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arXiv:2405.02434

Almost-idempotent quantum channels and approximate $C^*$-algebras

Alexei Kitaev

Published May 3, 2024• 0 citations
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Abstract

Let $Φ$ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that $Φ$ is $η$-idempotent, namely, $\|Φ^2-Φ\|_{\mathrm{cb}} \leη$, and construct an associated $\varepsilon$-$C^*$ algebra (of almost-invariant observables) for $\varepsilon=O(η)$. This type of structure has the axioms of a unital $C^*$ algebra but the associativity and other axioms involving the multiplication and the unit hold up to $\varepsilon$. We prove that any finite-dimensional $\varepsilon$-$C^*$ algebra $A$ is $O(\varepsilon)$-isomorphic to some genuine $C^*$ algebra $B$. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When $A$ comes from a finite-dimensional $η$-idempotent UCP map $Φ$, the $O(η)$-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel $Φ^*$ into a decoding channel, producing a state on $B$, and an encoding channel.

DOI
10.48550/arxiv.2405.02434

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Almost-idempotent quantum channels and approximate C∗C^*C∗-algebras
Let ΦΦΦ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that ΦΦΦ is ηηη-idempotent, namely, ∥Φ2−Φ∥cb≤η\|Φ^2-Φ\|_{\mathrm{cb}} \leη∥Φ2−Φ∥cb​≤η, and construct an associated ε\varepsilonε-C∗C^*C∗ algebra (of almost-invariant observables) for ε=O(η)\varepsilon=O(η)ε=O(η). This type of structure has the axioms of a unital C∗C^*C∗ algebra but the associativity and other axioms involving the multiplication and the unit hold up to ε\varepsilonε. We prove that any finite-dimensional ε\varepsilonε-C∗C^*C∗ algebra AAA is O(ε)O(\varepsilon)O(ε)-isomorphic to some genuine C∗C^*C∗ algebra BBB. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When AAA comes from a finite-dimensional ηηη-idempotent UCP map ΦΦΦ, the O(η)O(η)O(η)-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel Φ∗Φ^*Φ∗ into a decoding channel, producing a state on BBB, and an encoding channel.
Abstract. We construct explicit bistochastic entanglement-breaking maps on Mn whose completely bounded idempotence defects tend to zero, while their distances from all unital completely positive idempotents on Mn have lower bounds tending to one. This answers negatively Kitaev’s question on dimension-independent approximation by idempotent unital completely positive maps. The construction uses a regular simplex with n + 1 vertices, and the defect is exactly 2(n2 − n− 1)/n3. The obstruction persists under arbitrary finite matrix amplifications, with an asymptotic lower bound of 1/4 uniform in the auxiliary dimension. The proof combines a rank obstruction with the Choi–Effros theorem and an elementary rigidity result: two finite-dimensional C∗-algebras admitting unital completely positive approximate inverses with completely bounded error below 1/4 are ∗-isomorphic.
We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra B(H)\mathfrak{B}(\mathcal{H})B(H) of bounded operators on a separable Hilbert space H\mathcal{H}H ) to the space of all operators on H\mathcal{H}H relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation Φ\PhiΦ can be uniquely extended to a bounded linear operator on the Banach space of all G\sqrt{G}G​ -bounded operators on H\mathcal{H}H provided that the operation Φ\PhiΦ is GGG -limited: the predual operation Φ∗\Phi_{*}Φ∗​ maps the set of positive trace class operators ρ\rhoρ with finite value of TrρG\textrm{Tr}\rho GTrρG into itself. Assuming that GGG has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the GGG -limited property. Applications to the theory of Bosonic Gaussian channels are considered.
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Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on M⊗ˉN\mathcal{M}\bar{\otimes}\mathcal{N}M⊗ˉ​N but on the tensor product of M\mathcal{M}M with the \emph{opposite} algebra of N\mathcal{N}N, or equivalently with a commutant; returning to N\mathcal{N}N itself requires a transposition, and it has been observed that this step typically fails. We determine exactly when it does not fail. Our main result is that a Choi--Jamiołkowski type isomorphism which is natural in the first algebra exists if and only if the second algebra is ∗*∗-anti-isomorphic to itself. Naturality cannot be dropped: whenever M\mathcal{M}M is anti-isomorphic to itself such an isomorphism exists for trivial reasons, since the order structure of a predual is a Jordan invariant and cannot distinguish an algebra from its opposite. Consequently no such correspondence exists for the type III factors constructed by Connes. We show in addition that the domain admits no description in terms of properties of individual maps: the space of all normal maps with the operator norm is too large, and neither complete boundedness nor complete positivity cuts it down to the right size.
Provided a maximally entangled state vector in H⊗H{\mathcal{H}}\otimes{\mathcal{H}}H⊗H, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of CP(H,K){\mathsf CP}({\mathcal{H}},{\mathcal{K}})CP(H,K) (completely positive maps of density operators on H{\mathcal{H}}H tothose on K{\mathcal{K}}K) onto the density operators on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, which extends to a unitary map. Here, H{\mathcal{H}}H and K{\mathcal{K}}K are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both H{\mathcal{H}}H and K{\mathcal{K}}K which produces density operators on either L(H,K){\mathsf L}({\mathcal{H}},{\mathcal{K}})L(H,K) (Hilbert-Schmidt inner product) or K⊗H∗{\mathcal{K}}\otimes{\mathcal{H}}^*K⊗H∗ instead of K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H. In particular, no entangled state input is required. There are no such binatural maps carried on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, but those natural in only K{\mathcal{K}}K are canonically associated with unitary maps H→H∗{\mathcal{H}}\rightarrow{\mathcal{H}}^*H→H∗ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.