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Exact singlet verification accepts the spin singlet state with certainty, rejects every orthogonal state, and preserves the singlet on acceptance. Each nonzero accepting Kraus operator is then a scalar multiple of the singlet projector, whose operator Schmidt rank is four. The local Kraus rank bound from an entangled resource excludes a single shared Bell pair, which has Schmidt rank two. Two shared Bell pairs suffice by a Bell-state nondemolition measurement followed by reporting whether its label is the singlet. This protocol may disturb triplet coherence. The rank bound concerns the resource Schmidt rank, and is not an entropy lower bound of two bits for arbitrary resource states.

For $|s⟩=(|01⟩-|10⟩)/\sqrt{2}$ and any two-by-two unitary matrix, expand the two columns of $U$. The equal-index coefficients cancel, and the coefficient of $|01⟩-|10⟩$ is the determinant. Its modulus is one, so a common change of qubit basis preserves the spin singlet state up to a global phase. The density projector is exactly invariant.

Choose a shared random axis uniformly from $x,y,z$, perform its entanglement-assisted nondemolition parity measurement with one Bell pair, and accept the anticorrelated parity. The spin singlet state always passes unchanged. The averaged acceptance operator is $\Omega ={\sum }_j(I-{\sigma }_j\otimes {\sigma }_j)/6=P_s+(I-P_s)/3$, so an orthogonal triplet state passes with probability one third. This is a statistical verifier with a nonzero acceptance gap, rather than an exact single-shot singlet-versus-triplet decision. The Bell-pair cost of exact nondemolition singlet verification concerns the stronger zero-error task.

John Stewart Bell turned the debate about completing quantum mechanics into a constraint on observable correlations. In his 1964 analysis of a spin singlet state, each distant spin result depends on the local measurement direction and a shared hidden variable, but not on the other measurement direction. These assumptions constrain the joint correlations in a way incompatible with the quantum mechanics predictions. The result explains what an experiment can discriminate: it tests a class of local hidden-variable theories, rather than simply asking whether measurements are unpredictable.^[ref]

Author of the 1964 Bell theorem argument.

Only the two-qubit experiment is possible. Choose the spin singlet stateIt has eigenvalue $-1$ under each of $X\otimes X$, $Y\otimes Y$ and $Z\otimes Z$. Thus measuring either qubit in the same Pauli basis as the other always gives opposite eigenvalue labels. Each opposite-label pair occurs with probability $1/2$. More generally, collective-unitary covariance of the two-qubit singlet ensures opposite outcomes in any common orthonormal qubit basis.

For the sequential experiment, the first outcome is uniformly distributed and the Lüders rule prepares the corresponding eigenstate. Certain opposite outcomes in the computational basis force the fixed unitary matrix to have the formCertain opposite Hadamard basis outcomes require $⟨+|U|+⟩=(a+b)/2=0$, hence $b=-a$. Therefore $U=iaY$. But this operator preserves the two Pauli Y eigenstate rays: it gives the same Y-basis label on the second measurement, rather than the opposite one. This contradicts the third set of observations.

Equivalently, flipping all three bases would require $U^{†}XU=-X$, $U^{†}YU=-Y$, and $U^{†}ZU=-Z$. Conjugating $XY=iZ$ in these two ways gives both $iZ$ and $-iZ$, an impossibility. This three-axis obstruction to unitary qubit orthogonalization is a finite test of the obstruction behind the universal NOT gate.

The microscopic electric interaction is spin independent to leading order, but the Pauli exclusion principle ties the spatial symmetry of an electronic state to its spin symmetry. Different spin combinations therefore have different Coulomb and kinetic energies. Projecting those energies onto localized low-energy moments often gives an approximately rotationally invariant bilinear exchange Hamiltonian operator $-J{\mathbf{S}}_1\cdot {\mathbf{S}}_2$. For spin one half, the spin singlet state and triplet state energies are distinguished by ${\mathbf{S}}_1\cdot {\mathbf{S}}_2=-3/4$ and $1/4$, so any isotropic two-level spin singlet state–triplet state splitting can be written in this form plus a constant. More general spins or orbital degeneracy can require extra interactions.

An antiferromagnetic oxide commonly has superexchange through a ligand such as oxygen. Virtual hopping between half-filled metal orbitals lowers the energy of antiparallel spins more strongly because the corresponding intermediate doubly occupied configurations are allowed by the Pauli exclusion principle. The simple one-orbital Hubbard model limit gives $+4t^2{\mathbf{S}}_1\cdot {\mathbf{S}}_2/U$, equivalent to negative $J$ in the question's minus-sign convention. Bond angles and orbital occupancy can change the sign in more complicated oxides, so superexchange is not universally antiferromagnetic irrespective of geometry.

For quantum key distribution with Bell pairs, a source distributes one photon to Alice and its partner to Bob. One implementation is the Ekert protocol, using the spin singlet state $|{\psi }^{-}⟩=(|01⟩-|10⟩)/\sqrt{2}$. Each party independently chooses a polarization measurement setting after distribution. Equal settings give opposite outcomes; Bob reverses his outcome-to-bit assignment so these rounds give matching random bits. Other setting pairs test a Bell inequality, with ideal spin singlet state correlation $E(\mathbf{a},\mathbf{b})=-\mathbf{a}\cdot \mathbf{b}$ for the corresponding Bloch sphere axes. The optical polarizer angles correspond to twice those angles on the appropriate Bloch sphere great circle.

Over an authenticated public classical channel, Alice and Bob announce settings and detection times, select coincident detection rounds, and perform key sifting. They reveal a test sample to estimate the quantum bit error rate and the chosen Bell inequality signal, discarding all revealed bits. Excess disturbance causes an abort. Otherwise information reconciliation makes the retained strings agree and privacy amplification shortens them to remove an adversary's residual information, including information disclosed during reconciliation. The tested quantum entanglement supplies correlations unavailable to an independently prepared classical mixture; a security claim also requires stated detector and sampling assumptions.^[ref]

Each photon polarization analyzer uses a setting-dependent waveplate followed by a polarizing beam splitter. A single-photon detector on each output converts the routed photon into an electrical click, recording the outcome and its time. For example, a single-photon avalanche diode is a reverse-biased photodiode operated above breakdown: absorption creates a charge carrier, impact ionization triggers an avalanche, and the resulting pulse is registered. A quenching circuit lowers the bias to stop the avalanche and then resets the device. The detector dead time during recovery limits the count rate; dark counts produce false clicks, and imperfect detection loses rounds. The polarization information comes from the preceding analyzer, while the detector registers which output fired. Coincidence timing associates Alice's and Bob's records with the same emitted pair. ^[ref]

Use the time-ordered network ${\mathrm{CNOT}}_{1\to 2}$, ${\mathrm{CNOT}}_{2\to 1}$, ${\mathrm{CNOT}}_{1\to 2}$. On bits $(a,b)$ its successive values areThus the network exchanges the two computational-basis states. Since the gates are linear, this basis proof also establishes SWAP gate on arbitrary quantum states.

The Bell states $|{\Phi }^{\pm }⟩=(|00⟩\pm |11⟩)/\sqrt{2}$ and $|{\Psi }^{\pm }⟩=(|01⟩\pm |10⟩)/\sqrt{2}$ obeyThe antisymmetric spin singlet state alone has eigenvalue $-1$. Its standalone sign is a global phase; a Fredkin gate can make that sign observable relative to the control branch, as in part d.

For any joint lower-register state, the probability is $P_0=(1+⟨S⟩)/2$, where $S$ is its exchange operator. The Bell-state eigenvalues from part a therefore giveFor the spin singlet state, controlled exchange adds a minus sign to the control-1 branch, so the final Hadamard gate puts the ancilla in $|1⟩$. For the three symmetric states the ancilla returns to $|0⟩$.

The product-state overlap formula cannot be applied to arbitrary entangled lower inputs. All four Bell states have identical one-qubit reduced states $I/2$, yet the spin singlet state result differs maximally. The network measures exchange symmetry of the joint state, rather than fidelity of quantum states inferred from its two marginals.

Each source has density matrixAssume the two sources and the detector's random choices are independent. In the $p^2$ branch both resources are ideal. If either resource is $I_4/4$, the remote pair remains $I_4/4$ after any specified Bell-basis measurement outcome. An ideal detector and the correction of part (b) therefore yieldWhen the detector instead reports an independent uniformly random label, the corresponding four Pauli gates average the conditional pair to $I_4/4$. ThusHere $q$ is the probability of the reliable-detector branch, not its total chance of being correct: that chance is $q+(1-q)/4$.

For the usual two-setting CHSH inequality, the spin singlet state has correlation $-\mathbf{a}\cdot \mathbf{b}$ and the white-noise part has zero correlation, so $E(\mathbf{a},\mathbf{b})=-v\mathbf{a}\cdot \mathbf{b}$. With Alice's axes $z,x$ and Bob's axes $(z+x)/\sqrt{2},(z-x)/\sqrt{2}$, the absolute CHSH inequality combination is $2\sqrt{2}v$. These settings are optimal for an isotropic correlation tensor, givingThis is the threshold for the standard CHSH inequality with optimized projective polarization measurements. The unqualified phrase “the Bell inequality” needs this choice of test; entanglement alone is a weaker requirement, and no claim about every possible Bell inequality is needed for this calculation.

Use the resonant interaction picture and define $g>0$ by the one-atom Jaynes-Cummings model in part (a). Equal coupling to both atoms gives the Tavis-Cummings modelThe initial state has two excitations and is symmetric under atom exchange. Excitation conservation and this symmetry confine the evolution toThe antisymmetric atomic spin singlet state is uncoupled and unpopulated. Applying the annihilation operator and creation operator givesThe first off-diagonal entry combines two indistinguishable atomic emission paths; the second also includes the $\sqrt{2}$ enhancement for creating the second photon. Its eigenvalues are $0,\pm \sqrt{6}g$. Put $\Omega =\sqrt{6}g$ and $K=H_I/\hslash$, so $K^3={\Omega }^{2}K$. Its exponential isApplying this to $|A⟩$ yieldsAll three states have the same bare energy on resonance, so restoring the free Schrödinger-picture evolution adds only a common global phase. The expression starts at $|ee;0⟩$, has initial derivative $-i\sqrt{2}g|B⟩$, and its three squared probability amplitudes sum to one.

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.

In the two-body centre-of-mass frame, spatial reflection reverses the relative coordinate of a quark-antiquark pair. Its orbital state of angular momentum $L$ has orbital parity $(-1{)}^L$, as follows from $Y_{Lm}(-\hat{\mathbf{r}})=(-1{)}^L{Y}_{Lm}(\hat{\mathbf{r}})$. Opposite constituent intrinsic parities multiply to minus one. Thus the meson parity and charge conjugation rules begin withTo obtain charge conjugation, consider a self-conjugate flavor state and write its spin-orbital wavefunction using ordered creation operators $b_{\sigma }^{†}(\mathbf{p})d_{\tau }^{†}(-\mathbf{p})$. Charge conjugation interchanges the quark and antiquark operators. Reordering them back to quark-first order produces a fermionic sign minus one. Interchanging the two spin-$1/2$ labels has eigenvalue $-1$ in the spin singlet state ($S=0$) and $+1$ in the spin triplet ($S=1$), namely $(-1{)}^{S+1}$. Reversing the relative momentum contributes $(-1{)}^L$. ThereforeReciprocal phases for the two conjugate creation operators cancel, so the result is independent of that convention. This applies to a state that is mapped to itself by charge conjugation, not to every electrically neutral flavor state. For example a neutral meson with unequal quark and antiquark flavors can be exchanged with a different antimeson. Charged states have no individual $C$ eigenvalue.

Restrict flavor to the $u,d$ isospin doublet. Its product with the conjugate doublet decomposes as $2\otimes \overline{2}=3\oplus 1$. Convenient normalized flavor states, with irrelevant overall phase choices, areOrbital excitation normally costs energy, so the simplest low-lying quark model states have $L=0$. Coupling the two constituent spins gives $S=0$ or $S=1$, and then $J=S$. The isotriplet with $S=0$ is the pion triplet, while the isotriplet with $S=1$ is the rho meson triplet. The isosinglet with $S=1$ is the nonstrange component of the omega meson. Their quantum numbers areThe charged pions have $J^P=0^{-}$ and the charged rho mesons have $J^P=1^{-}$; charge conjugation exchanges the positive and negative members. The same tensor product also allows an $S=0$, $I=0$ pseudoscalar state, so symmetry alone does not assert that only the listed mesons exist or predict their masses. The physical isosinglet spectrum involves further dynamics and, when strange flavor is included, mixing.

For a vector-meson decay to two spinless pions, conservation of angular momentum forces their relative orbital value to be $L=1$. Their combined parity is $(-1{)}^2(-1{)}^L=-1$, and the neutral charge combination has $C=(-1{)}^L=-1$, matching both ${\rho }^0$ and $\omega$. Thus $C$ and $P$ alone do not distinguish these two decays. The decisive restriction is isospin together with Bose statistics.

Two isotriplet pions have $3\otimes 3=1\oplus 3\oplus 5$, or total $I=0,1,2$. Under exchange the isospin-$I$ state has sign $(-1{)}^{2-I}$: the $I=2$ highest state and its lowered partners are symmetric, the $I=1$ highest combination $(|+0⟩-|0+⟩)/\sqrt{2}$ is antisymmetric, and the singlet $(|+-⟩+|-+⟩-|00⟩)/\sqrt{3}$ is symmetric. The $L=1$ orbital state is antisymmetric. Since pions are spin-zero bosons, their total state must be symmetric, requiring the antisymmetric isospin sector $I=1$. This allows ${\rho }^0\to {\pi }^{+}{\pi }^{-}$ but excludes an isospin-conserving $\omega \to {\pi }^{+}{\pi }^{-}$ amplitude.

There is no analogous obstruction for three pions. In a Cartesian isospin basis $a,b,c$, their totally antisymmetric combination ${ϵ}_{abc}|{\pi }^a{\pi }^b{\pi }^c⟩$ is an $I=0$ state. It can multiply a totally antisymmetric momentum amplitude proportional in the rest frame toMomentum conservation ${\mathbf{p}}_1+{\mathbf{p}}_2+{\mathbf{p}}_3=0$ makes this amplitude change sign under every exchange of two pion labels. The product of the flavor and momentum factors therefore obeys Bose statistics. The cross product supplies $J=1$ and an even orbital parity; multiplying by the three negative pion intrinsic parities gives total $P=-1$. In the charged component, charge conjugation exchanges ${\pi }^{+}$ and ${\pi }^{-}$ and changes the momentum factor's sign, giving $C=-1$. Thus $\omega \to {\pi }^{+}{\pi }^{-}{\pi }^0$ is compatible with $I=0$ and $J^{PC}=1^{--}$.

The same distinction is encoded in G parity: $G=C(-1{)}^I$ for the neutral member gives $G_{\pi }=-1$, $G_{\rho }=+1$, $G_{\omega }=-1$. An $n$-pion state has $G=(-1{)}^n$, so the G-parity selection rule for pion multiplicities allows two pions for the rho and three for the omega. The printed prohibition of the omega two-pion mode is an exact-isospin strong-interaction selection rule. In the physical theory, isospin-breaking effects allow a small two-pion mode; it is not an absolute prohibition. The scope of C parity is also stated in the Particle Data Group quark-model review, and the physical two-pion mode is recorded in its omega listing.

Use the PDF's Bell state labels throughout this question: its ${\Phi }_{-}$ is the antisymmetric spin singlet state, whereas ${\Psi }_{\pm }$ are the even-parity states. WriteExpanding the antisymmetric combination cancels its $00$ and $11$ components and givesThe two normalized orthogonal vectors are the columns of a unitary matrix $U$. Its determinant $a_0{b}_1-a_1{b}_0$ has modulus one because $\det (U^{†}U)=|\det U{|}^2=1$. ThereforeThis proves collective-unitary covariance of the two-qubit singlet: a common basis change alters only its global phase, and leaves its density projector unchanged.

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.

Quantum teleportation transfers an arbitrary unknown qubit state to a separated receiver while using local operations and classical communication, rather than sending that input qubit. Alice and Bob first share one maximally entangled state of two qubits, such as a Bell pair; a shared spin singlet state is equivalent after a known local unitary operation.

Alice performs a joint Bell-basis measurement on the input qubit and her half of the pair. There are four possible outcomes. Bob's conditional qubit is the input state transformed by a known Pauli operator determined by that outcome. Alice sends the outcome as two classical bits, and Bob applies the inverse Pauli operator. Thus one shared Bell pair and two classical bits, together with local measurement and correction, transmit one unknown qubit exactly. The entangled resource is consumed.

The procedure does not produce a classical description of the unknown amplitudes and does not leave a second copy at Alice, so it respects the no-cloning theorem. Before receiving the classical outcome, Bob's reduced density matrix is $I/2$, independent of the input; this respects the no-communication theorem. The next part proves the full identity-channel action, including inputs entangled with another system.

Yes: deterministic entanglement swapping gives the required singlet. Denote Bob's two local qubits by $B_1,B_2$. Known local unitary operations first convert the two given spin singlet states into $|{\Phi }^{+}{⟩}_{A{B}_1}$ and $|{\Phi }^{+}{⟩}_{B_{2}C}$. For example applying $ZX$ to the second qubit of a singlet gives $|{\Phi }^{+}⟩$, where $X,Z$ are Pauli operators.

Use the binary-labelled Bell statesThey are ${\Phi }^{+},{\Phi }^{-},{\Psi }^{+},{\Psi }^{-}$ for labels $(0,0),(1,0),(0,1),(1,1)$. Reordering registers to group Bob's qubits, direct expansion givesAll four basis vectors are real in this convention; otherwise the paired expansion must account for complex conjugation. Bob performs the Bell-basis measurement on $B_1{B}_2$. This joint operation is local to him, so it is allowed by LOCC. Each outcome occurs with probability $1/4$ and leaves $AC$ in the corresponding Bell state.

Bob sends $(p,q)$ to Charlie. Charlie applies $Z^p{X}^{-q}$ to obtain $|{\Phi }^{+}{⟩}_{AC}$, and then $XZ$ to obtainThe correction works for every outcome, so no postselection or successful-outcome assumption is needed. This is entanglement swapping, equivalently teleportation of $B_1$ to $C$ while preserving its entanglement with $A$. If Bob's measurement result is ignored, the average $AC$ state is $I_4/4$; the protocol therefore does not supply entanglement usable without the classical outcome or violate the no-communication theorem.

Put $|0⟩=|{\uparrow }_z⟩$ and $|1⟩=|{\downarrow }_z⟩$. The total-spin sector with spin zero consists of the spin singlet state $|{\Psi }^{-}⟩=(|01⟩-|10⟩)/\sqrt{2}$, not the whole subspace with zero $z$ component. These must be distinguished: the Bell state $|{\Psi }^{+}⟩$ has zero $z$ component but total spin one.

There are two meanings of verification to separate here. A one-pair entanglement-assisted statistical singlet verification is possible: choose a shared random axis from $x,y,z$, measure that axis's two-spin parity by the meter circuit below, and accept anticorrelation. An $x$ or $y$ test uses local basis changes before and after the computational-axis circuit. The singlet always passes and is unchanged. Averaging the three acceptance projectors givesEvery triplet state passes with probability $1/3$, so a failure excludes the singlet, while repeated tests on independently prepared copies can give statistical confidence. A single pass does not certify the singlet with certainty.

Exact single-shot verification cannot be implemented with just one shared Bell pair. Here exact verification means a yes outcome with probability one on the spin singlet state, zero on its orthogonal complement, and preservation of the singlet on yes. If that stronger meaning is intended, the question needs an extra resource. The Bell-pair cost of exact nondemolition singlet verification gives a short proof. For a fully resolved tuple $\mu ,\nu$ of local measurement records, contraction of the shared $|{\Phi }^{+}⟩$ gives a system Kraus operatorLocal ancillas and locally adaptive operations are included in these operators. Refining any unobserved local environment gives the same form, so the operator Schmidt rank is at most two. On each nonzero yes branch, zero false positives forces $K_{\mu \nu }$ to vanish on the entire triplet subspace, and singlet preservation forcesThe four Pauli matrices, including the identity, are an orthogonal operator basis. Thus $P_s$ has operator Schmidt rank four, contradicting the rank-two bound. At least one yes branch is nonzero because the singlet must pass with certainty. Shared classical randomness cannot evade this branchwise argument.

With two shared Bell pairs, an explicit corrected protocol is available. On the first meter pair, apply local system-to-meter CNOT gates and measure both meters in the computational basis. If their records are $u,v$, the Kraus operator is $P_z/\sqrt{2}$, where $z=(-1{)}^{u\oplus v}$ and $P_z=(I+z{Z}_A{Z}_B)/2$. On the second pair, measure $X_A{X}_B$ by the same circuit conjugated with local Hadamard gates on the system. The two parities commute, and their joint projectors are the four Bell-state nondemolition measurement projectors. The singlet is exactly the outcome $(z,x)=(-1,-1)$ and remains unchanged. Local interactions and readouts fit within the stated interval; the combined verdict becomes available only after the records are compared by classical communication.

This corrected exact protocol resolves the triplet into three Bell states. It preserves the singlet and every Bell state, but generally destroys triplet superpositions. A binary Lüders rule measurement preserving all triplet coherence is a different operation: it violates the singlet-triplet measurement causality obstruction. The one-pair protocol used below measures a single parity and supplies the statistical test above; it does not give exact single-shot singlet verification.

The partial transpose of the spin singlet state projector is the Hermitian operatorFor every separable quantum state $\tau$, the positive partial transpose criterion and the trace-adjoint identity for partial transpose giveFor the specified Bell state, however,Thus $W$ is an entanglement witness detecting $|{\Phi }^{+}⟩$. In fact, it detects any state whose overlap with that Bell state exceeds $1/2$.

For the spin singlet state, measurements along the same axis are perfectly anticorrelated. By measuring either spin component of particle $B$, one can therefore predict with certainty the corresponding result for distant particle $A$. The Einstein–Podolsky–Rosen criterion of reality then says that every such spin component of $A$ is an element of physical reality, because the choice at $B$ can be made without disturbing $A$. Since the quantum state assigns no simultaneous sharp values to noncommuting spin components, the EPR argument concludes that the wavefunction is not a complete description, provided this locality premise is accepted.

Let $\lambda$ denote a proposed complete hidden state, and let $a,b$ be the chosen axes.

For two axes on each side, every $\lambda$ obeysMeasurement independence permits averaging the same distribution of $\lambda$ for all four setting pairs, producing the CHSH inequality

The singlet prediction isChoose coplanar unit vectorsThenand thereforeThus the predictions of quantum theory violate the conjunction of outcome determinism, parameter independence, and measurement independence. This is Bell theorem; the calculation alone does not select which premise a deeper theory must abandon.