Suppose Alice and Bob try to transform an entangled state shared between them into another one by local operations and classical communications. Then in general a certain amount of entanglement contained in the initial state will decrease in the process of transformation. However, an interesting phenomenon called partial entanglement recovery shows that it is possible to recover some amount of entanglement by adding another entangled state and transforming the two entangled states collectively.
In this paper we are mainly concerned with the feasibility of partial entanglement recovery. The basic problem we address is whether a given state is useful in recovering entanglement lost in a specified transformation. In the case where the source and target states of the original transformation satisfy the strict majorization relation, a necessary and sufficient condition for partial entanglement recovery is obtained. For the general case we give two sufficient conditions. We also give an efficient algorithm for the feasibility of partial entanglement recovery in polynomial time.
As applications, we establish some interesting connections between partial entanglement recovery and the generation of maximally entangled states, quantum catalysis, mutual catalysis, and multiple-copy entanglement transformation.
When an entangled state is transformed into another one with probability one by local operations and classical communication, the quantity of entanglement decreases. This letter shows that entanglement lost in the manipulation can be partially recovered by an auxiliary entangled pair. As an application, a maximally entangled pair can be obtained from two partially entangled pairs with probability one. Finally, this recovery scheme reveals a fundamental property of entanglement relevant to the existence of incomparable states.
Any deterministic bipartite entanglement transformation involving finite copies of pure states and carried out using local operations and classical communication (LOCC) results in a net loss of entanglement. We show that for almost all such transformations, partial recovery of lost entanglement is achievable by using $2 \times 2$ auxiliary entangled states, no matter how large the dimensions of the parent states are. For the rest of the special cases of deterministic LOCC transformations, we show that the dimension of the auxiliary entangled state depends on the presence of equalities in the m…
The need to retain the relative phases in quantum mechanics implies an addition law parametrized by a phase of two density operators required for the purification of a density matrix. This is shown with quantum tomography and the Wigner function. Entanglement is determined in terms of phase dependent multiplication.
In many applications entanglement must be distributed through noisy communication channels that unavoidably degrade it. Entanglement cannot be generated by local operations and classical communication (LOCC), implying that once it has been distributed it is not possible to recreate it by LOCC. Recovery of entanglement by purely local control is however not forbidden in the presence of non-Markovian dynamics, and here we demonstrate in two all-optical experiments that such entanglement restoration can even be achieved on-demand. First, we implement an open-loop control scheme based on a purely …
By using photon pairs created in parametric down conversion, we report on an experiment, which demonstrates that measurement can recover the quantum entanglement of two qubit system in a pure dephasing environment. The concurrence of the final state with and without measurement are compared and analyzed. Furthermore, we verify that recovered states can still violate Bell's inequality, that is, to say, such recovered states exhibit nonlocality. In the context of quantum entanglement, sudden death and rebirth provide clear evidence, which verifies that entanglement dynamics of the system is sens…
The task of preserving entanglement against noises is of crucial importance for both quantum communication and quantum information transfer. To this aim, quantum error correction (QEC) codes may be employed to compensate, at least partially, the detriments induced by environmental noise that can be modelled as a bit-flip or a phase-flip error channel. In this paper we investigate the effects of the simple three-qubit QEC codes to restore entanglement and nonlocality in a two-qubit system and consider two practical applications: superdense coding and quantum teleportation. Though the considered…
We extend a procedure exploiting spatial indistinguishability of identical particles to recover the spoiled entanglement between two qubits interacting with Markovian noisy environments. Here, the spatially localized operations and classical communication (sLOCC) operational framework is used to activate the entanglement restoration from the indistinguishable constituents. We consider the realistic scenario where noise acts for the whole duration of the process. Three standard types of noises are considered: a phase damping, a depolarizing, and an amplitude damping channel. Within this general…
We investigate the possibility to restore genuine tripartite entanglement under local amplitude damping. We show that it is possible to protect genuine entanglement using CNOT and Hadamard gates. We analyze several ordering of such recovery operations. We find that for recovery operations applied after exposing qubits to decoherence, there is no enhancement in lifetime of genuine entanglement. Actual retrieval of entanglement is only possible when reversal scheme is applied before and after the decoherence process. We find that retrieval of entanglement for mixture of $|\widetilde{W}\rangle$ s…
We extend the recent bounds of Sason and Verdú relating Rényi entropy and Bayesian hypothesis testing [arXiv:1701.01974] to the quantum domain and show that they have a number of different applications. First, we obtain a sharper bound relating the optimal probability of correctly distinguishing elements of an ensemble of states to that of the pretty good measurement, and an analogous bound for optimal and pretty good entanglement recovery. Second, we obtain bounds relating optimal guessing and entanglement recovery to the fidelity of the state with a product state, which then leads to tight t…
When Classical Correlations Certify Entanglement Recovery
The University of Tokyo（东京大学）；Nagoya University（名古屋大学）；Institute of Science Tokyo（东京科学大学）；Kyoto University（京都大学）；RIKEN Center for Quantum Computing (RQC)（理化学研究所量子计算中心）
We consider composability of quantum channels from a limited amount of entanglement via local operations and classical communication (LOCC). We show that any $k$-partially entanglement breaking channel can be composed from an entangled state with Schmidt number of $k$ via one-way LOCC. From the entanglement assisted construction we can reach an alternative definition of partially entanglement breaking channels.
Beyond Maximal Entanglement: Exact Resources for Multiparty Encrypted Quantum Cloning
S. N. Bose National Centre for Basic Sciences（S.N. 玻色基础科学国家中心）；Centre for Interdisciplinary Areas in Quantum Computing, Indian Institute of Technology Indore（印度印多尔理工学院量子计算跨学科领域中心）；Department of Physics, Indian Institute of Technology Indore（印度印多尔理工学院物理系）
Determination of the path taken by a quantum particle leads to a suppression of interference and to a classical behavior. We employ here a quantum 'which path' detector to perform accurate path determination in a two-path-electron-interferometer; leading to full suppression of the interference. Following the dephasing process we recover the interference by measuring the cross-correlation between the interferometer and detector currents. Under our measurement conditions every interfering electron is dephased by approximately a single electron in the detector - leading to mutual entanglement of …
Complementarity and entanglement are fundamental features of Quantum Mechanics that were recently related in triality equalities that involve quantum coherence, the wave aspect of a qubit, and quantum predictability and quantum entanglement, the particle aspect of a qubit. In this article, we give an operational connection between redictability and entanglement in entanglement swapping from initially partially entangled states. For this, we show that the predictability of the pre-measurement one-qubit density matrix is directly related to the probability of the partially entangled component of…
Using well known duality between quantum maps and states of composite systems we introduce the notion of Schmidt number of a quantum channel. It enables one to define classes of quantum channels which partially break quantum entanglement. These classes generalize the well known class of entanglement breaking channels.
Squashed entanglement [Christandl and Winter, J. Math. Phys. 45(3):829-840 (2004)] is a monogamous entanglement measure, which implies that highly extendible states have small value of the squashed entanglement. Here, invoking a recent inequality for the quantum conditional mutual information [Fawzi and Renner, Commun. Math. Phys. 340(2):575-611 (2015)] greatly extended and simplified in various work since, we show the converse, that a small value of squashed entanglement implies that the state is close to a highly extendible state. As a corollary, we establish an alternative proof of the fait…
We show a general approach for detecting genuine multipartite entanglement (GME) and partial inseparability in many-body-systems by means of macroscopic observables (such as the energy) only. We show that the obtained criteria, the "GME gap" and "the k-entanglement gap", detect large areas of genuine multipartite entanglement and partial entanglement in typical many body states, which are not detected by other criteria. As genuine multipartite entanglement is a necessary property for several quantum information theoretic applications such as e.g. secret sharing or certain kinds of quantum comp…
Partial teleportation of entanglement is to teleport one particle of an entangled pair through a quantum channel. This is conceptually equivalent to quantum swapping. We consider the partial teleportation of entanglement in the noisy environment, employing the Werner-state representation of the noisy channel for the simplicity of calculation. To have the insight of the many-body teleportation, we introduce the measure of correlation information and study the transfer of the correlation information and entanglement. We find that the fidelity gets smaller as the initial-state is entangled more f…
Bound entanglement with a nonpositive partial transposition (NPT) does not exist. For any NPT entangled state a distillation procedure can be based on a certain number of copies. This number is the minimal Schmidt rank of a pure state needed to witness the NPT entanglement under study.
It is known that the partial entanglement/separability violates distributive rules with respect to the operations of taking convex hull and intersection. In this note, we give criteria for three qubit partially entangled states arising from distributive rules, together with the corresponding witnesses. The criteria will be given in terms of diagonal and anti-diagonal entries. They actually characterize those partial entanglement completely when all the entries are zero except for diagonal and anti-diagonal entries. Important states like Greenberger-Horne-Zeilinger diagonal states fall down in …
It is challenging task to detect and measure genuine multipartite entanglement. We investigate the problem by considering the average based positive partial transposition(PPT) criterion and the realignment criterion. Sufficient conditions for detecting genuine tripartite entanglement are presented. We also derive lower bounds for the genuine tripartite entanglement concurrence with respect to the conditions. While the PPT criterion and the realignment criterion are powerful for detecting bipartite entanglement and for providing lower bounds of bipartite concurrences, our results give an effect…
We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT-operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this easily computable entanglement measure. As examples we discuss general Werner states and arbitrary bi-partite Gaussian states. Equipped with this result we then prove that for the anti-symmetric Werner state PPT-cost and PPT-entanglement of distillation coincide giving the first example of a truly mixed state for which entanglement ma…
The initial entanglement shared between inertial and accelerated observers degrades due to the influence of the Unruh effect. Here, we show that the Unruh effect can be completely eliminated by the technique of partial measurement. The lost entanglement could be entirely retrieved or even amplified, which is dependent on whether the optimal strength of reversed measurement is \emph{state-independent} or \emph{state-dependent}. Our work provides a novel and unexpected method to recover the lost entanglement under Unruh decoherence and exhibits the ability of partial measurement as an important …
We found three qubit Greenberger-Horne-Zeilinger diagonal states which tells us that the partial separability of three qubit states violates the distributive rules with respect to the two operations of convex sum and intersection. The gaps between the convex sets involving the distributive rules are of nonzero volume.
We construct seven kinds of three-qubit biseparable states to show that every class of biseparable states in the partial separability classification proposed by Szalay and Kökényesi [S. Szalay and Z. Kökényesi, Phys. Rev. A 86, 032341 (2012)] is nonempty.
We apply and extend the theory of universal recovery channels from quantum information theory to address the problem of entanglement wedge reconstruction in AdS/CFT. It has recently been proposed that any low-energy local bulk operators in a CFT boundary region's entanglement wedge can be reconstructed on that boundary region itself. Existing work arguing for this proposal relies on algebraic consequences of the exact equivalence between bulk and boundary relative entropies, namely the theory of operator algebra quantum error correction. However, bulk and boundary relative entropies are only a…
Complete complementarity relations, as e.g. $P(ρ_{A})^{2} + C(ρ_{A})^{2} + E(|Ψ\rangle_{AB})^{2}=1$, constrain the local predictability, $P$, and local coherence, $C$, and the entanglement, $E$, of bipartite pure states. For pairs of qubits prepared initially in a particular class of partially entangled pure states with null local coherence, these relations were used in Ref. [Phys. Lett. A, 451, 128414 (2022)] to provide an operational connection between local predictability of the pre-measurement states with the probability of the maximally entangled components of the states after the Bell-ba…
We construct tri-qubit genuinely entangled states which have positive partial transposes with respect to bi-partition of systems. These examples disprove a conjecture [L. Novo, T. Moroder and O. G\" uhne, Phys.Rev.A {88}, 012305 (2013)] which claims that PPT mixtures are necessary and sufficient for biseparability of three qubits.
We develop a theory of a quantifier of bipartite Gaussian entanglement called Gaussian intrinsic entanglement (GIE) which was proposed recently in [L. Mišta {\it et al.}, Phys. Rev. Lett. {\bf 117}, 240505 (2016)]. The GIE provides a compromise between computable and physically meaningful entanglement quantifiers and so far it was calculated for two-mode Gaussian states including all symmetric partial minimum-uncertainty states, weakly mixed asymmetric squeezed thermal states with partial minimum uncertainty, and weakly mixed symmetric squeezed thermal states. We improve the method of derivati…
We demonstrate the irreversibility of asymptotic entanglement manipulation under quantum operations that completely preserve the positivity of partial transpose (PPT), resolving a major open problem in quantum information theory. Our key tool is a new efficiently computable additive lower bound for the asymptotic relative entropy of entanglement with respect to PPT states, which can be used to evaluate the entanglement cost under local operations and classical communication (LOCC). We find that for any rank-two mixed state supporting on the $3\otimes3$ antisymmetric subspace, the amount of dis…
In this Letter, we show that the fulfillment of uncertainty relations is a sufficient criterion for a quantum-mechanically permissible state. We specifically construct two pseudo-spin observables for an arbitrary non-positive Hermitian matrix whose uncertainty relation is violated. This method enables us to systematically derive separability conditions for all negative partial-transpose states in experimentally accessible forms. In particular, generalized entanglement criteria are derived from the Schrodinger-Robertson inequalities for bipartite continuous-variable states.
We present a nonlocal entanglement concentration scheme for reconstructing some maximally entangled multipartite states from partially entangled ones by exploiting cross-Kerr nonlinearities to distinguish the parity of two polarization photons. Compared with the entanglement concentration schemes based on two-particle collective unitary evolution, this scheme does not require the parties to know accurately information about the partially entangled states--i.e., their coefficients. Moreover, it does not require the parties to possess sophisticated single-photon detectors, which makes this proto…
This paper defines the fidelity of recovery of a quantum state on systems $A$, $B$, and $C$ as a measure of how well one can recover the full state on all three systems if system $A$ is lost and a recovery operation is performed on system $C$ alone. The surprisal of the fidelity of recovery (its negative logarithm) is an information quantity which obeys nearly all of the properties of the conditional quantum mutual information $I(A;B|C)$, including non-negativity, monotonicity with respect to local operations, duality, invariance with respect to local isometries, a dimension bound, and continu…
We present the generalized version of Wang's protocol[A.M.Wang, Phys.Rev.A 74,032317 (2006)] for the remote implementation(sometimes referred to as quantum remote control) of partially unknown quantum operations. The protocol only requires no more than half of the entanglements used in Bidirectional Quantum State Teleportation. We also propose a protocol for another form of quantum remote control. It can remotely implement a unitary operation which is a combination of the projective representations of a group. Moreover, we prove that the Schmidt rank of the entanglements cannot not be less tha…
We provide an evidence for the validity of strong monogamy inequality of multi-party quantum entanglement using square of convex-roof extended negativity(SCREN). We first consider a large class of multi-qudit mixed state that are in a partially coherent superposition of a generalized W-class state and the vacuum, and provide some useful properties about this class of states. We show that monogamy inequality of multi-qudit entanglement in terms of SCREN holds for this class of states. We further show that SCREN strong monogamy inequality of multi-qudit entanglement also holds for this class of …
A general and computable criterion for k-(in)separability in continuous multipartite quantum systems is presented. The criterion can be experimentally implemented with a finite and comparatively low number of local observables. We discuss in detail how the detection quality can be optimised.
In this work we study the nature of correlations among mixed states in the setup of two symmetric strips. We use various tools to determine how the bulk geometry could be reconstructed from the boundary mixed information. These tools would be the modular Hamiltonian and modular flow, OPE blocks, quantum recovery channels such as Petz map, Uhlmann holonomy and Wilson lines. We comment on the similarities and connections between these approaches in our symmetric setup of a mixed system. Specially, we use parameters such as dissipation which is being modeled by the mass of graviton, and also the …
Quantum entanglement is a critical resource for quantum information and quantum computation. However, entanglement of a quantum system is subjected to change due to the interaction with the environment. One typical result of the interaction is the amplitude damping that usually results in the reduction of the entanglement. Here we propose a protocol to protect quantum entanglement from the amplitude damping by applying Hadamard and CNOT gates. As opposed to some recently studied methods, the scheme presented here does not require weak measurement in the reversal process, leading to a faster re…
If two separated observers are supplied with entanglement, in the form of $n$ pairs of particles in identical partly-entangled pure states, one member of each pair being given to each observer; they can, by local actions of each observer, concentrate this entanglement into a smaller number of maximally-entangled pairs of particles, for example Einstein-Podolsky-Rosen singlets, similarly shared between the two observers. The concentration process asymptotically conserves {\em entropy of entanglement}---the von Neumann entropy of the partial density matrix seen by either observer---with the yiel…
For the certification and benchmarking of medium-size quantum devices efficient methods to characterize entanglement are needed. In this context, it has been shown that locally randomized measurements on a multiparticle quantum system can be used to obtain valuable information on the so-called moments of the partially transposed quantum state. This allows one to infer some separability properties of a state, but how to use the given information in an optimal and systematic manner has yet to be determined. We propose two general entanglement detection methods based on the moments of the partial…
In order to classify partial entanglement of multi-partite states, it is natural to consider the convex hulls, intersections and differences of basic convex cones obtained from partially separable states with respect to partitions of systems. In this paper, we consider convex cones consisting of X-shaped three qubit states arising in this way. The class of X-shaped states includes important classes like Greenberger-Horne-Zeilinger diagonal states. We find all the extreme rays of those convex cones to exhibit corresponding partially separable states. We also give characterizations for those con…
We study the information content of the reduced density matrix of a region in quantum field theory that cannot be recovered from its subregion density matrices. We reconstruct the density matrix from its subregions using two approaches: scaling maps and recovery maps. The vacuum of a scale-invariant field theory is the fixed point of both transformations. We define the entanglement of scaling and the entanglement of recovery as measures of entanglement that are intrinsic to the continuum limit. Both measures increase monotonically under the renormalization group flow. This provides a unifying …
We demonstrate that even under positive partial transpose preserving operations in an asymptotic setting GHZ and W states are not reversibly interconvertible. We investigate the structure of minimal reversible entanglement generating set (MREGS) for tri-partite states under positive partial transpose (ppt) preserving operations. We demonstrate that the set consisting of W and EPR states alone cannot be an MREGS. In this context we prove the surprising result that the relative entropy of entanglement can be strictly sub-additive for certain pure tri-partite states which is crucial to keep open …
We present a protocol to simulate the correlations implied by nonmaximally entangled two qubit states. We extend this protocol to simulate the non-local part of these correlations. These protocols use single cbit communication and a single use of Millionaire box (M-box). To the best of our knowledge, these resources are weaker than those used in previous protocols using classical communication.
We explore the subtle relationships between partial separability and entanglement of subsystems in multiqubit quantum states and give experimentally accessible conditions that distinguish between various classes and levels of partial separability in a hierarchical order. These conditions take the form of bounds on the correlations of locally orthogonal observables. Violations of such inequalities give strong sufficient criteria for various forms of partial inseparability and multiqubit entanglement. The strength of these criteria is illustrated by showing that they are stronger than several ot…
We derive necessary and sufficient conditions for the approximate correctability of a quantum code, generalizing the Knill-Laflamme conditions for exact error correction. Our measure of success of the recovery operation is the worst-case entanglement fidelity of the overall process. We show that the optimal recovery fidelity can be predicted exactly from a dual optimization problem on the environment causing the noise. We use this result to obtain an easy-to-calculate estimate of the optimal recovery fidelity as well as a way of constructing a class of near-optimal recovery channels that work …
