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For bipartite pure source and target states with positive target entanglement entropy, the optimal rate of LOCC conversion with vanishing output error is the ratio of their entanglement entropies. Here vanishing error means that the trace distance between the output and $|\varphi ⟩⟨\varphi {|}^{\otimes m}$ tends to zero; a recorded failure probability must also tend to zero. Entanglement concentration first provides $nS({\rho }_A^{\psi })-o(n)$ Bell pairs, and entanglement dilution needs $mS({\rho }_A^{\varphi })+o(m)$ such pairs. This achieves the ratio. Conversely, refine the output into pure branches. Average monotonicity of pure-state entanglement entropy bounds their average entropy by the input entropy; continuity bound for quantum conditional entropy, with trivial conditioning, makes the output entropy loss per target tend to zero. This forbids a larger rate. The convention permits approximations for finite blocks: exact deterministic conversion of every finite block can obey stricter majorization constraints. The reversible concentration construction is given in the original concentration paper.

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.

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.

Exact quantum teleportation that preserves an arbitrary reference requires at least two classical bits from sender to receiver, even with preshared quantum entanglement. For the essential reduction, encode two classical bits in the four Bell states by Pauli gates on the sender's half of a Bell pair. Teleportation as an identity channel on a reference transfers this half while preserving the encoding. The receiver then has both halves and recovers both bits by a Bell-basis measurement, as in superdense coding.

The classical bit bound itself has a short no-communication theorem proof. For $N$ possible messages and $K$ possible classical transmissions, let ${\sigma }_{m,c}$ be the receiver's subnormalized conditional density matrix before decoding. The no-communication theorem gives ${\sum }_c{\sigma }_{m,c}=\tau$, independent of $m$, so $0\le {\sigma }_{m,c}\le \tau$. For each transmission $c$, let $\{E_{m|c}{\}}_m$ be the decoding POVM. Averaging over equiprobable messages,Perfect decoding of four messages requires $K\ge 4$, hence at least two classical bits. This applies to the preshared resources of the proposed quantum teleportation protocol as well as the Bell pair used for the encoding.

For a Bell pair $|{\varphi }_{+}⟩=(|00⟩+|11⟩)/\sqrt{2}$, Alice's basic measurement in the entrywise complex conjugate of any desired Bob orthonormal basis prepares each Bob pure state with probability $1/2$. Indeed, if $|u⟩={\sum }_{j=0}^1{u}_j|j⟩$, then $⟨\overline{u}|j⟩=u_j$, which proves the displayed contraction identity. The undisclosed-outcome density matrix stays $I/2$; quantum steering changes the conditional quantum state ensemble without violating the no-communication theorem.

Let $|{\Phi }_d⟩=d^{-1/2}{\sum }_{j=0}^{d-1}|j,j⟩$ and $1\le m\le d$. It can be converted deterministically by LOCC to $|{\Phi }_m⟩$. For each $m$-element subset $S$ of the local basis labels, let ${\Pi }_S$ be its projector and let Alice's Kraus operator be $K_S={\Pi }_S/\sqrt{\left(\genfrac{}{}{0ex}{}{d-1}{m-1}\right)}$. Every label occurs in $\left(\genfrac{}{}{0ex}{}{d-1}{m-1}\right)$ subsets, so ${\sum }_S{K}_S^{†}{K}_S=I$. Each outcome has probability $m/[d(\genfrac{}{}{0ex}{}{d-1}{m-1})]=1/\left(\genfrac{}{}{0ex}{}{d}{m}\right)$ and leaves the maximally entangled state $m^{-1/2}{\sum }_{j\in S}|j,j⟩$. Alice reports $S$, and both parties relabel these $m$ basis states to $0,\dots ,m-1$ by local unitary operations.

For $d=6$, $m=4$, only three outcomes are needed: partition the six labels into disjoint pairs $T_0,T_1,T_2$ and use $K_r=(I-{\Pi }_{T_r})/\sqrt{2}$. Each label is retained twice, proving completeness, and each outcome has probability $1/3$. Every branch leaves four equal Schmidt coefficients, hence two independent Bell pairs after binary relabelling.

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.

Entanglement dilution uses local operations and classical communication to prepare partially entangled states from maximally entangled states, such as Bell pairs. For many identical bipartite pure targets, the asymptotic cost in Bell pairs per target, allowing a vanishing approximation error, is their entanglement entropy in bits. The quantum typical subspace of the target's Schmidt coefficients has almost all its weight and dimension at most $2^{m(S+\delta )}$. A maximally entangled state of this dimension can be converted to the normalized truncated target by Nielsen's pure-state conversion theorem. The resulting state approaches the full target as the retained weight tends to one. Average monotonicity of pure-state entanglement entropy and entropy continuity give the converse lower bound. This is the reverse asymptotic task to entanglement concentration.

Entanglement distillation uses local operations and classical communication on many noisy entangled states to produce fewer high-fidelity Bell pairs. It is also called entanglement purification. The protocol may fail or abort; naming a purification step does not guarantee successful output for arbitrary inputs. If the output is an exact pure state of Bell pairs, every purification of a density operator factors between those pairs and the environment, so measuring each pair in the computational basis gives a shared uniform bit independent of an eavesdropper. Approximate privacy follows when the entire retained output is sufficiently close to that pure target. A separable quantum state cannot be distilled into a Bell pair by LOCC, because each local branch preserves separability.

A shared resource enables deterministic exact transmission of an arbitrary unknown qubit using only local operations and classical communication if and only if it can produce a shared Bell pair deterministically by such operations. Sufficiency follows from quantum teleportation. For necessity, prepare a Bell pair locally at the sender and apply the purported transmission protocol to one half: exact identity-channel action leaves a shared Bell pair between the retained reference and the receiver. A separable shared resource cannot satisfy this condition because local operations and classical communication preserve separability.

Share a GHZ state on $A,B,C$ and let the unknown input $Q$ be $|\chi ⟩$. Alice applies CNOT from $Q$ to $A$, then a Hadamard gate to $Q$, and measures $Q,A$ computationally, giving $s,t$. Charlie measures $C$ in the Hadamard basis, giving $c$. Bob's conditional state isup to a global phase. Alice's two bits and Charlie's one bit therefore specify Bob's inverse correction $Z^{s\oplus c}{X}^t$.

To prove the phase rule, write $|\chi ⟩=\alpha |0⟩+\beta |1⟩$. After Alice's result, $BC$ is $\alpha |tt⟩+(-1{)}^s\beta |(1\oplus t)(1\oplus t)⟩$. Taking Charlie's Hadamard basis partial inner product multiplies its second term relative to its first by $(-1{)}^c$. The remaining common phase is $(-1{)}^{ct}$, which is physically irrelevant. All eight result triples have probability $1/8$. Charlie's record localizes a known-phase Bell pair, allowing the standard Bell-basis teleportation identity to complete the transfer.

Suppose Alice holds a classical orthogonal flag $F$ for the phase of a Bell state:Her local controlled phase correction $U=I_{AB}\otimes |0⟩⟨0{|}_F+(Z_A\otimes I_B)\otimes |1⟩⟨1{|}_F$ maps this to $|{\Phi }^{+}⟩⟨{\Phi }^{+}{|}_{AB}\otimes [p|0⟩⟨0|+(1-p)|1⟩⟨1|{]}_F$. Thus Alice and Bob recover an exact Bell pair without revealing its bit value. Even if Eve knows the flag, purity decouples a subsystem from its purification, so the corrected pair can supply a private bit. Discarding the flag first can instead yield a separable quantum state, as happens at $p=1/2$.

Charlie's Hadamard basis measurement, Alice's Bell-basis measurement and communicated correction bits give deterministic GHZ-assisted quantum teleportation. Let the unknown input be $|\chi {⟩}_Q=\alpha |0⟩+\beta |1⟩$. Charlie measures $C$ in the Hadamard basis and labels $+$ by $c=0$, $-$ by $c=1$. The identityshows that $AB$ is conditionally a Bell pair with phase label $c$. Charlie sends that one classical bit to Bob. The procedure for localizing a Bell pair from a GHZ state supplies precisely the missing conditional information; forgetting $c$ leaves the mixed reduced state of the preceding part.

Alice performs the usual Bell-basis measurement on $QA$. Operationally, she applies ${\mathrm{CNOT}}_{Q\to A}$, applies $H_Q$, and measures $Q$ and $A$ in the computational basis, obtaining bits $s,t$. She sends both bits to Bob. To track the correction without a convention ambiguity, conditional on $s,t$ the normalized state on $BC$ isProjecting Charlie's factor onto his recorded Hadamard basis state gives Bob, up to the global phase $(-1{)}^{ct}$,Consequently the inverse Pauli correction isBob applies $X^t$ first and $Z^{s\oplus c}$ second. The measurements on Charlie and on Alice commute because they act on different systems, so deriving the conditional state with Alice's measurement first gives the same protocol as Charlie measuring first. Each triple $(s,t,c)$ has probability $1/8$, independent of the unknown input, and every triple succeeds after correction.

This uses one previously shared GHZ state, two classical bits from Alice and one from Charlie, with no quantum transmission during the transfer. The standard Bell-measurement/Pauli-correction mechanism was established in the original quantum teleportation proposal.^[ref] Here Charlie's record supplies the Bell-pair phase, giving a controlled quantum teleportation protocol. Alice's input is consumed by her measurement; Bob reconstructs its state without learning $\alpha ,\beta$ and without making a second surviving copy.

Combine the hypothetical protocol with superdense coding. Before a message is chosen, Alice and Bob share a further Bell pair $|{\varphi }_{+}{⟩}_{QR}$, with $Q$ at Alice and $R$ at Bob, in addition to any resources needed for the hypothetical quantum teleportation protocol. To encode two classical bits $u,v$, Alice applies the Pauli gates $Z_Q^u{X}_Q^v$, preparingThe four choices are the four mutually orthogonal Bell states. Alice now uses the hypothetical protocol to teleport $Q$ to Bob using only one classical bit. Its assumed preservation of the joint quantum state gives Bob both $B$ and $R$ inBob performs a Bell-basis measurement. Explicitly, a controlled-NOT gate $B\to R$, followed by a Hadamard gate on $B$, givesA computational-basis measurement therefore recovers both message bits with certainty after only one transmitted classical bit. The initial Bell pairs are shared before the message is chosen and carry no message themselves. This proves the classical communication lower bound for exact qubit teleportation: a perfect protocol preserving an external reference needs at least two transmitted classical bits.

Alice holds the input qubit $Q$ in $|\psi {⟩}_Q=a|0⟩+b|1⟩$ and her half $A$ of a shared Bell pair $|{\varphi }_{+}{⟩}_{AB}$; Bob holds $B$. Alice applies a controlled-NOT gate $Q\to A$, then a Hadamard gate on $Q$. In the system order $Q,A,B$, the initial quantum state isAfter these two quantum gates, direct expansion yields the quantum teleportation identityAlice performs computational-basis measurements on $Q,A$, and sends the two classical bits $m,n$ to Bob. Each outcome has probability $1/4$, independently of the input pure state. Bob applies the indicated Pauli gates: first $X^n$, then $Z^m$, so his total correction is $Z^m{X}^n$. In every branch,This proves exact quantum teleportation using one Bell pair and two classical bits. Alice's original qubit has been measured; the protocol transfers the quantum state without leaving her another copy.

Let the unknown input qubit be $|\psi {⟩}_S=\alpha |0⟩+\beta |1⟩$, with $|\alpha {|}^2+|\beta {|}^2=1$. Alice holds $S,A$, Bob holds $B$, and the shared Bell pair is $|{\varphi }^{+}{⟩}_{AB}$. All three-qubit strings below use the order $S,A,B$. Initially,The Bell-basis teleportation identity givesThus a perfect Bell state analyzer on Alice's two qubits identifies the Pauli operator affecting Bob's quantum state.

To display the intermediate three-qubit states explicitly, realize that Bell-basis measurement by a controlled-NOT gate $S\to A$, a Hadamard gate on $S$, and two computational-basis measurements. After the controlled-NOT gate,After the Hadamard gate,Measuring $S,A$ gives $(m,n)$ with probability $1/4$ for each outcome, independently of $\alpha ,\beta$. Conditional on that result, the normalized three-qubit post-measurement state isIf the Bell state analyzer is instead regarded as an ideal direct projective measurement preserving the measured qubits, replace $|mn{⟩}_{SA}$ here by the corresponding Bell state; Bob's factor is identical. A destructive optical analyzer records the same label but absorbs Alice's photons.

Alice sends the two classical bits $(m,n)$ to Bob. During transmission the above conditional three-qubit state is unchanged. Bob then applies the inverse Pauli operator $Z^m{X}^n$:The final conditional three-qubit state is therefore $|mn{⟩}_{SA}\otimes |\psi {⟩}_B$. The protocol transmits two classical bits per teleported qubit, consuming one shared Bell pair. Without Alice's record Bob averages over four Pauli operators and has density matrix $I/2$, so quantum teleportation respects the no-communication theorem. Alice's input has been measured and is no longer an independent copy, consistent with the no-cloning theorem.

In quantum dense coding, a shared Bell pair lets Alice send two classical bits by transmitting one qubit. Use photon polarization as the computational basis, $|0⟩=|H⟩$, $|1⟩=|V⟩$, and define the Bell statesAlice and Bob initially share $|{\varphi }^{+}{⟩}_{AB}$. Alice encodes a two-bit message $(m,n)$ by applying the Pauli operator $X^n{Z}^m$ to her qubit:The last minus sign is a global phase. Alice transmits her photon to Bob, who now holds both qubits and uses a Bell-basis measurement to identify the message. The four Bell states are orthogonal, so an ideal Bell state analyzer decodes them without error. The resource cost is one previously shared Bell pair and one transmitted qubit per two classical bits; the shared quantum entanglement is consumed. Before transmission Bob's reduced density matrix is $I/2$ for every message, consistent with the no-communication theorem.

The BBO crystal supplies the polarization-entangled photon pair. An ultraviolet laser pumps this nonlinear optical crystal; spontaneous parametric down-conversion produces signal and idler photons with ${\omega }_p={\omega }_s+{\omega }_i$ and phase matching ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. In a noncollinear type-II phase matching arrangement, collect the intersections of the two emission cones. The alternatives $|H{⟩}_A|V{⟩}_B$ and $|V{⟩}_A|H{⟩}_B$ must be indistinguishable in arrival time, spectrum and spatial mode. Compensation of birefringence and adjustment of the relative quantum phase then give $(|HV⟩+|VH⟩)/\sqrt{2}$; a local Pauli X gate converts it to $|{\varphi }^{+}⟩$. Half-wave plates and other waveplates implement Alice's Pauli gates. The BBO is the pair source; the message is encoded subsequently by the local optical elements.^[ref]

A basic passive linear optics Bell state analyzer with no extra resources identifies only two of the four polarization Bell states unambiguously. Thus the ideal two-bit capacity presupposes complete Bell-basis measurement; that basic optical implementation alone does not attain it deterministically.

In entanglement swapping, start with independent Bell pairs $|{\varphi }^{+}{⟩}_{12}|{\varphi }^{+}{⟩}_{34}$. Send qubits 2 and 3 to a central Bell state analyzer, retaining 1 and 4 at separate endpoints. Reordering the tensor product factors to group 23 and 14 gives the identityConsequently each ideal Bell-basis measurement outcome has probability $1/4$ and projects 14 onto the corresponding Bell state, even though those two photons have never interacted. Communicating the two-bit result permits an endpoint Pauli gate to convert every outcome into the same chosen Bell pair. Without that record, the endpoints have ${\rho }_{14}=I_4/4$, a product state, rather than a known entangled state.

The optical Bell state analyzer overlaps photons 2 and 3 at a balanced beam splitter, with matched spatial modes, spectra and arrival times, and resolves the output photon polarizations using polarizing beam splitters and single-photon detectors. Two-photon interference maps Bell states to different output patterns. The antisymmetric $|{\psi }^{-}⟩$ gives one photon in each output with opposite polarizations. The symmetric $|{\psi }^{+}⟩$ gives two opposite-polarization photons in the same output; the two polarization detectors there can resolve this pattern. The $|{\varphi }^{\pm }⟩$ states give same-polarization pairs and remain mutually ambiguous in this basic setup. The detected pattern therefore heralds a particular endpoint Bell state. With ideal detection this arrangement succeeds unambiguously on two of four equally likely inputs, with probability $1/2$; the successful events can be selected. A complete analyzer would accept all four outcomes. The endpoint quantum entanglement is conditional on the measurement record, so it enables no instantaneous signalling.^[ref]

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]

A realized message $x$ has self-information $I(x)=-{\log }_2{p}_x$. The average information per source message is its Shannon entropy,The average reaches ${\log }_{2}3\simeq 1.58496$ bits per message at $p_A=p_B=p_C=1/3$. Individual rare realizations can have larger self-information; this bound concerns the average.

The given strings form a prefix code with lengths ${\ell }_A=1$ and ${\ell }_B={\ell }_C=2$. To attain the entropy bound without redundancy, the resulting long bit stream must carry one bit of information per bit: unbiased bits with no predictable dependence. For an independent source, the dyadic source matched to a complete prefix code hasThese probabilities make the encoded stream equivalent to parsing independent fair bits using the code tree. Its expected codeword length and source entropy agree:Here “optimal” means attaining the Shannon bound. If it means only an optimal prefix code among symbol-by-symbol binary prefix codes, the probabilities are not unique: for three positive-probability symbols, this tree is optimal whenever $p_A$ is a largest probability. Huffman coding need not attain entropy exactly for nondyadic probabilities.

Concatenating the specified source-message encodings givesPack consecutive bits in pairs, rather than allocating a fresh Bell pair to each variable-length source word:Under the mapping in part 1, transmitThe block uses six message-bearing qubit transmissions and six preshared Bell pairs. Source-message boundaries are recovered by parsing the decoded prefix bit stream.