Quantum teleportation resource accounting

Alice and Bob initially share the Bell state with positive zero-zero and one-one amplitudes. Let the independent input on C have amplitudes alpha and beta. Expand the three-qubit product in the Bell basis of C tensor A. Collecting the B amplitudes gives four possible conditional vectors, related to the input by identity, Z, X, and XZ. Each branch has amplitude factor one half and probability one quarter, independent of the unknown normalized input.

Use the definitions $|\Phi^\pm\rangle=(|00\rangle\pm|11\rangle)/\sqrt2$ and $|\Psi^\pm\rangle=(|01\rangle\pm|10\rangle)/\sqrt2$ on the stated factor order. In particular, the minus Psi branch gives XZ times the input, rather than ZX under this convention. The two differ only by a global sign, but preserving one convention makes corrections checkable and prevents relative-sign mistakes when the protocol is embedded in a larger circuit.

Alice sends a two-bit label for her four-valued Bell result. Bob applies the corresponding inverse: identity, Z, X, or ZX. Operator products act right to left, so the last correction means first X and then Z. The corrected output equals the input for every alpha and beta. Linearity also preserves arbitrary correlations with an untouched reference, which is a stronger requirement than reproducing the probabilities of one chosen input basis.

Before Bob receives the label, his state is the average over the four equally probable Pauli branches. This average is the maximally mixed qubit, independent of the input. Alice cannot choose her Bell result to encode a desired instantaneous signal. The classical message does not describe the continuous amplitudes either; it merely identifies which of four known corrections is needed. Information becomes usable only after that message reaches Bob through an ordinary causal communication channel.

The resource pair is consumed: after Alice's Bell measurement, A is part of the measured pair C:A rather than still sharing the original entanglement with B. Establishing that pair beforehand can itself require a quantum channel, purification, storage, and synchronization. Treating those costs as free is a resource convention, not a demonstration that classical communication alone transmits an arbitrary unknown state. Bob still needs a physical qubit and controlled local operations.

The input's quantum information is transferred, not its matter or energy. Alice's measured qubits do not retain a second independent copy. A postselected experiment that reports only one Bell outcome implements a conditional protocol with an associated success probability; it is not automatically the deterministic four-outcome scheme. Noisy entanglement, incomplete Bell measurement, miscalibrated Pauli operations, and mismatched reference frames change the realized channel. Their effect must be quantified separately from this exact ideal identity.