19.1 Joint Bell measurement and classical feed-forward
19.1.1 Combining an unknown qubit, a shared Bell pair, and two classical bits
Direct classical transmission of an unknown single-qubit state \(|\psi\rangle_S = \alpha|0\rangle + \beta|1\rangle\) (\(|\alpha|^2 + |\beta|^2 = 1\)) is impossible: measuring a single copy yields at most one bit of information and collapses the state, while sending the continuous complex amplitudes \((\alpha, \beta)\) would require infinitely many classical bits. In 1993, Bennett, Brassard, Crépeau, Jozsa, Peres, and Wootters showed that if Alice and Bob pre-share a maximally entangled Bell pair \(|\Phi^+\rangle_{AB} = (|00\rangle_{AB} + |11\rangle_{AB})/\sqrt{2}\), Alice can transfer \(|\psi\rangle_S\) intact to Bob’s register \(B\) by performing a joint Bell-basis measurement on \((S, A)\) and sending Bob the two-bit classical outcome \((m_1, m_2) \in \{0,1\}^2\).
19.1.2 Why teleportation respects both no-cloning and no-signalling
Quantum teleportation never violates the no-cloning theorem because Alice’s joint Bell projection entangles \(S\) and \(A\) and erases the original state on \(S\) at the exact stage where the conditional state appears on \(B\)—at no point do two copies of \(|\psi\rangle\) coexist. Equally important, teleportation never violates relativistic no-signalling: before Bob receives the two classical bits \((m_1, m_2)\) over a subluminal communication channel, his local qubit \(B\) is in the maximally mixed state \(\rho_B = \frac{1}{2}I\), completely independent of \((\alpha, \beta)\).
19.1.3 Inspecting conditional branch states before and after correction
Each of the four equally likely Bell outcomes \((m_1, m_2) \in \{00, 01, 10, 11\}\) recorded by Alice projects Bob’s qubit into a known Pauli-rotated image \(X^{m_2} Z^{m_1}|\psi\rangle_B\) of the input state. Once the classical bits arrive at Bob’s station, applying the conditional inverse unitary \(Z^{m_1} X^{m_2}\) aligns all four branches into the exact target state \(|\psi\rangle_B\).
Visual animation · Section 19.1
Joint Bell measurement and classical feed-forward
Interactive visual model for 19.1 Joint Bell measurement and classical feed-forward (Combining an unknown qubit, a shared Bell pair, and two classical bits · Why teleportation respects both no-cloning and no-signalling · Inspecting conditional branch states before and after correction). Use Animate or Step to follow the physical state evolution and quantitative predictions.
19.2 Four Pauli-frame branches and entanglement swapping
19.2.1 Decomposing the three-qubit state into four Bell-measurement branches
Define the four orthonormal Bell states on Alice’s two qubits \((S, A)\):
In the circuit realization of the Bell analyzer (a \(\mathrm{CNOT}_{S\to A}\) followed by a Hadamard gate \(H_S\) and computational-basis measurements yielding bits \(m_1\) on \(S\) and \(m_2\) on \(A\)), the four Bell states map to \(|m_1 m_2\rangle \in \{|00\rangle, |10\rangle, |01\rangle, |11\rangle\}\). Rewriting the initial three-qubit product state \(|\psi\rangle_S \otimes |\Phi^+\rangle_{AB}\) in the Bell basis of \((S, A)\) yields the algebraic identity
19.2.2 Uniform marginal state prior to classical message arrival
Each Bell branch occurs with probability \(P(m_1, m_2) = \| \frac{1}{2} X^{m_2} Z^{m_1}|\psi\rangle \|^2 = 1/4\), regardless of \(\alpha\) and \(\beta\). If Bob does not learn \((m_1, m_2)\)—or discards the classical message—his ensemble density matrix is the average over all four Pauli frames:
This Pauli-twirl identity proves algebraically that without the two classical feed-forward bits, zero information about \(|\psi\rangle\) is accessible at Bob’s location.
19.2.3 Entangling independent pairs via intermediate Bell projection
If the input qubit \(S\) is itself entangled with a reference qubit \(R\)—for example, in a Bell state \(|\Phi^+\rangle_{RS}\)—while \((A, B)\) forms an independent Bell pair \(|\Phi^+\rangle_{AB}\), performing the joint Bell measurement on \((S, A)\) plus Pauli correction on \(B\) projects the distant qubits \((R, B)\) into the entangled state \(|\Phi^+\rangle_{RB}\), even though qubits \(R\) and \(B\) never interacted directly. This protocol, known as entanglement swapping, is the foundational building block of quantum repeaters.
Visual animation · Section 19.2
Four Pauli-frame branches and entanglement swapping
Interactive visual model for 19.2 Four Pauli-frame branches and entanglement swapping (Decomposing the three-qubit state into four Bell-measurement branches · Uniform marginal state prior to classical message arrival · Entangling independent pairs via intermediate Bell projection). Use Animate or Step to follow the physical state evolution and quantitative predictions.
19.3 Classical measure-and-prepare benchmarks and heralding
19.3.1 Comparing average teleportation fidelity against the two-thirds classical bound
Suppose Alice and Bob share no entanglement and Alice instead measures the unknown qubit \(|\psi\rangle\) along some axis and sends the classical outcome to Bob, who prepares a guessed state \(\rho_{\mathrm{guess}}\). Averaged uniformly over all pure states on the Bloch sphere (Haar measure), any such classical measure-and-prepare strategy achieves an average state fidelity of at most
When Alice and Bob instead use a noisy shared Werner resource \(\rho_p = p|\Phi^+\rangle\langle\Phi^+| + \frac{1-p}{4}I_4\) (which has singlet/Bell fraction \(F_{\mathrm{Bell}} = (3p+1)/4\)) and ideal local operations, the average teleportation fidelity across the Bloch sphere is
19.3.2 Accounting for probabilistic Bell analyzers and postselection rates
In linear-optical photonic teleportation without auxiliary photons or nonlinearities, a single 50:50 beam splitter can distinguish at most two of the four Bell states (typically \(|\Psi^-\rangle\) and \(|\Psi^+\rangle\) via two-photon HOM interference, with maximum success probability \(1/2\)). Experimental reports must therefore distinguish heralded conditional fidelity (conditioned on a successful Bell-analyzer click before Bob’s detection) from postselected coincidence detection that destroys the output photon.
19.3.3 Verifying unknown-state transfer across noncommuting bases
Testing teleportation solely on computational-basis inputs \(\{|0\rangle, |1\rangle\}\) cannot distinguish quantum teleportation from a classical CNOT or measure-in-\(Z\) channel, which achieves \(F = 1\) on \(\{|0\rangle, |1\rangle\}\) but only \(F = 1/2\) on equatorial states \(\{|+\rangle, |-\rangle, |+i\rangle, |-i\rangle\}\). A complete verification measures the output fidelity across all six Pauli eigenstates (or performs full quantum process tomography) to prove that \(\bar{F} > 2/3\).
Visual animation · Section 19.3
Classical measure-and-prepare benchmarks and heralding
Interactive visual model for 19.3 Classical measure-and-prepare benchmarks and heralding (Comparing average teleportation fidelity against the two-thirds classical bound · Accounting for probabilistic Bell analyzers and postselection rates · Verifying unknown-state transfer across noncommuting bases). Use Animate or Step to follow the physical state evolution and quantitative predictions.
Explore the Apparatus
Choose the polar and azimuthal angles \((\theta, \phi)\) of the input state \(|\psi\rangle_S = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle\) and the purity \(p\) of the shared Bell pair. Inspect all four Bell-measurement branches \((m_1, m_2) \in \{00, 01, 10, 11\}\) before and after the conditional Pauli feed-forward \(Z^{m_1}X^{m_2}\), and compare the average output fidelity \(\bar{F} = (1+p)/2\) against the \(2/3\) classical measure-and-prepare threshold.
Interactive model · Idealized predictions
Four messages, one recovered state
On a small screen, swipe the diagram to inspect its components. Moving marks illustrate the model; they are not measured trajectories.
Read the values as a table
What is transferred when a quantum state is teleported? The text and worked examples remain available without JavaScript. Animation starts only when requested; Step gives a motion-free way to follow the model.
Worked Check
Problem: Alice teleports the state \(|\psi\rangle_S = \frac{3}{5}|0\rangle + \frac{4i}{5}|1\rangle\) using an ideal shared Bell pair \(|\Phi^+\rangle_{AB}\). Write down the unnormalized branch state on Bob’s qubit when Alice’s Bell circuit reports \((m_1, m_2) = (1, 1)\) (corresponding to \(|\Psi^-\rangle_{SA}\)), identify the conditional recovery gate Bob must apply, and verify that the recovered state equals \(|\psi\rangle_B\). Next, if the shared Bell pair is degraded by depolarizing noise to \(p = 0.72\), find the average teleportation fidelity \(\bar{F}\).
Solution:
1. For outcome \((m_1, m_2) = (1, 1)\), Bob’s pre-correction state is
Quantum teleportation transfers an unknown state \(|\psi\rangle\) using a shared Bell pair and two classical bits from Alice’s Bell-basis measurement so Bob can apply the matching Pauli correction \(X^{b_2}Z^{b_1}\).
Without Alice’s two classical bits, Bob’s local density matrix is the maximally mixed state \(I/2\) regardless of \(|\psi\rangle\), preserving relativistic causality.
Entanglement swapping teleports an entangled share itself, projecting two independent pairs \(AB\) and \(CD\) into an entangled pair \(AD\) conditioned on the middle Bell-state outcome.
Practice Problems
Problem 19.1: Fidelity when classical feed-forward is omitted
Suppose Alice performs an ideal Bell measurement on \(|\psi\rangle_S \otimes |\Phi^+\rangle_{AB}\), but the classical communication line is disconnected so Bob always applies the identity operation \(I_B\). What is Bob’s unconditional output state \(\rho_B\), and what is its fidelity \(F = \langle\psi|\rho_B|\psi\rangle\) with the input state \(|\psi\rangle\)?
Hint
Use the Pauli-twirl average \(\rho_B = \frac{1}{2}I\).
Solution
Averaging over the four equally probable outcomes \((00, 01, 10, 11)\) gives \(\rho_B = \frac{1}{2}I\). For any normalized pure state \(|\psi\rangle\), the fidelity is \(F = \langle\psi|\frac{1}{2}I|\psi\rangle = 1/2 = 0.50\), equivalent to random guessing.
Problem 19.2: Minimum Bell-state fidelity to beat the classical bound
A noisy two-qubit resource state has maximally entangled fraction \(F_{\mathrm{Bell}} = \langle\Phi^+|\rho_{AB}|\Phi^+\rangle\) and isotropic noise on the remaining three Bell states. Express the average teleportation fidelity \(\bar{F}\) in terms of \(F_{\mathrm{Bell}}\) and determine the threshold \(F_{\mathrm{Bell}}\) needed to exceed \(\bar{F}_{\mathrm{classical}} = 2/3\).
Hint
Use \(\bar{F} = (2F_{\mathrm{Bell}} + 1)/3\).
Solution
From \(\bar{F} = (2F_{\mathrm{Bell}} + 1)/3 > 2/3\), we obtain \(2F_{\mathrm{Bell}} > 1\), so \(F_{\mathrm{Bell}} > 1/2 = 0.50\). Any isotropic state with Bell fidelity above \(0.50\) is entangled and yields nonclassical average teleportation fidelity.
Problem 19.3: Classical dephasing channel masquerading as teleportation
Suppose a flawed protocol replaces the shared Bell pair with the classically correlated separable state \(\rho_{\mathrm{sep}} = \frac{1}{2}|00\rangle\langle 00| + \frac{1}{2}|11\rangle\langle 11|\). What output fidelity is obtained for computational-basis inputs \(\{|0\rangle, |1\rangle\}\), what fidelity is obtained for equatorial inputs \(\{|+\rangle, |-\rangle\}\), and what is the six-state average fidelity?
Hint
Note that \(\rho_{\mathrm{sep}} = \frac{1}{2}(|\Phi^+\rangle\langle\Phi^+| + |\Phi^-\rangle\langle\Phi^-|)\) applies a random \(Z\) flip with probability \(1/2\) to Bob’s output.
Solution
Because \(\rho_{\mathrm{sep}}\) lacks off-diagonal coherence between \(|00\rangle\) and \(|11\rangle\), the teleportation channel reduces to full \(Z\)-basis dephasing \(\mathcal{E}(\rho) = \frac{1}{2}\rho + \frac{1}{2}Z\rho Z\). It preserves \(|0\rangle\) and \(|1\rangle\) with fidelity \(F_Z = 1\), but maps \(|\pm\rangle\) and \(|\pm i\rangle\) to \(\frac{1}{2}I\) with fidelity \(F_X = F_Y = 1/2\). The six-state (and Bloch-sphere) average fidelity is \((1 + 1 + 0.5 + 0.5 + 0.5 + 0.5)/6 = 2/3\), never exceeding the classical bound.
Key Takeaways
Quantum teleportation transfers an unknown state \(|\psi\rangle_S\) using a shared Bell pair \(|\Phi^+\rangle_{AB}\), a joint Bell measurement on \((S,A)\), and two bits \((m_1, m_2)\) of classical feed-forward that select the Pauli correction \(Z^{m_1}X^{m_2}\).
Before the classical bits arrive, Bob’s marginal density matrix is \(\rho_B = \frac{1}{2}I\) for every input state, strictly preserving no-signalling; the joint measurement destroys \(|\psi\rangle\) at Alice’s station, preserving no-cloning.
Beating the maximum classical measure-and-prepare average fidelity \(\bar{F}_{\mathrm{classical}} = 2/3\) across noncommuting bases requires shared entanglement with Bell fidelity \(F_{\mathrm{Bell}} > 1/2\).