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Finite Sector Channel Optimality

Abstract

The common residual spectra determine the exact channel error for sector coarse-graining under arbitrary local quantum operations and finite shared classical mixtures.

Theorem 1.1 (The residual Gram minimum determines the optimum).

Lean statement: D5/S3/Quantum/Entanglement/FiniteSectorChannelOptimality.result

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/FiniteSectorChannelOptimality.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

The logical sectors form a nonempty finite set. Each sector has a positive target rank and a common finite list of nonnegative residual spectral values. The list is decreasing and sums to one. The source encoding repeats each residual coefficient across the target coordinates, with the normalization given by the sector rank. The target encoding is flat on those coordinates.

The residual Gram kernel pairs the square roots of the two sectors’ spectra. Its minimum quadratic form is taken over all probability vectors on the logical sectors. The optimum unhalved diamond error is twice one minus this minimum. The same value is the infimum over arbitrary products of local completely positive trace-preserving maps and over finite probability mixtures of such products. An actual product channel attains the value.

Both encodings are actual isometric quantum channels. Each arbitrary local channel pair has an actual joint realization, and each finite shared classical mixture has an actual mixture realization. These realizations agree on every input matrix. The diamond error ranges over every finite passive reference and every joint density input, so the reference can retain correlations with the logical sector.

Complete positivity makes the channel’s Choi matrix positive. Its spectral decomposition gives a finite Kraus family, and trace preservation makes that family complete. Stacking the Kraus matrices gives the finite isometric environment used in the channel constructions and competitor bounds.

Tracing out the residual coordinate on each side gives the attaining product, using the same local splitting channel on both sides. Its partial-trace action is specified on every physical input matrix. On every encoded logical matrix, the joint action is multiplication by the residual Gram kernel followed by the target encoding. In particular, each logical basis sector produces its exact target pure state. The same encoding and joint channel have the stated diamond error and both stated infima. The kernel is positive semidefinite, has diagonal one, and has entries at most one. Pure joint inputs give a rank-one positive matrix minus a positive matrix of equal trace. The positive spectral part has rank at most one, which bounds the trace norm by the maximal complementary quadratic form. Spectral convex decomposition extends this bound to every density input.

A common correlated reference test gives the lower bound against each arbitrary local competitor. The singular-value prefix constraints of the same Stinespring realization control its overlap with the flat target. The same test also controls finite shared classical mixtures, since its target overlap is linear in the output state. Compactness of the sector probability simplex supplies a minimizing test and identifies both infima with the error of the residual-trace product.

For arbitrary positive flat source and target ranks, exact output on every basis sector is possible if and only if each source rank is a positive integer multiple of its target rank. Necessity follows from the actual local Stinespring isometries: purity forces the joint amplitude to factor through the target vector, and the resulting environment projection has integer rank. Sufficiency splits each source coordinate into its target and residual coordinates by a sector-dependent finite equivalence, constructs a common local isometry, and traces out its environment.

References