Amending coherence-breaking iterates with one unitary

Abstract

Every incoherent trace-preserving Kraus channel of finite coherence-breaking index at least two admits one unitary conjugation repeated between its successive applications.

All matrices are complex and use the fixed incoherent basis indexed by Fin d. The superscript H denotes conjugate transpose, tr is the matrix trace, and a superscript [m] denotes m-fold function iteration. Subtraction in an iteration count is natural-number subtraction.

Definition 1.1 (Erasing the diagonal).

$$\forall d\in \mathbb{N},\forall X\in \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\forall i\in \mathrm{Fin}\left(d\right),\forall j\in \mathrm{Fin}\left(d\right),((i=j)\Rightarrow (\mathrm{offDiag}\left(X\right)\left(i,j\right)=0))\wedge ((i\ne j)\Rightarrow (\mathrm{offDiag}\left(X\right)\left(i,j\right)=X\left(i,j\right)))$$

Formalization. D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.offDiag (✓ std3).

Source. Repository-derived.

Commentary.

The complex linear map offDiag replaces each diagonal entry by zero and preserves every other matrix entry.

Definition 1.2 (Incoherent Kraus operators).

$$\forall d\in \mathbb{N},\forall r\in \mathbb{N},\forall K\in \mathrm{Fin}\left(r\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\mathrm{IsIncoherentKraus}\left(K\right)\iff (\forall s\in \mathrm{Fin}\left(r\right),\forall j\in \mathrm{Fin}\left(d\right),\exists k\in \mathrm{Fin}\left(d\right),\forall i\in \mathrm{Fin}\left(d\right),(i\ne k)\Rightarrow (K\left(s,i,j\right)=0))$$

Formalization. D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.IsIncoherentKraus (✓ std3).

Citation. Long-Mei Yang; Bin Chen; Tao Li; Shao-Ming Fei; Zhi-Xi Wang (2018). Amending Coherence-Breaking Channels via Unitary Operations. DOI: 10.1007/s10773-018-3890-4. URL: https://arxiv.org/abs/1810.09642v1.

Commentary.

Section 2 calls a Kraus representation incoherent when each individual Kraus operator preserves incoherence on every incoherent input. Equivalently, each column of every Kraus operator has support in at most one row: each basis vector is sent to a scalar multiple of a basis vector. This predicate imposes no trace-preservation condition.

Definition 1.3 (Coherence-breaking maps).

$$\forall d\in \mathbb{N},\forall Phi\in \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\mathrm{IsCoherenceBreaking}\left(Phi\right)\iff (\forall rho\in \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),(\mathrm{PosSemidef}\left(rho\right))\Rightarrow ((\mathrm{tr}\left(rho\right)=1)\Rightarrow (\forall i\in \mathrm{Fin}\left(d\right),\forall j\in \mathrm{Fin}\left(d\right),(i\ne j)\Rightarrow (Phi\left(rho\right)\left(i,j\right)=0))))$$

Formalization. D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.IsCoherenceBreaking (✓ std3).

Citation. Long-Mei Yang; Bin Chen; Tao Li; Shao-Ming Fei; Zhi-Xi Wang (2018). Amending Coherence-Breaking Channels via Unitary Operations. DOI: 10.1007/s10773-018-3890-4. URL: https://arxiv.org/abs/1810.09642v1.

Commentary.

The output of every positive semidefinite matrix of trace one has zero entries away from the diagonal. This is the coherence-breaking condition of Section 2. The predicate itself is defined for every matrix-valued function; in the conjecture it is applied to quantum channels and their composites.

Definition 1.4 (The same conjugation between successive channels).

$$\forall d\in \mathbb{N},\forall r\in \mathbb{N},\forall K\in \mathrm{Fin}\left(r\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\forall U\in \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\forall n\in \mathbb{N},\mathrm{amended}\left(K,U,n\right)=(\mathrm{krausApply}\left(K\right))\circ (((LinearMap.mulLeftRight\left(\mathbb{C},(U,(U{)}^H)\right))\circ (\mathrm{krausApply}\left(K\right)){)}^{[n-1]})$$

Formalization. D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.amended (✓ std3).

Citation. Long-Mei Yang; Bin Chen; Tao Li; Shao-Ming Fei; Zhi-Xi Wang (2018). Amending Coherence-Breaking Channels via Unitary Operations. DOI: 10.1007/s10773-018-3890-4. URL: https://arxiv.org/abs/1810.09642v1.

Commentary.

The composition in Section 5 has n copies of the channel and n minus one copies of the same conjugation when n is at least one. The definition also includes n = 0 through natural-number subtraction. For the finite dimensions used here, the Kraus action and matrix conjugation are given by the following formulas. No normalization of K or unitarity of U is needed to define these maps.

$$\forall d\in \mathbb{N},\forall r\in \mathbb{N},\forall K\in \mathrm{Fin}\left(r\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\forall X\in \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\mathrm{krausApply}\left(K,X\right)=\sum _{(s:\mathrm{Fin}\left(r\right))}(((K\left(s\right))\cdot (X))\cdot ((K\left(s\right){)}^H))$$

$$\forall d\in \mathbb{N},\forall U\in \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\forall X\in \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),LinearMap.mulLeftRight\left(\mathbb{C},(U,(U{)}^H)\right)\left(X\right)=((U)\cdot (X))\cdot ((U{)}^H)$$

Definition 1.5 (The finite-index amendment conjecture).

$$\mathrm{claim}\iff (\forall d\in \mathbb{N},\forall r\in \mathbb{N},\forall K\in \mathrm{Fin}\left(r\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),(\mathrm{IsKrausChannel}\left(K\right))\Rightarrow ((\mathrm{IsIncoherentKraus}\left(K\right))\Rightarrow (\forall n\in \mathbb{N},(2\le n)\Rightarrow ((\mathrm{IsCoherenceBreaking}\left((\mathrm{krausApply}\left(K\right){)}^{[n]}\right))\Rightarrow ((\forall k\in \mathbb{N},(1\le k)\Rightarrow ((k<n)\Rightarrow (\neg (\mathrm{IsCoherenceBreaking}\left((\mathrm{krausApply}\left(K\right){)}^{[k]}\right)))))\Rightarrow (\exists U\in \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),(U\in Matrix.unitaryGroup\left(\mathrm{Fin}\left(d\right),\mathbb{C}\right))\wedge (\neg (\mathrm{IsCoherenceBreaking}\left(\mathrm{amended}\left(K,U,n\right)\right)))))))))$$

Formalization. D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.claim (✓ std3).

Citation. Long-Mei Yang; Bin Chen; Tao Li; Shao-Ming Fei; Zhi-Xi Wang (2018). Amending Coherence-Breaking Channels via Unitary Operations. DOI: 10.1007/s10773-018-3890-4. URL: https://arxiv.org/abs/1810.09642v1.

Commentary.

Section 5 conjectures amendment for incoherent trace-preserving Kraus channels with coherence-breaking index n at least two. The n-th iterate is coherence-breaking, and none of the iterates with 1 <= k < n is coherence-breaking, as in the minimum defining the index in Section 2, Eq. (1). There is one unitary U whose conjugation, inserted in every intervening slot, makes the composite fail to be coherence-breaking. Trace preservation is the Kraus completeness equality below.

$$\forall d\in \mathbb{N},\forall r\in \mathbb{N},\forall K\in \mathrm{Fin}\left(r\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\mathrm{IsKrausChannel}\left(K\right)\iff (\sum _{(s:\mathrm{Fin}\left(r\right))}(((K\left(s\right){)}^H)\cdot (K\left(s\right)))=1)$$

Theorem 1.6 (A permutation with fourth-root phases amends the channel).

$$\mathrm{claim}$$

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

Resolves. Problems/yang-chen-li-fei-wang-2018-coherence-breaking-amendment (proved) by D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.result.

Source. Repository-derived.

Acknowledgement. Long-Mei Yang; Bin Chen; Tao Li; Shao-Ming Fei; Zhi-Xi Wang (2018). Amending Coherence-Breaking Channels via Unitary Operations. DOI: 10.1007/s10773-018-3890-4. URL: https://arxiv.org/abs/1810.09642v1.

Commentary.

Write Phi for the Kraus channel, let P erase the diagonal entries of a matrix, and put T = P composed with Phi. Incoherence implies that T kills every diagonal matrix, so T = T composed with P. Choose a permutation sigma and phases omega(c) in {1,i,-1,-i}; the unitary U sends the c-th basis vector to omega(c) times the sigma(c)-th basis vector. Its conjugation Lambda commutes with P. The off-diagonal part of the amended composite is (T Lambda)^(n-1) T.

If that composite were coherence-breaking, this linear map would vanish on every density matrix. Density matrices span the Hermitian matrices over the reals, and every complex matrix is a Hermitian matrix plus i times a Hermitian matrix, so the map would vanish identically. Multiplication on the right by Lambda then gives (T Lambda)^n = 0. Nilpotency forces Tr(T Lambda) = 0, where Tr is the trace of a linear endomorphism of the matrix space.

Minimality of n >= 2 implies that Phi itself is not coherence-breaking, hence T is nonzero. Since T kills diagonal matrices and has only off-diagonal outputs, there are a != b and k != l for which the (k,l) entry of T(E_ab) is nonzero. Choose sigma with sigma(k) = a and sigma(l) = b. The weighted sum of Tr(T Lambda) over all fourth-root phase assignments, with weight conjugate(omega(k)) omega(l), is 4^d times that nonzero entry. Orthogonality of the phase characters therefore supplies a unitary for which Tr(T Lambda) is nonzero. This contradicts the trace forced by coherence-breaking and proves the conjecture.

References

• Truth anchor: D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.IsCoherenceBreaking
• Truth anchor: D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.IsIncoherentKraus
• Truth anchor: D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.amended
• Truth anchor: D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.claim
• Truth anchor: D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.offDiag
• Truth anchor: D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.result
• Dependency: D5/S3/Quantum/Entanglement/AbsoluteSeparability/GurvitsBarnumMoments
• Dependency: D5/S3/Quantum/Matrix/MonomialDiagonalPreserving
• Dependency: D5/S3/Quantum/QuantumChannels/ConcealmentKernelNecessityRefutation
• Dependency: D5/S3/Quantum/QuantumChannels/OutputMaximalCoherenceTensorizationRefutation