Fidelity and Bures distance

For positive density matrices, the square root means the unique positive matrix square root obtained spectrally. The operator between the outer roots in the definition is positive, even when the two states do not commute. The equivalent trace-norm expression is useful computationally because it involves singular values. Ordinary matrix multiplication cannot usually replace these square roots. In particular, the Hilbert–Schmidt overlap of two mixed states is generally neither fidelity convention.

If one state is a pure projector, squared fidelity becomes the probability of accepting that projector when measuring the other state. If both are pure, it is the squared absolute inner product. A global phase cancels, whereas a relative phase can change the overlap. If the states commute, calculate root fidelity by summing square roots of matched classical probabilities, then square that entire sum. Squaring each term before summing would remove the cross terms and give a different quantity.

Choose a reference factor large enough to purify either state. A purification is a normalized vector on system tensor reference whose partial trace over the reference gives the specified state. Uhlmann's theorem says root fidelity is the largest absolute overlap between such purifications. Optimization over the unused reference is essential: two arbitrary purifications need not achieve it. This is a theorem about states on a declared tensor product, rather than an assertion that a mixed state secretly has one preferred wavefunction.

The Bures distance used here is the square root of twice one minus root fidelity. The related Bures angle is the inverse cosine of root fidelity; it is a different function. For normalized states, zero Bures distance means equality, and orthogonal supports give distance square root of two. The formula has no spatial units. It measures separation in state space, so interpreting it as the physical travel distance of a particle or as elapsed evolution time would require additional structure.

A deterministic channel cannot decrease fidelity between two input hypotheses: processing can make states harder to distinguish. Unitary processing preserves it. This direction agrees with trace-distance contraction, although one measure increases and the other decreases. For two pure states the upper trace-distance bound is an equality. Mixed states generally require the inequality, so obtaining a single fidelity number does not determine the exact discrimination error.

State fidelity also differs from a guarantee about a quantum device on every input. A gate can reproduce one test state perfectly and fail badly on another, or fail on correlations with a reference. Average fidelity, worst-case fidelity, and entanglement-sensitive channel metrics answer different questions. Before comparing quoted values, identify the target, input ensemble, convention, and optimization. An uncertainty in estimating fidelity is another issue entirely; the exact algebraic definitions do not themselves provide statistical confidence intervals.