ with equal probabilities. Among the stated $N\ge 4$, simultaneous eigenstates therefore occur precisely at $N=5,7,9,\dots$.

For odd $N$, certainty of each $O_j$ again permits remote prediction of both local $X$ and $Y$. Writing ${\lambda }_N=(-1{)}^{(N+1)/2}$, the local hidden-variable theory would require $x_j{\prod }_{k\ne j}{y}_k={\lambda }_N$ for all $j$. Their product ishence ${\prod }_j{x}_j={\lambda }_N$, since $N-1$ is even and $N$ is odd. Quantum theory requires the same product to be $-1$. The contradiction occurs when ${\lambda }_N=+1$, giving the infinite familyFor $N\equiv 1(\mathrm{mod}4)$, these particular certain product constraints are consistent; for example $x_j=-1$, $y_j=+1$ satisfies them. This distinguishes the GHZ contradiction for N congruent to 3 modulo 4 from the larger set of odd $N$ for which simultaneous eigenstates exist.

Combining the EPR criterion of reality with perfect predictive correlations and locality motivates pre-existing values for the locally selectable observables. The general local hidden-variable theory additionally assumes a complete shared variable $\lambda$ and conditional factorizationThe second condition is measurement independence. The reality criterion alone does not prove this factorization for arbitrary imperfect correlations: it motivates the local completion whose consequences are being tested.

A deterministic local hidden-variable model may be used without loss of generality by putting local random seeds into $\lambda$. For each $\lambda$, write its setting responses as $A,A^{\prime },B,B^{\prime }\in \{-1,1\}$. Integrating with the same setting-independent distribution and using the triangle inequality givesExactly one of $B-B^{\prime }$ and $B+B^{\prime }$ vanishes and the other has magnitude two. Hence the requested CHSH bound is two. No assumption about the spacelike quantum state enters this local-model derivation.

The equal mixture of $|{\Phi }^{+}⟩$ and $|{\Phi }^{-}⟩$ cancels their off-diagonal coherence, leaving the separable quantum stateA trivial LHV model chooses a shared random bit $j$, uniformly zero or one, and prepares both local qubits in $|j⟩$. Each party uses the local Born rule for its own chosen spin direction, with independent local random seeds if a deterministic description is desired. This reproduces the entire joint measurement distribution, not merely the $z$-axis correlation.

Since the state has an explicit local hidden-variable theory for these local measurements, it cannot violate the CHSH inequality, by the derivation in part (b). Classical correlation of the shared preparation bit is sufficient; the cancellation of coherence removes the entanglement.

The EPR criterion of reality says that a physical quantity has an element of reality if its value can be predicted with certainty without physically disturbing the system. The proposed criterion distinguishes what the system possesses from what one happens to measure.

A local hidden-variable theory supplements the preparation by a variable $\lambda$, drawn from a distribution $\mu (\lambda )$ independent of the later measurement settings. Its locality assumption is conditional factorization:Alice's local response does not depend on Bob's setting $y$, and Bob's does not depend on Alice's $x$. Correlations can arise from the shared past variable $\lambda$. In a deterministic local hidden-variable model, the response probabilities are point masses at functions $a=A_x(\lambda )$ and $b=B_y(\lambda )$. In particular, all possible local settings have definite values for a fixed $\lambda$, even when only one setting is used in a trial.

The setting-independence assumption is needed for comparing these predetermined values across different experiments. Conditional factorization is stronger than quantum no-signalling: no-signalling constrains observed marginal probabilities, whereas local hidden-variable theory constrains their decomposition at fixed $\lambda$. This distinction is what Bell theorem tests.

Fix a hidden variable in the deterministic local hidden-variable model, so that every $A_j,B_j$ is an integer. Let $[x{]}_d$ be the representative of $x$ modulo $d$ in $\{0,\dots ,d-1\}$. The terms in the chained modular Bell inequality alternate between the two parties. Before reduction their sum telescopes:After reduction, the sum $T$ is a nonnegative integer congruent to $-1$ modulo $d$. For $d\ge 2$, the smallest possible such integer is $d-1$. Thus $T\ge d-1$ for each hidden variable separately. Averaging over the setting-independent distribution givesHere each expectation is the expectation value of the reduced random variable, not the residue of its expectation. Each term can be measured using one setting at each site. The proof uses their common deterministic assignments rather than any joint quantum measurement of incompatible local settings. Stochastic local hidden-variable theories satisfy the same bound: include their local random seeds in $\lambda$ and average the resulting deterministic assignments.

The printed final coefficient in the definition of the average is typographically incomplete. The expectation used here is the usual $\mathbb{E}X={\sum }_{x=0}^{d-1}xP(X=x)$, with final coefficient $d-1$.

The Schmidt decomposition theorem states that a normalized vector in a finite-dimensional tensor product ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$ has the form ${\sum }_{j=1}^r{s}_j|u_j⟩|v_j⟩$, where the two families are orthonormal, $s_j>0$, ${\sum }_j{s}_j^2=1$, and $r\le \min (\dim {\mathcal{H}}_A,\dim {\mathcal{H}}_B)$. Extending each family to an orthonormal basis for two qubits givesThe nonzero Schmidt coefficients can be made positive by absorbing phases into the basis vectors. Both are positive exactly when the pure state is entangled. A product state has Schmidt rank one, so one coefficient is zero; the source's assertion of two positive coefficients for every pure state needs this exception.

Choose each local orthonormal basis independently as the computational basis. In that basis ${\sigma }_z=|0⟩⟨0|-|1⟩⟨1|$. This is a local change of coordinates, implemented by separate unitary matrices, rather than a physical restriction on the original quantum state. The associated Pauli operators supply the other two local axes. Up to an irrelevant common phase, each unitary change of qubit basis corresponds to a rotation of its Bloch sphere.

Put $s=2{\lambda }_0{\lambda }_1$. The Schmidt-basis Pauli correlation tensor is diagonal. The Pauli operators ${\sigma }_x\otimes {\sigma }_x$ exchange $|00⟩$ and $|11⟩$, while ${\sigma }_y\otimes {\sigma }_y$ do so with minus signs, and ${\sigma }_z\otimes {\sigma }_z$ leaves both fixed. HenceMixed components vanish: those containing one $z$ and one transverse Pauli operator map the occupied basis vectors outside their span, while the $xy$ and $yx$ matrix elements are purely imaginary and cancel for real Schmidt coefficients. By bilinearity, for arbitrary real vectors,For a CHSH inequality test take unit quantum measurement axesThese CHSH axes for an entangled pure two-qubit state giveThe local bound is two: for each hidden state, $A(B+B^{\prime })+A^{\prime }(B-B^{\prime })$ is $\pm 2$ when all four outcomes are $\pm 1$, and averaging cannot increase its absolute value. Thus every entangled pure two-qubit state violates a CHSH inequality, the content of Gisin's theorem. A product state has $s=0$ and does not violate it, so the unqualified final claim in the question is false for that case. The maximum $2\sqrt{2}$ occurs for equal Schmidt coefficients. This excludes local hidden-variable theories satisfying measurement independence, but does not permit faster-than-light signalling: the local reduced density matrix and its quantum measurement probabilities are unchanged by the remote choice of axis.

The Einstein–Podolsky–Rosen criterion of reality says that if, without disturbing a system, one can predict a physical quantity with certainty, then an element of physical reality corresponds to that quantity.

For two settings on each wing and outcomes in $\{-1,1\}$, the CHSH inequality for every local hidden-variable theory is

If a successful preparation flag is fixed before freely selected Bell inequality settings, conditioning a local hidden-variable theory on that flag only replaces its hidden-variable distribution by the conditional distribution. The later response functions remain local and the distribution is independent of the later settings, so the CHSH inequality still holds. Setting-dependent rejection during the Bell measurements does not satisfy this argument.