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arXiv:1910.04647 · Quantum

Entanglement-breaking superchannels

Senrui Chen, Eric Chitambar

Published Jul 16, 2020• 15 citations
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Abstract

In this paper we initiate the study of entanglement-breaking (EB) superchannels. These are processes that always yield separable maps when acting on one side of a bipartite completely positive (CP) map. EB superchannels are a generalization of the well-known EB channels. We give several equivalent characterizations of EB supermaps and superchannels. Unlike its channel counterpart, we find that not every EB superchannel can be implemented as a measure-and-prepare superchannel. We also demonstrate that many EB superchannels can be superactivated, in the sense that they can output non-separable channels when wired in series.We then introduce the notions of CPTP- and CP-complete images of a superchannel, which capture deterministic and probabilistic channel convertibility, respectively. This allows us to characterize the power of EB superchannels for generating CP maps in different scenarios, and it reveals some fundamental differences between channels and superchannels. Finally, we relax the definition of separable channels to include (p,q) -non-entangling channels, which are bipartite channels that cannot generate entanglement using p - and q -dimensional ancillary systems. By introducing and investigating k -EB maps, we construct examples of (p,q) -EB superchannels that are not fully entanglement breaking. Partial results on the characterization of (p,q) -EB superchannels are also provided.

DOI
10.22331/q-2020-07-16-299

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In this paper we initiate the study of entanglement-breaking (EB) superchannels. These are processes that always yield separable maps when acting on one side of a bipartite completely positive (CP) map. EB superchannels are a generalization of the well-known EB channels. We give several equivalent characterizations of EB supermaps and superchannels. Unlike its channel counterpart, we find that not every EB superchannel can be implemented as a measure-and-prepare superchannel. We also demonstrate that many EB superchannels can be superactivated, in the sense that they can output non-separable channels when wired in series.We then introduce the notions of CPTP- and CP-complete images of a superchannel, which capture deterministic and probabilistic channel convertibility, respectively. This allows us to characterize the power of EB superchannels for generating CP maps in different scenarios, and it reveals some fundamental differences between channels and superchannels. Finally, we relax the definition of separable channels to include (p,q) -non-entangling channels, which are bipartite channels that cannot generate entanglement using p - and q -dimensional ancillary systems. By introducing and investigating k -EB maps, we construct examples of (p,q) -EB superchannels that are not fully entanglement breaking. Partial results on the characterization of (p,q) -EB superchannels are also provided.
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.
Abstract Entanglement breaking (EB) channels, as completely positive and trace-preserving linear operators, sever the entanglement between the input system and other systems. In the realm of infinite-dimensional systems, a related concept known as strongly EB (SEB) channels emerges. This paper delves into characterizations of SEB channels, delineating sufficient conditions for a channel to be classified as SEB, especially with respect to the commutativity of its range. Moreover, we demonstrate that every closed self-adjoint subspace of trace-zero operators, with the trace norm, is the null space of a SEB channel.
We introduce a 3-parameter class of maps (1) acting on a bipartite system which are a natural generalisation of the depolarizing channel (and include it as a special case). Then, we find the exact regions of the parameter space that alternatively determine a positive, completely positive, entanglement-breaking, or entanglement-annihilating map. This model displays a much richer behaviour than the one shown by a simple depolarizing channel, yet it stays exactly solvable. As an example of this richness, positive partial transposition but not entanglement-breaking maps is found in Theorem 2. A simple example of a positive yet indecomposable map is provided (see the Remark at the end of Section IV). The study of the entanglement-annihilating property is fully addressed by Theorem 7. Finally, we apply our results to solve the problem of the entanglement annihilation caused in a bipartite system by a tensor product of local depolarizing channels. In this context, a conjecture posed in the work of Filippov [J. Russ. Laser Res. 35, 484 (2014)] is affirmatively answered, and the gaps that the imperfect bounds of Filippov and Ziman [Phys. Rev. A 88, 032316 (2013)] left open are closed. To arrive at this result, we furthermore show how the Hadamard product between quantum states can be implemented via local operations.
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.
Quantum superchannels are maps whose input and output are quantum channels. Rather than taking the domain to be the space of all linear maps we motivate and define superchannels on the operator system spanned by quantum channels. Extension theorems for completely positive maps allow us to apply the characterisation theorem for superchannels to this smaller set of maps. These extensions are non unique, showing two different superchannels act the same on all input quantum channels, and so this new definition on the smaller domain captures more precisely the action of superchannels as transformations between quantum channels. The non uniqueness can affect the auxilliary dimension needed for the characterisation as well as the tensor product of the superchannels.
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.