Entanglement preserving maps and the universality of finite time disentanglement (original) (raw)

Bipartite entanglement-annihilating maps: Necessary and sufficient conditions

We fully characterize bipartite entanglement-annihilating (EA) channels that destroy entanglement of any state shared by subsystems and, thus, should be avoided in any entanglement-enabled experiment. Our approach relies on extending the problem to EA positive maps, the cone of which remains invariant under concatenation with partially positive maps. Due to this invariancy, positive EA maps adopt a well characterization and their intersection with completely positive tracepreserving maps results in the set of EA channels. In addition to a general description, we also provide sufficient operational criteria revealing EA channels. They have a clear physical meaning since the processes involved contain stages of classical information transfer for subsystems. We demonstrate the applicability of derived criteria for local and global depolarizing noises, and specify corresponding noise levels beyond which any initial state becomes disentangled after passing the channel. The robustness of some entangled states is discussed.

Eventually entanglement breaking maps

Journal of Mathematical Physics, 2018

We analyze linear maps on matrix algebras that become entanglement breaking after composing a finite or infinite number of times with themselves. This means that the Choi matrix of the iterated linear map becomes separable in the tensor product space. If a linear map becomes entanglement breaking after finitely many iterations, we say the map has a finite index of separability. In particular we show that every unital PPT-channel has a finite index of separability and that the class of unital channels that have finite index of separability is a dense subset of the unital channels. We construct concrete examples of maps which are not PPT but have finite index of separability. We prove that there is a large class of unital channels that are asymptotically entanglement breaking. This analysis is motivated by the PPT-squared conjecture made by M. Christandl that says every PPT channel, when composed with itself, becomes entanglement breaking.

On the quantification of entanglement in infinite-dimensional quantum systems

2001

We investigate entanglement measures in the infinite-dimensional regime. First, we discuss the peculiarities that may occur if the Hilbert space of a bi-partite system is infinitedimensional, most notably the fact that the set of states with infinite entropy of entanglement is trace-norm dense in state space, implying that in any neighbourhood of every product state lies an arbitrarily strongly entangled state. The starting point for a clarification of this counterintuitive property is the observation that if one imposes the natural and physically reasonable constraint that the mean energy is bounded from above, then the entropy of entanglement becomes a trace-norm continuous functional. The considerations will then be extended to the asymptotic limit, and we will prove some asymptotic continuity properties. We proceed by investigating the entanglement of formation and the relative entropy of entanglement in the infinite-dimensional setting. Finally, we show that the set of entangled states is still tracenorm dense in state space, even under the constraint of a finite mean energy.

On the Relation between States and Maps in Infinite Dimensions

Open Systems & Information Dynamics, 2007

Relations between states and maps, which are known for quantum systems in finitedimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators \({\mathcal{L}}_2({\mathcal{H}}_2 , {\mathcal{H}}_1 )\) and the corresponding tensor products \({\mathcal{H}}_1\otimes{\mathcal{H}}_2^*\) of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map \({\mathcal{C}} : {\mathcal{L}}_1({\mathcal{L}}_2({\mathcal{H}}_2, {\mathcal{H}}_1)) \to {\mathcal{L}}_\infty({\mathcal{L}}({\mathcal{H}}_2), {\mathcal{L}}_1({\mathcal{H}}_1))\) from trace-class operators on \({\mathcal{L}}_2 ({\mathcal{H}}_2, {\mathcal{H}}_1)\) (with the nuclear norm) into compact operators mapping the space of all bounded operators on \({\mathcal{H}}_2\) into trace class operators on \({\mathcal{H}}_1\) (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.

Classification of nonasymptotic bipartite pure-state entanglement transformations

Physical Review A, 2002

Let {|ψ , |φ } be an incomparable pair of states (|ψ |φ), i.e., |ψ and |φ cannot be transformed to each other with probability one by local transformations and classical communication (LOCC). We show that incomparable states can be multiple-copy transformable, i.e., there can exist a k, such that |ψ ⊗k+1 → |φ ⊗k+1 , i.e., k + 1 copies of |ψ can be transformed to k + 1 copies of |φ with probability one by LOCC but |ψ ⊗n |φ ⊗n ∀n ≤ k. We call such states k-copy LOCC incomparable. We provide a necessary condition for a given pair of states to be k-copy LOCC incomparable for some k. We also show that there exist states that are neither k-copy LOCC incomparable for any k nor catalyzable even with multiple copies. We call such states strongly incomparable. We give a sufficient condition for strong incomparability. We demonstrate that the optimal probability of a conclusive transformation involving many copies, pmax |ψ ⊗m → |φ ⊗m can decrease exponentially with the number of source states m, even if the source state has more entropy of entanglement. We also show that the probability of a conclusive conversion might not be a monotonic function of the number of copies. Fascinating developments in quantum information theory [1] and quantum computing [2] during the past decade has led us to view entanglement as a valued physical resource. Consequently, recent studies have largely been devoted towards its quantification in appropriate limits (finite or asymptotic), optimal manipulation, and transformation properties under local operations and classical communication (LOCC) [3, 4, 5, 6, 7, 8]. Since the specific tasks that can be accomplished with entanglement as a resource is closely related to its transformation properties, it is of importance to know what transformations are allowed under LOCC. Suppose Alice and Bob share a pure state |ψ (source state), which they wish to convert to another entangled state |φ (target state) under LOCC. A necessary and sufficient condition for this transformation to be possible with certainty (denoted by |ψ → |φ) has been obtained by Nielsen [3]. If such a deterministic transformation is not possible but |ψ has at least as many Schmidt coefficients as |φ , then one

On Positive Maps, Entanglement and Quantization

Open Systems & Information Dynamics (OSID), 2004

We outline the scheme for quantization of classical Banach space results associated with some prototypes of dynamical maps and we describe the quantization of correlations. A relation between these two areas is discussed.

Unitary quantum gates, perfect entanglers, and unistochastic maps

Physical Review A, 2013

Non-local properties of ensembles of quantum gates induced by the Haar measure on the unitary group are investigated. We analyze the entropy of entanglement of a unitary matrix U equal to the Shannon entropy of the vector of singular values of the reshuffled matrix. Averaging the entropy over the Haar measure on U (N 2) we find its asymptotic behaviour. For two-qubit quantum gates we derive the induced probability distribution of the interaction content and show that the relative volume of the set of perfect entanglers reads 8/3π ≈ 0.85. We establish explicit conditions under which a given one-qubit bistochastic map is unistochastic, so it can be obtained by partial trace over a one-qubit environment initially prepared in the maximally mixed state.

On Quantum Correlations and Positive Maps

Letters in Mathematical Physics, 2004

We present a discussion on local quantum correlations and their relations with entanglement. We prove that vanishing coefficient of quantum correlations implies separability. The new results on locally decomposable maps which we obtain in the course of proof also seem to be of independent interest.

Trace decreasing quantum dynamical maps: Divisibility and entanglement dynamics

2021

Trace decreasing quantum operations naturally emerge in experiments involving postselection. However, the experiments usually focus on dynamics of the conditional output states as if the dynamics were trace preserving. Here we show that this approach leads to incorrect conclusions about the dynamics divisibility, namely, one can observe an increase in the trace distance or the system-ancilla entanglement although the trace decreasing dynamics is completely positive divisible. We propose solutions to that problem and introduce proper indicators of the information backflow and the indivisibility. We also review a recently introduced concept of the generalized erasure dynamics that includes more experimental data in the dynamics description. The ideas are illustrated by explicit physical examples of polarization dependent losses.

Absolutely separating quantum maps and channels

We consider quantum maps resulting in absolutely separable output states that remain separable under arbitrary unitary transformations U U †. Such absolutely separating maps affect the states in a way, when no Hamiltonian dynamics can make them entangled afterwards. We study general properties of absolutely separating maps and channels with respect to bipartitions and mul-tipartitions and show that absolutely separating maps are not necessarily entanglement breaking. 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.