The Pure-State Handshake

Abstract

A pure state sandwiches any matrix to a scalar multiple of itself.

Theorem 1.1 (The pure-state sandwich collapses to a scalar).

$$\begin{gathered} \rho X\rho =\mathrm{inner}(v,Xv)\cdot \rho , \\ \rho \rho =\rho ,\mathrm{inner}(v,Xv)=\mathrm{Tr}(X\rho ) \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Quantum/PureState/PureStateHandshake.pure_state_handshake (✓ std3). ∎

Source. Repository-derived.

Commentary.

For a normalized amplitude vector v (inner product of v with itself equal to 1), the rank-one density matrix rho = |v><v| — the outer product with entries rho i j = v i times conjugate of v j — is idempotent (rho times rho = rho), so a pure state is its own square root.

The handshake is the middle identity: for ANY matrix X, sandwiching X between two copies of rho collapses to a scalar multiple, rho X rho = <v, X v> times rho, and that scalar equals the density-matrix expectation Tr(X rho). Specializing X to an inverse state gives the mechanism behind the pure-state divergence handshake. The load-bearing new content is this sandwich-collapse identity — there is no library lemma for rho X rho with a general middle matrix — while the idempotency and the expectation-equals-trace fact are its supporting glue. Only the normalization <v,v> = 1 is used, and only for idempotency; the handshake and the trace identity hold for every v and every X, with no positivity or invertibility hypothesis.

Only the algebraic handshake mechanism is recorded here. The downstream conclusion — that the Belavkin-Staszewski and max divergences of a pure state against sigma both equal the logarithm of <v, sigma-inverse v> — is not covered by this statement.

References

• Truth anchor: D5/S3/Quantum/PureState/PureStateHandshake.pure_state_handshake