arXiv:2211.02401 · Proceedings of the Royal Society of Edinburgh Section A Mathematics

Coupling capacity in C*-algebras

Adam Skalski, Ivan G. Todorov, Lyudmila B. Turowska

Published Sep 7, 2023• 1 citation

Abstract

Given two unital C*-algebras equipped with states and a positive operator in the enveloping von Neumann algebra of their minimal tensor product, we define three parameters that measure the capacity of the operator to align with a coupling of the two given states. Further, we establish a duality formula that shows the equality of two of the parameters for operators in the minimal tensor product of the relevant C*-algebras. In the context of abelian C*-algebras, our parameters are related to quantitative versions of Arveson's null set theorem and to dualities considered in the theory of optimal transport. On the other hand, restricting to matrix algebras we recover and generalize quantum versions of Strassen's theorem. We show that in the latter case our parameters can detect maximal entanglement and separability.

DOI: 10.1017/prm.2023.81

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arXiv:2211.02401 · Proceedings of the Royal Society of Edinburgh Section A Mathematics

Coupling capacity in C*-algebras

Adam Skalski, Ivan G. Todorov, Lyudmila B. Turowska

Published Sep 7, 2023• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.1017/prm.2023.81

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Entanglement in C $^*$ -algebras: tensor products of state spacesEntanglement in C$^*$-algebras: tensor products of state spaces

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Let $O\mapsto A(O)$ be a local Haag--Kastler net and let $A^d(O)=A(O')'$ be its dual net. Normal unital completely positive maps on $B(H)$ that fix $A(O')$ pointwise are exactly the channels with Kraus operators in $A^d(O)$ . Thus Haag duality at $O$ is equivalent to equality with the $A(O)$ -inner channels, while essential duality is equivalent to order-independence of the corresponding causal-complement channel classes at spacelike separation. For self-adjoint generators $a\in M$ and $b\in N$ , the small-parameter order defect has coefficient $\operatorname{diam}σ(-i[a,b])$ . Uniform optimization gives $Γ(M,N)=2Δ_{\mathrm{sa}}(M,N)$ , and reversible inner-channel cb balls recover the same invariant through $\lim_{\varepsilon\downarrow0}Ω^{\mathrm{rev}}_\varepsilon(M,N)/\varepsilon^2=Γ(M,N)/4$ . The full inner-channel class admits a normalized coefficient with the same commutativity zero set. With a positive reference observable $G$ , the mean-input-energy-constrained coefficient $Γ_{G,E}$ has the same zero criterion and increases to $Γ$ as $E\to\infty$ , while no model-independent recovery rate follows from the general von Neumann-algebraic hypotheses. For bosonic second-quantization nets the spacelike defect is either $0$ or $4$ ; generalized free-field examples realize both branches. In the translation-covariant case with positive one-particle time generator, failure of essential duality admits extremal bounded witness flows that are energy-limited relative to the second-quantized time-translation energy.Let $O\mapsto A(O)$ be a local Haag--Kastler net and let $A^d(O)=A(O')'$ be its dual net. Normal unital completely positive maps on $B(H)$ that fix $A(O')$ pointwise are exactly the channels with Kraus operators in $A^d(O)$. Thus Haag duality at $O$ is equivalent to equality with the $A(O)$-inner channels, while essential duality is equivalent to order-independence of the corresponding causal-complement channel classes at spacelike separation. For self-adjoint generators $a\in M$ and $b\in N$, the small-parameter order defect has coefficient $\operatorname{diam}σ(-i[a,b])$. Uniform optimization gives $Γ(M,N)=2Δ_{\mathrm{sa}}(M,N)$, and reversible inner-channel cb balls recover the same invariant through $\lim_{\varepsilon\downarrow0}Ω^{\mathrm{rev}}_\varepsilon(M,N)/\varepsilon^2=Γ(M,N)/4$. The full inner-channel class admits a normalized coefficient with the same commutativity zero set. With a positive reference observable $G$, the mean-input-energy-constrained coefficient $Γ_{G,E}$ has the same zero criterion and increases to $Γ$ as $E\to\infty$, while no model-independent recovery rate follows from the general von Neumann-algebraic hypotheses. For bosonic second-quantization nets the spacelike defect is either $0$ or $4$; generalized free-field examples realize both branches. In the translation-covariant case with positive one-particle time generator, failure of essential duality admits extremal bounded witness flows that are energy-limited relative to the second-quantized time-translation energy.

quant-pharXiv:2605.00075May 1, 20260 citations

Entanglement in C

^*

-algebras: tensor products of state spaces

We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C

^*

-algebras. Relatedly, we study entanglement in (infinite dimensional) C

^*

-algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C

^*

-algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C

^*

-algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C

^*

-algebras. We show that the minimal and maximal tensor products of state spaces of C

^*

-algebras agree precisely when one of the two C

^*

-algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C

^*

-algebras. Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C

^*

-algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C

^*

-algebras. This enables a systematic way of determining the trace simplex of a tensor product of C

^*

-algebras.

Using the natural duality between linear functionals on tensor products of

C^*

-algebras with the trace class operators on a Hilbert space

H

and linear maps of the

C^*

-algebra into

B(H)

, we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF

C^*

-algebras.

Let

O\mapsto A(O)

be a local Haag--Kastler net and let

A^d(O)=A(O')'

be its dual net. Normal unital completely positive maps on

B(H)

that fix

A(O')

pointwise are exactly the channels with Kraus operators in

A^d(O)

. Thus Haag duality at

O

is equivalent to equality with the

A(O)

-inner channels, while essential duality is equivalent to order-independence of the corresponding causal-complement channel classes at spacelike separation. For self-adjoint generators

a\in M

and

b\in N

, the small-parameter order defect has coefficient

\operatorname{diam}σ(-i[a,b])

. Uniform optimization gives

Γ(M,N)=2Δ_{\mathrm{sa}}(M,N)

, and reversible inner-channel cb balls recover the same invariant through

\lim_{\varepsilon\downarrow0}Ω^{\mathrm{rev}}_\varepsilon(M,N)/\varepsilon^2=Γ(M,N)/4

. The full inner-channel class admits a normalized coefficient with the same commutativity zero set. With a positive reference observable

G

, the mean-input-energy-constrained coefficient

Γ_{G,E}

has the same zero criterion and increases to

Γ

E\to\infty

, while no model-independent recovery rate follows from the general von Neumann-algebraic hypotheses. For bosonic second-quantization nets the spacelike defect is either

0

4

; generalized free-field examples realize both branches. In the translation-covariant case with positive one-particle time generator, failure of essential duality admits extremal bounded witness flows that are energy-limited relative to the second-quantized time-translation energy.