Bell state ($|{\Phi }^{\pm }⟩,|{\Psi }^{\pm }⟩$)

The four Bell states form an orthonormal maximally entangled basis for two qubits.

A Bell pair consists of two qubits prepared jointly in a Bell state. It is a maximally entangled two-qubit resource with Schmidt rank two. Two independent Bell pairs have resource Schmidt rank four across the same party partition.

For real $-1\le \alpha \le 1$, this two-qubit density operator has equal $00$ and $11$ populations and off-diagonal coherences $\alpha /2$. It is a pure state exactly when $\alpha =\pm 1$, and a separable quantum state exactly when $\alpha =0$. Its partial transpose has eigenvalues $1/2,1/2,\alpha /2,-\alpha /2$, directly witnessing entanglement for every nonzero admissible coherence.

A Bell-basis measurement is the four-outcome projective measurement whose projectors are onto the four Bell states. It jointly measures two qubits and is the sender's measurement in quantum teleportation.

The Bell states are joint eigenstates of commuting $Z_A{Z}_B$ and $X_A{X}_B$. Two shared Bell state meter pairs realize their two entanglement-assisted nondemolition parity measurements using spacelike local circuits. Their projector products give the four rank-one Bell state outcomes and preserve each input Bell state. The globally identified outcome requires later local operations and classical communication.

Superdense coding uses a shared Bell state to communicate two classical bits by transmitting one qubit. The sender selects one of the four Bell states with a local Pauli operation, and the receiver performs a Bell-basis measurement.

The two-spin singlet isSpin measurements along axes $\mathbf{a}$ and $\mathbf{b}$ have correlation $E(\mathbf{a},\mathbf{b})=-\mathbf{a}\cdot \mathbf{b}$, and equal axes give perfect anticorrelation.

For a bipartite density operator ${\rho }_{AB}$, the reduced state of subsystem $A$ is the partial trace ${\rho }_A={\mathrm{Tr}}_B{\rho }_{AB}$. Every measurement performed only on $A$ has outcome probabilities determined entirely by ${\rho }_A$.

A trace-preserving operation on subsystem $B$ cannot change the reduced density matrix of a disjoint subsystem $A$ when its unobserved outcomes are averaged. Local measurements on $A$ therefore cannot reveal which operation was chosen on $B$.

A bipartite density operator is a product state when it factors as ${\rho }_A\otimes {\rho }_B$. A pure product state factors as $|{\psi }_A⟩\otimes |{\psi }_B⟩$; its reduced density matrices have rank one and zero entanglement entropy. Mixed product states can have nonzero marginal Von Neumann entropy.

Write a two-qubit pure state as ${\sum }_{i,j=0}^1{c}_{ij}|ij⟩$ and form its coefficient matrix $C=(c_{ij})$. The state is a product state exactly when $C$ has rank one, equivalently when

A bipartite density operator is entangled when it is not a separable quantum state. For a pure state, this is equivalent to not being a product state, or to either reduced density matrix having rank greater than one.

For a normalized pure state of two qubits with coefficient matrix $C$, the concurrence is $2|\det C|$. It ranges from zero for a product state to one for a maximally entangled state.

The state $a|00⟩+b|11⟩$ is entangled exactly when $ab\ne 0$. Its reduced density matrix has nonzero eigenvalues $|a{|}^2$ and $|b{|}^2$.

Entanglement concentration probabilistically converts partially entangled pure states into fewer maximally entangled states by local operations and classical communication.

The Bell states $|{\Phi }^{+}⟩=(|00⟩+|11⟩)/\sqrt{2}$ and $|{\Phi }^{-}⟩=(|00⟩-|11⟩)/\sqrt{2}$ both have reduced state $I/2$ on either qubit. No measurement on one qubit alone can distinguish their relative sign.

Local operations and classical communication allow separated parties to measure or transform their own subsystems and exchange ordinary classical messages.

An exact deterministic LOCC conversion of a bipartite pure state to another is possible exactly when the input vector of squared Schmidt coefficients is majorized by the output vector. Sort the two probability vectors decreasingly and pad with zeros to a common length before applying majorization. A maximally entangled state may therefore be converted to a product state, while the reverse conversion is forbidden. The criterion does not assert conversion by postselection or with a catalyst.

Two parties can distinguish $|{\Phi }^{+}⟩$ from $|{\Phi }^{-}⟩$ by both measuring in the Hadamard basis and comparing outcomes: equal outcomes identify $|{\Phi }^{+}⟩$, while unequal outcomes identify $|{\Phi }^{-}⟩$.

Quantum teleportation transfers an unknown qubit using one shared Bell pair, a Bell-basis measurement by the sender, two classical bits, and a Pauli correction by the receiver.

Two initial maximally entangled pairs $A{B}_1$ and $B_{2}C$ can be turned into an entangled pair $AC$ by a generalized Bell basis measurement on $B_1{B}_2$, followed by a local correction at an endpoint using its classical label. This is qudit teleportation of $B_1$, including its correlations with $A$, to $C$. The initial two pairs are consumed. Without the classical label, averaging the conditional endpoint states need not leave entanglement.

A sender and receiver share $n^{-1/2}{\sum }_j|jj⟩$. The sender measures the input and their resource half in the generalized Bell basis. Outcome $(r,s)$ gives the receiver $X^s{Z}^{-r}|\chi ⟩/n$, with probability $1/n^2$. After classical communication of the outcome, the correction $Z^r{X}^{-s}$ restores the unknown quantum state. The same calculation preserves correlations with a reference, as in teleportation as an identity channel on a reference.

Ideal corrected quantum teleportation transfers an input system to the receiver while preserving its entire joint density operator with an arbitrary reference. Expand a joint pure state as ${\sum }_j|j⟩|r_j⟩$. Every teleportation branch acts with the same known correction on each system basis vector and does nothing to the reference vectors; undoing that correction restores the complete sum. Linearity extends the result to mixed states. The sender's measured carrier no longer retains the original reference correlations.

Apply a Controlled-Z gate to input $a|0⟩+b|1⟩$ and a fresh $|+⟩$ quantum ancilla. Measuring the input in the equatorial qubit measurement basis at angle $\alpha$, with outcome $s$, leaves the unnormalized outputEach branch has probability $1/2$, independently of the input. The logical output is $J(\alpha )|\psi ⟩$ with known Pauli frame $X^s$; the input qubit has been measured, so this does not clone it.

A two-vertex graph state starts with two $|+⟩$ states joined by a Controlled-Z gate. Measuring the first vertex in the $X$ basis leaves the second in $X^{s}H|+⟩=X^s|0⟩$, where $s$ is the measurement outcome. Thus a known zero input can be supplied to a measurement wire using only a graph-state resource and a known Pauli frame. This preparation link precedes the links implementing the desired logical J gate; later measurement angles and output-bit corrections absorb its byproduct.

Two angle-zero one-bit teleportations with outcomes $p,q$ implement $X^{q}H{X}^{p}H=X^q{Z}^p$ on an arbitrary input, up to global phase. A Controlled-Z gate with a fixed $|1⟩$ quantum ancilla implements $Z$, so $Z^p$ removes the known $Z$ factor. The result is $X^q$ up to phase. With probability one half the desired Pauli X gate is applied; otherwise the input is unchanged. Repeating until $q=1$ gives an exact heralded correction with two attempts on average and almost-sure termination. It has no finite worst-case measurement count. A Pauli frame avoids this repeat-until-success procedure when only logical action or classical output statistics are required.

For the printed equatorial-basis convention,It is the Hadamard gate after a phase gate. The identities $J(\alpha )Z=XJ(\alpha )$ and $J(\alpha )X=e^{i\alpha }ZJ(-\alpha )$ follow by multiplying their matrices. For $\alpha =0,\pi /2$ these are Clifford gates. In particular $J(-\pi /2)=XJ(\pi /2)$, permitting a sign change of the measurement angle to be absorbed into a Pauli frame.

The J gate is $J(\alpha )=HP(\alpha )$, where $P(\alpha )=\mathrm{diag}(1,e^{i\alpha })$: the phase gate acts first and the Hadamard gate acts second. Unitary invariance of the operator norm reduces its angle error to the single nonzero diagonal difference of the phase gate. Its norm is $|e^{i\delta }-1|=2|\sin (\delta /2)|\le |\delta |$. The quantum circuit gate-error telescoping bound then makes an angle tolerance of $ϵ/k$ sufficient for $k$ such imperfect gates and exact other gates.

For $|{\beta }_{xz}⟩=(Z^z{X}^x\otimes I)|{\beta }_{00}⟩$, an arbitrary qubit obeysA Bell-basis measurement reveals $(x,z)$, after which the receiver applies the inverse Pauli correction.

A deterministic finite-dimensional universal programmable quantum gate cannot implement every unitary exactly: programs for physically distinct unitaries would have to be mutually orthogonal, but a finite-dimensional program register contains only finitely many mutually orthogonal states.

When the shared resource is $|{\Psi }^{+}⟩$, Bell outcomes ${\Phi }^{+},{\Phi }^{-},{\Psi }^{+},{\Psi }^{-}$ leave the receiver with $X|\alpha ⟩,XZ|\alpha ⟩,|\alpha ⟩,Z|\alpha ⟩$, respectively, up to global phases.

Quantum dense coding uses one shared Bell pair and one transmitted qubit to communicate two classical bits. Alice applies one of four Pauli operators to her qubit, producing four orthogonal Bell states that Bob can distinguish jointly. The transmitted qubit alone is maximally mixed for every message and reveals no information to an interceptor.