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arXiv:quant-ph/0302031 · Reviews in Mathematical Physics

Entanglement Breaking Channels

Michał Horodecki, Peter W. Shor, Mary Beth Ruskai

Published Aug 1, 2003• 602 citations
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Abstract

This paper studies the class of stochastic maps, or channels, for which (I⊗Φ)(Γ) is always separable (even for entangled Γ). Such maps are called entanglement breaking, and can always be written in the form Φ(ρ)=∑ k R k Tr F k ρ where each R k is a density matrix and F k >0. If, in addition, Φ is trace-preserving, the {F k } must form a positive operator valued measure (POVM). Some special classes of these maps are considered and other characterizations given. Since the set of entanglement-breaking trace-preserving maps is convex, it can be characterized by its extreme points. The only extreme points of the set of completely positive trace preserving maps which are also entanglement breaking are those known as classical-quantum or CQ. However, for d≥3, the set of entanglement breaking maps has additional extreme points which are not extreme CQ maps.

DOI
10.1142/s0129055x03001709

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This paper studies the class of stochastic maps, or channels, for which (I⊗Φ)(Γ) is always separable (even for entangled Γ). Such maps are called entanglement breaking, and can always be written in the form Φ(ρ)=∑ k R k Tr F k ρ where each R k is a density matrix and F k >0. If, in addition, Φ is trace-preserving, the {F k } must form a positive operator valued measure (POVM). Some special classes of these maps are considered and other characterizations given. Since the set of entanglement-breaking trace-preserving maps is convex, it can be characterized by its extreme points. The only extreme points of the set of completely positive trace preserving maps which are also entanglement breaking are those known as classical-quantum or CQ. However, for d≥3, the set of entanglement breaking maps has additional extreme points which are not extreme CQ maps.
This paper has been removed and is superceded by a pair of papers on stochastic maps, or channels, which break entanglement (EBT). quant-ph/0302031 General Entanglement Breaking Channels by Michael Horodecki, Peter W. Shor and Mary Beth Ruskai includes the results originally in section 3, as well as a number of new results about the extreme points. (The remarks in previously in section 3 about "sign change condition" are incorrect and have been omitted from quant-ph/0302031.) The new paper contains a to the conjecture that all extreme points of EBT maps are CQ when d > 2. quant-ph/0302032 Qubit Entanglement Breaking Channels by Mary Beth Ruskai contains the results oroginally in section 2. The main result is that for qubits, EBT maps are precisely the convex hull of those known as classical-quantum channels. The complete positivity conditions in a canonical parameterization are reviewed and used to formulate EBT constraints.
The set of Entanglement Saving (ES) quantum channels is introduced and characterized. These are completely positive, trace preserving transformations which when acting locally on a bipartite quantum system initially prepared into a maximally entangled configuration, preserve its entanglement even when applied an arbitrary number of times. In other words, a quantum channel ψ is said to be ES if its powers ψn are not entanglement-breaking for all integers n. We also characterize the properties of the Asymptotic Entanglement Saving (AES) maps. These form a proper subset of the ES channels that is constituted by those maps that not only preserve entanglement for all finite n but which also sustain an explicitly not null level of entanglement in the asymptotic limit n → ∞. Structure theorems are provided for ES and for AES maps which yield an almost complete characterization of the former and a full characterization of the latter.
Abstract One of the fundamental questions in quantum information theory is determining entanglement of quantum states, which is generally an NP-hard problem. In this paper, we prove that all PPT ( π ― A ⊗ π B ) -invariant quantum states are separable if and only if all extremal unital positive ( π B , π A ) -covariant maps are decomposable where π A , π B are unitary representations of a compact group and π A is irreducible. Moreover, an extremal unital positive ( π B , π A ) -covariant map L is decomposable if and only if L is completely positive or completely copositive. We then apply these results to prove that all PPT quantum channels of the form Φ ( ρ ) = a Tr ( ρ ) d Id d + b ρ + c ρ T + ( 1 − a − b − c ) diag ( ρ ) are entanglement-breaking, and that there is no A-BC PPT-entangled ( U ⊗ U ― ⊗ U ) -invariant tripartite quantum state. The former strengthens some conclusions in (Vollbrecht and Werner 2001 Phys. Rev. A 64 062307; Kopszak et al 2020 J. Phys. A: Math. Theor. 53 395306), and the latter resolves some open questions raised in (Collins et al 2018 Linear Algebra Appl. 5
A unit-preserving and completely positive linear map, or a channel, Λ:A→Ain between C*-algebras A and Ain is called entanglement-breaking (EB) if ω◦(Λ⊗idB) is a separable state for any C*-algebra B and any state ω on the injective C*-tensor product Ain⊗B. In this paper, we establish the equivalence of the following conditions for a channel Λ with a quantum input space and with a general outcome C*-algebra, generalizing the known results in finite dimensions: (i) Λ is EB; (ii) Λ has a measurement-prepare form (Holevo form); (iii) n copies of Λ are compatible for all 2 ≤ n < ∞; (iv) countably infinite copies of Λ are compatible. By using this equivalence, we also show that the set of randomization-equivalence classes of normal EB channels with a fixed input von Neumann algebra is upper and lower Dedekind-closed, i.e., the supremum or infimum of any randomization-increasing or decreasing net of EB channels is also EB. As an example, we construct an injective normal EB channel with an arbitrary outcome operator algebra M acting on an infinite-dimensional separable Hilbert space by using the coherent states and the Bargmann measure.