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The original two-swap channel

Abstract

The commutator controls the original qutrit channel on the full complex traceless space.

Theorem 1.1 (Both bounds include commuting installations).

Proof. Machine-checked in Lean as D5/S3/Quantum/QuantumChannels/OriginalQutritGap.original_two_sided (✓ std3). ∎

Source. Repository-derived.

Commentary.

U exchanges coordinates 1 and 2, W exchanges 0 and 1, and xi=(1,1,1)/sqrt(3). Put Pxi=xi xi*, D(X)=(X+W X W*)/2 and Lambda(X)=U D(X) U*. For every complex unitary 3 by 3 matrix N, Theta_N=Lambda squared composed with Ad_N composed with Lambda squared. The carrier H0 consists of all complex trace-zero 3 by 3 matrices. Its Hilbert-Schmidt norm is transported by CartesianVariance.vectorize; r_N is the induced operator norm of Theta_N on H0. Delta_N is the squared Hilbert-Schmidt norm of N Pxi-Pxi N.

The unit vector v has zero diagonal and all six off-diagonal entries 1/sqrt(6). The exact energy identity for the original A=Lambda squared gives contraction by at most one half on the orthogonal complement of v in H0. Both A and its adjoint fix v. For R=Ad_N and eta squared equal to 1-abs(inner(v,Rv)) squared, the two losses in A R A yield eta squared/3 at most 1-r_N squared at most eta squared. The same N occurs in the middle action and in the commutator.

The centered identity Pxi=I/3+(sqrt(6)/3)v connects these estimates to the original projection. Hermiticity makes t=inner(v,Rv) real; positivity of trace(Pxi N Pxi N*) gives t at least -1/2. Direct unitary norm invariance gives delta_N=(4/3)(1-t), hence 0 at most delta_N at most 2 and eta squared=(3 delta_N/4)(2-3 delta_N/4). These identities include delta_N=0 and N xi orthogonal to xi. No division by delta_N or exclusion of either endpoint is used.

The direct scalar consequence is r_N to the power n at most (1-delta_N/8) to the power n/2, at most exp(-n delta_N/16), for integers n at least zero. This supplies a decay envelope for a fixed installed N. Physical consumers still require the original memoryless lift, exact law (101.5), the same installed N and J J N J J chronology, coherent archive access, two-sided archive alignment and complete late measurements with three outcomes. The original block still uses four instruments, one N and four records. The population statement restricted to d different from 2/3 remains separate; d=2/3 stays allowed in the universal matrix theorem. The estimate supplies no preparation-time lower bound, calibration procedure, common physical realization or uniform finite cost over all installations.

References