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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]

For spacelike-separated measurement regions, a sufficient specification $\lambda$ of their relevant past should screen each outcome from additional information in the other wing. In a two-setting experiment a Bell-local model has $P(A,B\mid a,b,\lambda )=P(A\mid a,\lambda )P(B\mid b,\lambda )$. Together with measurement independence and bounded outcomes, this implies the CHSH inequality. Bell local causality is stronger than quantum no-signalling and is not an assumption that every outcome was predetermined.

Bell's theorem shows that no theory assigning outcomes through local hidden variables can reproduce every quantum correlation.

The force of the Bell theorem is that a disagreement about a local hidden-variable theory becomes a constraint on frequencies in a Bell test. Quantum entanglement alone does not guarantee a violation for every measurement in quantum mechanics: the quantum state, analyzer settings and treatment of missed events all matter. A violation of the CHSH inequality tests the conjunction of Bell local causality, measurement independence and the stated outcome and sampling assumptions; it remains compatible with quantum no-signalling.

Classical communication transmits a distinguishable classical record, such as measurement bits, between parties. In quantum teleportation it tells the receiver which local correction to apply. Quantum no-signalling prevents shared entanglement alone from transmitting the record. A classical message cannot propagate across a spacelike separation in a relativistic protocol.

A rank-one projective measurement whose Bob basis is computational when Alice is zero and rotated by $\theta$ when Alice is one can signal. Bob starts in zero. Alice zero leaves Bob zero; Alice one dephases Bob in the rotated basis, giving the displayed nonzero flip probability for $0<\theta \le \pi /4$. This violates quantum no-signalling. The conclusion applies to a measurement distinguishing the four product-basis states, rather than a degenerate observable that does not resolve them.

Outcome independence requires $P(A,B\mid a,b,\lambda )=P(A\mid a,b,\lambda )P(B\mid a,b,\lambda )$. It says that, once settings and the proposed complete state are fixed, learning the distant outcome does not change the local outcome probability. Combining it with parameter independence gives Bell factorization. Entangled quantum states generally fail outcome independence even though their observable marginal distributions obey quantum no-signalling.

Charlie cannot send a bit to Alice with the stated preparation, fixed interaction and final local measurements. Write $V={\mathrm{CNOT}}_{B\to A}$ and $W={\mathrm{CNOT}}_{B\to C}$, so the complete gate is $U=WV$ with the rightmost gate acting first. Alice has finished interacting before the gate involving Charlie acts.

For arbitrary product input density matrices, put ${\tau }_{AB}=V({\rho }_A\otimes {\rho }_B)V^{†}$. The final reduced density matrix of Alice isThe equality follows because a unitary operator on the traced-out systems leaves the partial trace unchanged. The right-hand side is independent of Charlie's preparation. Therefore every outcome probability of every local measurement by Alice is independent of Charlie's message. Quantum no-signalling applies after Alice's last interaction, even though phase kickback can allow communication between the two qubits of an individual CNOT gate.

In superdense coding, Alice applies a local Pauli operator to her half of the shared Bell state. Choose a phase bit $u$ and a parity bit $v$, and encode them using $X^v{Z}^u$ on Alice's qubit. With the standard definitions of the Bell states,Alice sends her qubit to Bob. He now holds both halves and performs a Bell-basis measurement, recovering $uv$ without error. Equivalently, a CNOT gate with Alice's qubit as control, followed by a Hadamard gate on that qubit, maps these states to their computational labels up to global phases. Four distinguishable alternatives give two classical bits per transmitted qubit, with one shared entangled pair consumed per use.

The initial entanglement distribution is independent of the later message and can be completed beforehand. Bob's local reduced density matrix before Alice's transmission is $I/2$ for every encoding, so quantum no-signalling prevents it from carrying her choice. This does not make the entangled pair free: if a qubit transmission is used to distribute it, counting that transmission as well gives two qubit transmissions for the two-bit message. The quoted dense-coding rate counts the subsequent message-bearing transmission with entanglement already shared.

For a spin singlet state, quantum spin measurements along unit vectors $\mathbf{a},\mathbf{b}$ predict $E(\mathbf{a},\mathbf{b})=-\mathbf{a}\cdot \mathbf{b}$. Take ${\mathbf{a}}_0=\mathbf{z}$, ${\mathbf{a}}_1=\mathbf{x}$, ${\mathbf{b}}_0=(\mathbf{z}+\mathbf{x})/\sqrt{2}$ and ${\mathbf{b}}_1=(\mathbf{z}-\mathbf{x})/\sqrt{2}$. These giveThe violation therefore excludes the whole conjunction of the sufficient assumptions, for any model reproducing these correlations. It does not identify which premise fails without further commitments about causal explanation, state completeness and measurement setting generation.

A defensible verdict is to retain ordinary independent setting preparation and reject Bell local causality, specifically its classical screening-off factorization. In standard quantum theory, taking the pure entangled state as the complete state preserves parameter independence: each local marginal is $1/2$, regardless of the distant setting. Yet outcome independence fails. For equal spin axes, $P(A=+,B=-)=1/2$, whereas $P(A=+)P(B=-)=1/4$. The distant outcome changes the conditional prediction even after the quantum state is specified. This failure is compatible with quantum no-signalling, since conditioning on that outcome requires learning a result which cannot itself be controlled.

Adding further past variables does not restore the entire Bell conjunction while retaining the violating correlations; the CHSH inequality already allowed arbitrary such $\lambda$. Alternatively, one may reject measurement independence, permit retrocausal setting dependence in the relevant past state, or adopt another non-Bell-local description. Those are logically possible responses, but each must explain its replacement probability and causal model rather than treating the inequality as a failed algebraic theorem. A deterministic theory is not exempt from this requirement.

The assumption of measurement independence is supported by the purpose and design of independently varied settings, but cannot be proved for every conceivable hidden variable merely by inspecting observed setting frequencies. Conversely, a rejection of Bell factorization does not alone establish a controllable superluminal signal or select a unique ontology. The reasoned verdict is against Bell's screening-off locality, conditional on the other stated premises; the unconditional conclusion concerns their conjunction.

For a bipartite state ${\rho }_{AB}$, let a local operation on A have Kraus operators $M_{\alpha }$ satisfying ${\sum }_{\alpha }{M}_{\alpha }^{†}{M}_{\alpha }=I$. Without selecting its outcome, the state becomesFor every remote observable $B$, cyclicity of the trace givesTherefore the remote partial trace is unchanged:This proves quantum no-signalling for local trace-preserving quantum channels, including choices between different such operations. It makes no assumption that ${\rho }_{AB}$ is separable.

Conditioning on one outcome instead uses an unnormalized state with only that outcome's Kraus operators and then divides by its probability. Its remote conditional state can change, yielding quantum steering. However, the outcome cannot be selected deterministically by choosing the measurement setting, and the remote observer needs an ordinary message identifying the selected ensemble. Ignoring the outcome restores the unchanged marginal. No-signalling thus concerns operationally accessible unconditional statistics, not equality of every conditional probability.

The proof presupposes that the intervention is genuinely confined to A and is trace-preserving when outcomes are ignored. A nonlocal operation or setting-dependent postselection does not meet those premises. In field theory local algebras need not come with a simple finite-dimensional tensor factorization; the corresponding statement is expressed using commuting local algebras and local operations, as below.

Locality in quantum theory is not one interchangeable condition. Spatial concentration of a wave packet concerns a state description. Quantum no-signalling restricts operational changes of remote statistics. Microcausality concerns commutation of spacelike observable algebras. The spectrum condition constrains energy-momentum and stability, while primitive causality concerns determination of future-domain expectations by initial-region data. The distinctions are clearest when all five are discussed; none should be replaced silently by the Bell factorization examined elsewhere.

Wave-function tails and entangled correlations are not by themselves controllable signals. Conversely, absence of signals does not by itself supply a local initial-value dynamics or positive-energy spectrum. Each claim must specify the observable algebras, allowed operations and relevant causal regions.

Primitive causality is an initial-region completeness requirement. A state's full restriction to a suitable region $O$ determines expectations of observables in its future domain of dependence. The domain of dependence is not the whole causal future: it contains points whose past-inextendible causal curves all meet the specified initial data region. Merely being reachable by one causal curve from $O$ is not enough.

A standard local time-slice axiom makes this precise when $O$ contains a neighbourhood of a Cauchy slice of its causal development:with the regions and algebra identifications understood in the specified local net. Definitions applying primitive causality to all bounded open regions can be stronger than a time-slice property stated only for suitable causally convex Cauchy neighbourhoods; these formulations should not be conflated without extra hypotheses.

For a local hyperbolic Klein–Gordon equation, a later field is determined by the field and its normal derivative, or canonical momentum, on a Cauchy slice. Its value can be written using the causal fundamental solution, with integration supported where the past light cone meets the initial slice. If that support lies in the initial patch, the later smeared field is generated by the patch's initial field and momentum operators. Their algebra therefore generates the observables of the patch's domain of dependence. This is the local differential-equation mechanism behind the time-slice property.

The required information is the entire restricted quantum state: it assigns expectations to every observable and product in the initial algebra. Mean values of the initial field and momentum alone do not determine their fluctuations or higher correlations and thus are insufficient. Knowledge of the restricted state fixes later probability distributions and correlations; it does not fix which individual outcome a future measurement will produce.

Finally, primitive causality concerns propagation of information already encoded in initial data, while microcausality controls compatibility of spacelike measurements and quantum no-signalling controls possible interventions. The spectrum condition adds a separate positivity restriction. A causal initial-value evolution, positive energy and no-signalling are complementary requirements, not equivalent definitions of quantum locality.

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.

Use the nonnegative representative of reduction modulo four. The four product eigenstates and eigenvalues arebecause $-2\equiv 2(\mathrm{mod}4)$. Consequently the observable is $I+Z_A{Z}_B$. It has the even sector $\mathrm{span}\{|00⟩,|11⟩\}$ with value two and the odd sector $\mathrm{span}\{|01⟩,|10⟩\}$ with value zero.

The entanglement-assisted nondemolition parity measurement needs just the single shared $|{\Phi }^{+}⟩$ provided in the question. Apply ${\mathrm{CNOT}}_{A\to d_1}$ and ${\mathrm{CNOT}}_{B\to d_2}$. For a computational-basis input $|ab⟩$, the meter becomesMeasuring the meters gives $u\oplus v=a\oplus b$. For an arbitrary coherent system input, its conditional Kraus operator isIt preserves every superposition within the measured sector, so this realizes the Lüders rule for the two degenerate eigenvalues. The answer is two for equal meter bits, zero for unequal meter bits. Each individual meter bit is uniform; only later comparison reveals the parity, respecting quantum no-signalling. This construction is independent of the defective one-pair singlet-verification request in part (a).

Interpret the requested measurement as distinguishing all four stated orthogonal eigenstates, hence as their rank-one projective measurement. If they have indistinguishable degenerate eigenvalues, this assumption need not hold; the identity observable, for example, cannot reveal the basis and gives no contradiction.

Let Bob initially prepare $|0{⟩}_B$. Alice encodes a bit by preparing $|0{⟩}_A$ or $|1{⟩}_A$, without changing Bob's initial reduced density matrix. If Alice prepares zero, the global input is the first eigenstate, so nondemolition leaves Bob in $|0⟩$ with certainty. If Alice prepares one, the rank-one Lüders rule, after discarding the outcome, dephases Bob in the rotated basisHis output is ${\rho }_B={\cos }^2\theta |u⟩⟨u|+{\sin }^2\theta |v⟩⟨v|$. A computational-basis measurement then givesIt is positive for every $0<\theta \le \pi /4$. Repeating the experiment would transmit Alice's choice across a spacelike interval, violating quantum no-signalling. The controlled-basis measurement causality obstruction therefore excludes the entire nonzero interval, including $\pi /4$.

At $\theta =0$, the basis is the computational product basis, up to an irrelevant sign on the fourth vector. Alice and Bob each measure $Z$ locally, then compare their results later. Each product eigenstate is preserved. Thus being a product basis is insufficient for an instantaneous quantum nondemolition measurement: a remote party's choice of which local basis is measured can still cause signalling.

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.

Write the system's $z$ basis as $|0⟩,|1⟩$, and use a separate pair of meter qubits in the Bell state $|{\Phi }^{+}⟩$. Alice applies a CNOT gate from system $A$ to her meter qubit; Bob simultaneously applies a CNOT gate from $B$ to his meter qubit. Flipping neither or both meter qubits preserves $|{\Phi }^{+}⟩$, while flipping exactly one gives $|{\Psi }^{+}⟩$. Hence the entanglement-assisted nondemolition parity measurement interaction produceswhere $P_{\mathrm{e}}=|00⟩⟨00|+|11⟩⟨11|$ and $P_{\mathrm{o}}=|01⟩⟨01|+|10⟩⟨10|$. Each party now measures only their meter qubit in the $z$ basis. If their binary records are $u,v$, the system Kraus operator isUnequal records verify zero total $z$ spin, since $S_z^{\mathrm{tot}}=\hslash (Z_A+Z_B)/2$ vanishes precisely on the odd sector. The probability of success is $⟨\psi |P_{\mathrm{o}}|\psi ⟩$, and the successful conditional state is $P_{\mathrm{o}}|\psi ⟩/\sqrt{⟨P_{\mathrm{o}}⟩}$. Every zero-total-$z$-spin state is left unchanged, including any coherent superposition of $|01⟩$ and $|10⟩$. Similarly the even-sector coherence is preserved. This is a quantum nondemolition measurement of the parity, rather than separate measurements of both system spins.

All quantum operations and local meter measurements can finish within the spacelike time window. Nevertheless each local meter record is individually uniform: $K_{u0}^{†}{K}_{u0}+K_{u1}^{†}{K}_{u1}=I/2$. The verification result is obtained only by comparing the records using local operations and classical communication. Thus “instantaneous” refers to the local completion of the joint measurement instrument, not instant access to its nonlocal outcome; quantum no-signalling remains intact.

At $\theta =0$ the four eigenstates are the computational product basis, with irrelevant signs on two vectors. Alice and Bob measure their own system qubits in $Z$. These local projective measurements preserve each product eigenstate; their pair of records identifies the global outcome after local operations and classical communication.

At $\theta =\pi /4$ the four states are the Bell states. They are simultaneous eigenstates of the commuting Pauli operators $Z_A{Z}_B$ and $X_A{X}_B$: ${\Phi }^{+},{\Phi }^{-},{\Psi }^{+},{\Psi }^{-}$ have respective pairs $(+,+),(+,-),(-,+),(-,-)$. Their nonlocal parity measurements can be performed without directly distinguishing the local system spins.

Use the first shared Bell state pair to perform the entanglement-assisted nondemolition parity measurement of $Z_A{Z}_B$. Use the second shared pair for $X_A{X}_B$: both parties apply a local Hadamard gate to their system qubit, execute the same local system-to-meter CNOT gates and $z$-meter measurements, then undo the Hadamard gates. This measures $X_A{X}_B$ because $HZH=X$. The two system parity projectors commute, since anticommutation at both sites cancels:Each $P_{z,x}$ is the corresponding rank-one Bell state projector. Every complete tuple of four local meter records has system Kraus operator $P_{z,x}/2$ for its two parities. Summing the four record tuples compatible with $(z,x)$ gives the ideal outcome map $\rho \mapsto P_{z,x}\rho P_{z,x}$. The protocol is a Bell-state nondemolition measurement: an input Bell state is preserved, while an arbitrary input is projected onto the reported Bell state with the Born rule probability.

The local circuits need no adaptive communication between the laboratories, so both parties can finish inside the specified time window. Global identification of $(z,x)$ still requires later local operations and classical communication. This endpoint protocol respects quantum no-signalling, unlike the hypothetical intermediate-angle instrument in part (i).

Assume the proposed device distinguishes the four displayed eigenstates, as an ideal rank-one projective measurement. This assumption matters: if all four had the same eigenvalue, the identity observable would admit that eigenbasis but its Lüders rule measurement would do nothing and could not signal. The printed eigenvectors alone do not exclude this degeneracy.

For the complete resolving measurement, let $c=\cos \theta$, $s=\sin \theta$, and ${\Pi }_j$ be the four rank-one projectors. If the nonlocal outcome is ignored, the nonselective projective measurement channel is ${\mathcal{D}}_{\theta }(\rho )={\sum }_j{\Pi }_j\rho {\Pi }_j$. Each listed state's local reduced density matrix is diagonal in $Z_A$, with $Z_A$ expectation $\cos 2\theta$ for either plus state and $-\cos 2\theta$ for either minus state. ConsequentlyCompute $C_{\theta }$ in its even and odd two-dimensional blocks: both have diagonal entries $\cos 2\theta ,-\cos 2\theta$ and off-diagonal entries $\sin 2\theta$. ThusandTo signal, prepare Alice in $|+⟩$ and Bob initially in $|+⟩$. Bob encodes a bit by either doing nothing or applying a local Pauli Z gate, which changes his state to $|-⟩$. Alice's input reduced density matrix is identical in both cases, but after the hypothetical instantaneous measurement her local $Z_A$ expectation isThe two probabilities for Alice's outcome $0$ are $(1\pm \sin 2\theta \cos 2\theta )/2$, so their difference is $\sin 2\theta \cos 2\theta$. It is strictly positive for $0<\theta <\pi /4$. Repeated trials let Alice infer Bob's bit while their operations are still spacelike, violating quantum no-signalling and relativistic causality. The relativistic causality constraint on an ideal nonlocal measurement therefore permits onlywithin the specified interval. The argument requires no rapid communication of the hypothetical nonlocal outcome: Alice reads her own changed local statistics. It rules out the full ideal instrument, not merely the later classical comparison of locally obtained records.

Use one-bit remote preparation of real qubit states. Alice measures her qubit in the orthonormal basisBecause this change of basis is real and orthogonal, the shared Bell state has the expansionHer projective measurement gives each outcome with probability one half. She sends $m=0$ for the first outcome and $m=1$ for the second. Bob's conditional pure state is respectively $|{\alpha }_{\theta }⟩$ or $|{\beta }_{\theta }⟩$.

For $m=0$ Bob does nothing. For $m=1$ he applies the fixed unitary operatorwhere $Y$ is the Pauli Y gate. Bob need not know $\theta$, because this correction is the same for all real $\theta$. Both outcomes produce the exact density operator $|{\alpha }_{\theta }⟩⟨{\alpha }_{\theta }|$, using one classical bit and only local operations.

This remote state preparation consumes the shared entangled state. Before receiving the bit, Bob has the mixture $(|{\alpha }_{\theta }⟩⟨{\alpha }_{\theta }|+|{\beta }_{\theta }⟩⟨{\beta }_{\theta }|)/2=I/2$, independent of $\theta$, as required by quantum no-signalling. The real-state restriction is what permits a fixed unitary correction of the orthogonal outcome.