Superdense coding and classical capacity

Alice holds A and Bob holds B of a shared positive-Phi Bell pair. To send bits a and b, Alice applies $X^bZ^a$ to A. The four choices give the four Bell states up to physically irrelevant global signs. The first bit controls relative phase and the second controls computational-basis parity. These labels are encoded in the joint state, not in a locally readable change of Alice's or Bob's individual qubit.

Alice then physically sends A through an ideal qubit channel. Bob, now holding both factors, can make a complete Bell-basis measurement and identify the two bits exactly. Before transmission his B marginal is I/2 for every message. Even Alice's transmitted marginal is I/2 for every message, so inspecting that particle alone cannot identify the label. Reliable decoding uses correlations with the previously shared half, which is precisely the additional resource.

An explicit decoder applies controlled-NOT with A as control and B as target, followed by a Hadamard gate on A, and measures both in the computational basis. The controlled-NOT converts Bell parity into B's ordinary bit. The Hadamard converts the remaining relative sign into A's bit. For our encoding convention the outcomes are a on A and b on B, with a possible overall phase that never changes their probabilities.

The resource pair does not survive as a reusable Bell pair after complete decoding. Counting only the final forward transmission can be appropriate when entanglement is supplied in advance, but it is an entanglement-assisted rate. If distributing the initial pair also costs one forward qubit, two quantum transmissions have carried two classical bits over the complete cycle. Storage errors, a restricted Bell analyzer, or a noisy forward channel require a different performance analysis; the ideal two-bit result alone supplies no engineering error model.

For an ensemble of isolated qubit messages without a shared quantum reference, the Holevo quantity is at most one bit. Increasing the number of nonorthogonal signal states does not permit arbitrary recovery of their continuous coordinates from a single carrier. In dense coding the receiver's ensemble consists of four orthogonal pure states in dimension four, whose uniform average is the identity divided by four. Its Holevo quantity is two bits and the Bell measurement actually attains it.

A Holevo upper bound is not always attained by a measurement on one signal. Channel coding can use long blocks and collective decoding, so a general classical capacity question requires an asymptotic definition and an encoding class. Shared entanglement changes that capacity problem. With a nonmaximally entangled pair, local Pauli encodings need not produce four orthogonal states; their overlap reflects the unequal Schmidt weights. The maximally entangled resource is therefore doing quantitative work, beyond merely providing any correlation between laboratories.