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arXiv:1510.06466

Entanglement preserving maps and the universality of finite time disentanglement

Raphael C. Drumond, Cristhiano Duarte, Marcelo Terra Cunha, Maria Carolina Nemes

Published Oct 22, 2015• 0 citations
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Abstract

In this paper we investigate how common is the phenomenon of Finite Time Disentanglement (FTD) with respect to the set of quantum dynamics of bipartite quantum states with finite dimensional Hilbert spaces. Considering a quantum dynamics from a general sense, as just a continuous family of Completely Positive Trace Preserving maps (parametrized by the time variable) acting on the space of the bipartite systems, we conjecture that FTD does not happen only when all maps of the family are induced by local unitary operations. We prove that is the case for dynamics where all maps are induced by unitaries and, for pairs of qubits, where all maps are unital. Moreover, we prove some results about unitaries/CPTP maps preserving product/pure states.

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In this paper we investigate how common is the phenomenon of Finite Time Disentanglement (FTD) with respect to the set of quantum dynamics of bipartite quantum states with finite dimensional Hilbert spaces. Considering a quantum dynamics from a general sense, as just a continuous family of Completely Positive Trace Preserving maps (parametrized by the time variable) acting on the space of the bipartite systems, we conjecture that FTD does not happen only when all maps of the family are induced by local unitary operations. We prove that is the case for dynamics where all maps are induced by unitaries and, for pairs of qubits, where all maps are unital. Moreover, we prove some results about unitaries/CPTP maps preserving product/pure states.
Absolutely separable states ϱ remain separable under arbitrary unitary transformations . By example of a three qubit system we show that in a multipartite scenario neither full separability implies bipartite absolute separability nor the reverse statement holds. The main goal of the paper is to analyze quantum maps resulting in absolutely separable output states. Such absolutely separating maps affect the states in a way, when no Hamiltonian dynamics can make them entangled afterwards. We study the general properties of absolutely separating maps and channels with respect to bipartitions and multipartitions and show that absolutely separating maps are not necessarily entanglement breaking. We examine the stability of absolutely separating maps under a tensor product and show that is absolutely separating for any N if and only if Φ is the tracing map. Particular results are obtained for families of local unital multiqubit channels, global generalized Pauli channels, and combination of identity, transposition, and tracing maps acting on states of arbitrary dimension. We also study the interplay between local and global noise components in absolutely separating bipartite depolarizing maps and discuss the input states with high resistance to absolute separability.
In this paper we study quantum dynamical mappings called also quantum processes.The set of values of such mapping is a one-parameter family of completely positive trace-preserving linear operators defined on a finite-dimensional Hilbert space.In quantum information theory such operators are referred to as quantum channels.An important concept for quantum dynamical mappings is their divisibility.There are different types of this concept.The present paper deals with so-called completely positive divisible quantum processes.For two such processes, which are bijective and satisfy a commutativity condition, we construct a compound quantum process.It is shown that this compound quantum process is also completely positive divisible.Endowing a set of quantum channels with the norm topology, we consider continuous quantum processes and continuous completely positive evolutions.The latter are defined as two-parameter families of quantum channels satisfying additional properties.We prove that a continuous bijective completely positive divisible quantum process generates a continuous completely positive evolution.In order to illustrate the considered concepts and the results on them, we provide examples of quantum dynamical mappings with values in the set of qubit channels.In particular, a completely positive divisible compound quantum process is constructed for two bijective commuting quantum processes.Geometric and physical interpretations of this compound quantum process are given.
Consider a bipartite quantum system S=AB such that each part interacts only with its local environment. Under such circumstances, one expects that the entanglement between parts A and B does not increase after the time evolution. In fact, this is the case if the reduced dynamics of the system is given by EA⊗EB\mathcal{E}_{A}\otimes \mathcal{E}_{B}EA​⊗EB​, where EA\mathcal{E}_{A}EA​ and EB\mathcal{E}_{B}EB​ are quantum channels, i.e., completely positive trace-preserving maps. But, the reduced dynamics of the system may be given by a map as ΨA⊗ΨBΨ_{A}\otimes Ψ_{B}ΨA​⊗ΨB​, where ΨAΨ_{A}ΨA​ and ΨBΨ_{B}ΨB​ are local non-positive maps. Then, the entanglement between A and B can exceed its initial value, as was shown in the case studied by Jordan et al. [Phys. Rev. A 76, 022102 (2007)]. In this paper, we first explore the general circumstances under which one can find such cases as they found. Next, we introduce another general procedure which leads to local non-positive maps that exhibit entanglement exceeding. Furthermore, we demonstrate that this apparent entanglement increase in the two above procedures corresponds physically to the retrieval of pre-existing inaccessible entanglement stored across the global system-environment degrees of freedom, rather than genuine generation of the entanglement ex nihilo\textit{ex nihilo}ex nihilo.
Completely positive trace-preserving maps SSS, also known as quantum channels, arise in quantum physics as a description of how the density operator ρρρ of a system changes in a given time interval, allowing not only for unitary evolution but arbitrary operations including measurements or other interaction with an environment. It is known that if the Hilbert space H\mathscr{H}H that ρρρ acts on is finite-dimensional, then every SSS must have a fixed point, i.e., a density operator ρ0ρ_0ρ0​ with S(ρ0)=ρ0S(ρ_0)=ρ_0S(ρ0​)=ρ0​. In infinite dimension, SSS need not have a fixed point in general. However, we prove here the existence of a fixed point under a certain additional assumption which is, roughly speaking, that SSS leaves invariant a certain set of density operators with bounded ``cost'' of preparation. The proof is an application of the Schauder-Tychonoff fixed point theorem. Our motivation for this question comes from a proposal of Deutsch for how to define quantum theory in a space-time with closed timelike curves; our result supports the viability of Deutsch's proposal.
For a fixed pair of input and output states in the space HA of a system A, a quantum channel, i.e., a linear, completely positive and trace-preserving map, between them is not unique, in general. Here, this point is discussed specifically for a decoherence channel, which maps from a pure input state to a completely decoherent state like the thermal state. In particular, decoherence channels of two different types are analyzed: one is unital and the other is not, and both of them can be constructed through reduction of B in the total extended space HA⊗HB, where HB is the space of an ancillary system B that is a replica of A. The nonuniqueness is seen to have its origin in the unitary symmetry in the extended space. It is shown in an example of a two-qubit system how such symmetry is broken in the objective subspace HA due to entanglement between A and B. A comment is made on possible relevance of the present work to nanothermodynamics in view of quantum Darwinism.
We consider the decay of quantum entanglement quantified by the concurrence of a pair of two-level systems, each of which is interacting with a reservoir at finite temperature TTT. For a broad class of initially entangled states, we demonstrate that the system always becomes disentangled in a finite time, i.e., ``entanglement sudden death'' occurs. This class includes all states which previously had been found to have long-lived entanglement in zero-temperature reservoirs. Our general result is illustrated by an example.