Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

Unitary and doubly extreme decomposition of a qutrit channel

Abstract

Every equal-modulus Haagerup–Musat–Ruskai qutrit channel is a mixture of a unitary conjugation and a rank-three channel extreme in both the unital and trace-preserving completely positive maps.

Definition 1.1 (The four HMR operators).

Formalization. D5/S3/Quantum/QuantumChannels/QutritChannelUnitaryExtremeDecomposition.hmrKraus (✓ std3).

Citation. Uffe Haagerup; Magdalena Musat; Mary Beth Ruskai (2021). Extreme Points and Factorizability for New Classes of Unital Quantum Channels. DOI: 10.1007/s00023-021-01071-y. URL: https://arxiv.org/abs/2006.03414v4.

Commentary.

The four matrices are the operators A₁, A₂, A₃, A₄ of Theorem UCPT, HMR2021J.tex lines 280–285, in that order. The vector notation ![A₁,A₂,A₃,A₄] is a function on Fin 4, with indices zero through three. The star on a complex entry is complex conjugation. This definition allows arbitrary complex α and β; the source’s channel theorem assumes |α|² + |β|² = 1.

Definition 1.2 (The HMR matrix map).

Formalization. D5/S3/Quantum/QuantumChannels/QutritChannelUnitaryExtremeDecomposition.hmrChannel (✓ std3).

Citation. Uffe Haagerup; Magdalena Musat; Mary Beth Ruskai (2021). Extreme Points and Factorizability for New Classes of Unital Quantum Channels. DOI: 10.1007/s00023-021-01071-y. URL: https://arxiv.org/abs/2006.03414v4.

Commentary.

This is the complex linear map Φ in HMR2021J.tex line 286. MatrixMap.of_kraus K L sends ρ to the sum of K(k)ρL(k). Both families here are k ↦ A(k), so the action is one half of the sum of A(k)*ρA(k). In particular, the adjoints occur on the left. The scalar one half is complex. The definition imposes no modulus condition.

Definition 1.3 (Preserving the identity).

Formalization. D5/S3/Quantum/QuantumChannels/QutritChannelUnitaryExtremeDecomposition.IsUnital (✓ std3).

Citation. Uffe Haagerup; Magdalena Musat; Mary Beth Ruskai (2021). Extreme Points and Factorizability for New Classes of Unital Quantum Channels. DOI: 10.1007/s00023-021-01071-y. URL: https://arxiv.org/abs/2006.03414v4.

Commentary.

A complex linear map on three-by-three matrices is unital precisely when it maps the identity matrix to the identity matrix. Together with complete positivity this is the UCP condition used in HMR2021J.tex lines 377–383.

Theorem 1.4 (The decomposition for all equal-modulus parameters).

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

Resolves. Problems/haagerup-musat-ruskai-2021-qutrit-channel-unitary-extreme-decomposition (proved) by D5/S3/Quantum/QuantumChannels/QutritChannelUnitaryExtremeDecomposition.result.

Source. Repository-derived.

Acknowledgement. Uffe Haagerup; Magdalena Musat; Mary Beth Ruskai (2021). Extreme Points and Factorizability for New Classes of Unital Quantum Channels. DOI: 10.1007/s00023-021-01071-y. URL: https://arxiv.org/abs/2006.03414v4.

Commentary.

For every pair of complex parameters of norm 1/√2, the channel is a convex combination of a unitary conjugation and a completely positive, unital, trace-preserving map ψ. Its Choi matrix has natural-valued rank three, and ψ is separately extreme in the UCP maps and in the CPT maps, where convexity is over the real scalars. The equality holds for every complex matrix ρ. MatrixMap.choi_matrix is the Choi matrix operation; MatrixMap.IsCompletelyPositive quantifies over every finite amplification, and MatrixMap.IsTracePreserving requires preservation of the trace of every matrix. Complex.ofReal displays both real-to-complex coefficient casts. The source’s question is HMR2021J.tex lines 387–390; its positive real parameter example is at lines 366–383. The construction below always takes lam = 1/4.

First consider any three matrices y(i) and their map Ψ(ρ) = Σᵢ y(i)ρy(i). If Ψ = tθ + sη with t > 0, s ≥ 0 and θ, η completely positive, then C(Ψ) − tC(θ) = sC(η) is positive semidefinite. Positivity implies ker C(Ψ) ⊆ ker C(θ). Let V have columns given by the entries of y(i), so C(Ψ) = VV*. Factor C(θ) = BB*. The kernel inclusion gives B* = QC(Ψ) for a matrix Q, hence C(θ) = VHV* with H = VQQV. Inverting the Choi correspondence gives θ(ρ) = Σᵢⱼ Hᵢⱼ y(i)*ρy(j).

Suppose Σᵢ y(i)y(i) = I and the nine products y(i)y(j) are complex linearly independent. Unitality of θ then makes H equal to I by uniqueness of coefficients, so θ = Ψ. This proves extremality among UCP maps. For CPT maps, assume Σᵢ y(i)y(i) = I and independence of the nine products y(i)y(j). Trace preservation, cyclicity of trace and the nondegenerate trace pairing give Σᵢⱼ Hᵢⱼ y(j)y(i)* = I. Coefficient uniqueness applied to the transpose of H again gives H = I. Independence of y(i)y(j) also implies independence of the three y(i), then of their adjoints and of the columns of V. Thus rank C(Ψ) = rank(VV) = rank V = 3.

Put q = √2, viewing the real entries below as complex numbers. Choose the three y(i), in order, to be 1/(2√3) times the matrices [[−1,−q,1],[q,0,−q],[−1,q,1]], [[1,q,1],[q,0,q],[1,q,1]], and [[1,−q,1],[−q,0,q],[−1,q,−1]]. Entrywise multiplication verifies both sums of products equal I. For each of the two nine-product families, explicit dual coordinate matrices recover each coefficient from any linear combination of the products: the sum of the entrywise products with the chosen dual matrix is that coefficient. Applying these identities to a zero linear combination proves both independence assertions. These calculations use (√2)² = 2, (√3)² = 3 and √3 ≠ 0. Consequently the map Ψ₀ associated with these three matrices has both extremality properties and Choi rank three.

Write Φ₀ for the channel at α = β = 1/√2, and set U₀ = 1/2 [[1,−q,−1],[q,0,q],[−1,−q,1]]. Direct calculation of its nine entries proves U₀U₀ = I. Expanding the two Kraus sums entry by entry proves, for every ρ, Φ₀(ρ) = (1/4)U₀ρU₀ + (3/4)Ψ₀(ρ). These matrices are the source’s operators 2X₂ and (2/√3)(X₁,X₃,X₄), respectively, from HMR2021J.tex lines 370–381.

For general α and β of the required norms, set u = √2 α and v = √2 β. Both have norm one. Choose a fifth root r with r⁵ = uv² and set s = r²/v. Norm multiplicativity gives |r| = |s| = 1, and algebra gives rs² = u and r²/s = v. The diagonal matrices D = diag(r,1,r⁻¹) and E = diag(s,1,s⁻¹) are unitary. Substituting these identities in the nine entries of the Kraus sums gives Φα,β(ρ) = EΦ₀(DρD)E.

The transformation T(φ)(ρ) = Eφ(DρD)E preserves complete positivity by composition of Kraus maps. Unitarity gives unitality, and cyclicity of trace gives trace preservation. Its inverse uses D* and E*. Together these transformations form a real linear equivalence mapping each of the UCP and CPT sets onto itself; the image theorem for extreme points therefore preserves each extremality property. In Choi coordinates C(T(φ)) = QC(φ)Q*, where Q has entries Q((a,i),(b,j)) = conjugate(E(b,a)) conjugate(D(i,j)). The rank of a product cannot exceed that of its middle factor. Applying this inequality to T and its inverse proves equality of Choi ranks.

Transport the positive real parameter decomposition by T. The unitary term becomes conjugation by U = DU₀E, and the other term becomes ψ = T(Ψ₀), which retains complete positivity, unitality, trace preservation, rank three and both extremality properties. With lam = 1/4 the two nonnegative weights are 1/4 and 3/4, and their sum is one. The phase identity identifies the transported channel with Φα,β, giving the stated equality on every ρ.

References