Michal Horodecki, Pawel Horodecki, Ryszard Horodecki
We apply the inseparability criterion for $2 \times 2$ systems, local filtering and Bennett et al. purification protocol [Phys. Rev. Lett. {\bf 76}, 722 (1996)] to show how to distill {\it any} inseparable $2\times 2$ system. The extended protocol is illustrated geometrically by means of the state parameters in the Hilbert-Schmidt space.
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We prove that all inseparable Gaussian states of two modes can be distilled into maximally entangled pure states by local operations. Using this result we show that a bipartite Gaussian state of arbitrarily many modes can be distilled if and only if its partial transpose is not positive.
We present a family of 3--qubit states to which any arbitrary state can be depolarized. We fully classify those states with respect to their separability and distillability properties. This provides a sufficient condition for nonseparability and distillability for arbitrary states. We generalize our results to $N$--particle states.
Genuineness and distillability of entanglement play a key role in quantum information tasks, and they are easily disturbed by the noise. We construct a family of multipartite states without genuine entanglement and distillability sudden death across every bipartition, respectively. They are realized by establishing the noise as the multipartite high dimensional Pauli channels. Further, we construct a locally unitary channel as another noise such that the multipartite Greenberger-Horne-Zeilinger state becomes the D{ü}r's multipartite state. We also show that the quantum information masking stil…
The relation between Bell inequalities with two two-outcome measurements per site and distillability is analyzed in systems of an arbitrary number of quantum bits. We observe that the violation of any of these inequalities by a quantum state implies that pure-state entanglement can be distilled from it. The corresponding distillation protocol may require that some of the parties join into several groups. We show that there exists a link between the amount of the Bell inequality violation and the size of the groups they have to form for distillation. Thus, a strong violation is always sufficien…
Bipartite quantum states are classified into three categories: separable states, bound entangled states, and free entangled states. It is of great importance to characterize these families of states for the development of quantum information science. In this paper, I show that the separable states and the bound entangled states have a common spectral property. More precisely, I prove that for undistillable -- separable and bound entangled -- states, the eigenvalue vector of the global system is majorized by that of the local system. This result constitutes a new sufficient condition for distil…
We construct a class of entangled states in $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$ quantum systems with $dim\mathcal{H}_{A}=dim\mathcal{H}_{B}=dim\mathcal{H}_{C}=2$ and classify those states with respect to their distillability properties. The states are bound entanglement for the bipartite split$(AB)-C$. The states are NPT entanglement and $1$-copy undistillable for the bipartite splits $A-(BC)$ and $B-(AC)$. Moreover, we generalize the results of $2\otimes2\otimes2$ systems to the case of $2n\otimes 2n\otimes2n$ systems.
Entanglement and nonlocality are both fundamental aspects of quantum theory, and play a prominent role in quantum information science. The exact relation between entanglement and nonlocality is however still poorly understood. Here we make progress in this direction by showing that, contrary to what previous work suggested, quantum nonlocality does not imply entanglement distillability. Specifically, we present analytically a 3-qubit entangled state that is separable along any bipartition. This implies that no bipartite entanglement can be distilled from this state, which is thus fully bound e…
An important open problem in quantum information theory is the question of the existence of NPT bound entanglement. In the past years, little progress has been made, mainly because of the lack of mathematical tools to address the problem. (i) In an attempt to overcome this, we show how the distillability problem can be reformulated as a special instance of the separability problem, for which a large number of tools and techniques are available. (ii) Building up to this we also show how the problem can be formulated as a Schmidt number problem. (iii) A numerical method for detecting distillabil…
We show that all quantum states that do not have a positive partial transpose are distillable via channels, which preserve the positivity of the partial transpose. The question whether NPT bound entanglement exist is therefore closely related to the connection between the set of separable superoperators and PPT-preserving maps.
We present an abstract formulation of the so-called Innsbruck-Hannover programme that investigates quantum correlations and entanglement in terms of convex sets. We present a unified description of optimal decompositions of quantum states and the optimization of witness operators that detect whether a given state belongs to a given convex set. We illustrate the abstract formulation with several examples, and discuss relations between optimal entanglement witnesses and n-copy non-distillable states with non-positive partial transpose.
We present a lower bound of concurrence for arbitrary dimensional bipartite quantum states. This lower bound may be used to improve all the known lower bounds of concurrence. Moreover, the lower bound gives rise to an operational sufficient criterion of distillability of quantum entanglement. The significance of our result is illustrated by quantitative evaluation of entanglement for entangled states that fail to be identified by the usual concurrence estimation method, and by showing the distillability of mixed states that can not be recognized by other distillability criteria.
We consider quantum systems composed of $N$ qubits, and the family of all Bell's correlation inequalities for two two-valued measurements per site. We show that if a $N$-qubit state $ρ$ violates any of these inequalities, then it is at least bipartite distillable. Indeed there exists a link between the amount of Bell's inequality violation and the degree of distillability. Thus, we strengthen the interpretation of Bell's inequalities as detectors of useful entanglement.
A multipartite quantum state violates a Bell inequality asymptotically if, after jointly processing by general local operations an arbitrarily large number of copies of it, the result violates the inequality. In the bipartite case we show that asymptotic violation of the CHSH-inequality is equivalent to distillability. Hence, bound entangled states do not violate it. In the multipartite case we consider the complete set of full-correlation Bell inequalities with two dichotomic observables per site, also called WWZB-inequalities. We show that asymptotic violation of any of these inequalities by…
The hypothetical nonlocal box (\textsf{NLB}) proposed by Popescu and Rohrlich allows two spatially separated parties, Alice and Bob, to exhibit stronger than quantum correlations. If the generated correlations are weak, they can sometimes be distilled into a stronger correlation by repeated applications of the \textsf{NLB}. Motivated by the limited distillability of \textsf{NLB}s, we initiate here a study of the distillation of correlations for nonlocal boxes that output quantum states rather than classical bits (\textsf{qNLB}s). We propose a new protocol for distillation and show that it asym…
Quantum mechanics is already 100 years old, but remains alive and full of challenging open problems. On one hand, the problems encountered at the frontiers of modern theoretical physics like Quantum Gravity, String Theories, etc. concern Quantum Theory, and are at the same time related to open problems of modern mathematics. But even within non-relativistic quantum mechanics itself there are fundamental unresolved problems that can be formulated in elementary terms. These problems are also related to challenging open questions of modern mathematics; linear algebra and functional analysis in pa…
Distillation of multipartite entangled states for arbitrary subsets of parties in noisy quantum networks of increasing size
本文研究含噪量子网络中任意参与方子集的多粒子纯态蒸馏问题，基于图论连通性参数给出部分可蒸馏性的必要与充分条件及显式网络构造。
The dynamical behavior of interacting systems plays a fundamental role for determining quantum correlations, such as entanglement. In this Letter, we describe temporal quantum effects of the inseparable evolution of composite quantum states by comparing the trajectories to their classically correlated counterparts. For this reason, we introduce equations of motions describing the separable propagation of any interacting quantum system, which are derived by requiring separability for all times. The resulting Schroedinger-type equations allow for comparing the trajectories in a separable configu…
There is an ongoing effort to quantify entanglement of quantum pure states for systems with more than two subsystems. We consider three approaches to this problem for three-qubit states: choosing a basis which puts the state into a standard form, enumerating ``local invariants,'' and using operational quantities such as the number of maximally entangled states which can be distilled. In this paper we evaluate a particular standard form, the {\it Schmidt form}, which is a generalization of the Schmidt decomposition for bipartite pure states. We show how the coefficients in this case can be para…
Localizability of entanglement in fully inseparable states is a key ingredient of assisted quantum information protocols as well as measurement-based models of quantum computing. We investigate the existence of fully inseparable states with nonlocalizable entanglement, that is, with entanglement which cannot be localized between any pair of subsystems by any measurement on the remaining part of the system. It is shown, that the nonlocalizable entanglement occurs already in suitable mixtures of a three-qubit GHZ state and white noise. Further, we generalize this set of states to a two-parametri…
We show that the no-signaling principle can be violated with classical inseparable beams in the presence of a parity-time (PT) symmetric subsystem. Thus, the problems associated to PT-symmetric quantum theories recently discovered by Lee et al. [Phys. Rev. Lett. 112, 130404 (2014)] are not exclusive to quantum mechanics, but already exist in the classical case. The possibility to implement local optical PT-symmetric subsystems via light-matter interactions enables the experimental exploration of local PT symmetry and subtle quantum concepts via classical analogues.
Criteria for distillability, and the property of having a positive partial transpose, are introduced for states of general bipartite quantum systems. The framework is sufficiently general to include systems with an infinite number of degrees of freedom, including quantum fields. We show that a large number of states in relativistic quantum field theory, including the vacuum state and thermal equilibrium states, are distillable over subsystems separated by arbitrary spacelike distances. These results apply to any quantum field model. It will also be shown that these results can be generalized t…
When a quantum system is distributed to spatially separated parties, it is natural to consider how the system evolves when the parties perform local quantum operations with classical communication (LOCC). However, the structure of LOCC channels is exceedingly complex leaving many important physical problems unsolved. In this paper we consider generalized resource theories of entanglement based on different relaxations to the class of LOCC. The behavior of various entanglement measures is studied under non-entangling channels, as well as the newly introduced classes of dually non-entangling and…
Entanglement distillation is a fundamental task in quantum information processing. It not only extracts entanglement out of corrupted systems but also leads to protecting systems of interest against intervention with environment. In this work, we consider a realistic scenario of entanglement distillation where noisy quantum operations are applied. In particular, the two-way distillation protocol that tolerates the highest error rate is considered. We show that among all types of noise there are only four equivalence classes according to the distillability condition. Since the four classes are …
On the Preservation and Manifestation of Quantum Entanglement
本文利用扩展最小作用原理提出，纠缠粒子在类空间分离后，其不可分割关联通过对局域真空涨落带来的信息度量取极值而保持并显现，并以高斯波包自由粒子为例说明。
It is known that he bipartite quantum states, with rank strictly smaller than the maximum of the ranks of its two reduced states, are distillable by local operations and classical communication. Our first main result is that this is also true for NPT states with rank equal to this maximum. (A state is PPT if the partial transpose of its density matrix is positive semidefinite, and otherwise it is NPT.) This was conjectured first in 1999 in the special case when the two local ranks are equal. Our second main result provides a complete solution of the separability problem for bipartite states of…
A solution to 2-copy distillability of Werner states
研究维纳态双拷贝可蒸馏性问题，通过证明任意维度维纳态双拷贝可蒸馏当且仅当单拷贝可蒸馏，回答了长期问题，是确定非正定部分转置态是否可蒸馏的重要一步。
Within entanglement theory there are criteria which certify that some quantum states cannot be distilled into pure entanglement. An example is the positive partial transposition criterion. Here we present, for the first time, the analogous thing for secret correlations. We introduce a computable criterion which certifies that a probability distribution between two honest parties and an eavesdropper cannot be (asymptotically) distilled into a secret key. The existence of non-distillable correlations with positive secrecy cost, also known as bound information, is an open question. This criterion…
On the two-copy distillability of Werner states and a new partial trace inequality
该研究证明了四维Werner态ρ(4,-1/2)不是双拷贝可蒸馏的，并揭示了Werner态在不同拷贝数下的可蒸馏区域重合的条件。
Noise-robust 1-copy distillation protocol for all distillable Bell-diagonal qutrits
本文提出了一种抗噪声的1拷贝提纯协议，解决贝尔-对角三能级态的可提纯问题，证明违反PPT标准是1可提纯的必要充分条件，并展示了其对量子态白噪声的鲁棒性。
Non-local correlations are not only a fascinating feature of quantum theory, but an interesting resource for information processing, for instance in communication-complexity theory or cryptography. An important question in this context is whether the resource can be distilled: Given a large amount of weak non-local correlations, is there a method to obtain strong non-locality using local operations and shared randomness? We partly answer this question by no: CHSH-type non-locality, the only possible non-locality of binary systems, which is not super-strong, but achievable by measurements on ce…
In recent quantum optical continuous-variable experiments, the number of fully inseparable light modes has drastically increased by introducing a multiplexing scheme either in the time domain or in the frequency domain. Here, modifying the time-domain multiplexing experiment reported in Nature Photonics 7, 982 (2013), we demonstrate successive generation of fully inseparable light modes for more than one million modes. The resulting multi-mode state is useful as a dual-rail CV cluster state. We circumvent the previous problem of optical phase drifts, which has limited the number of fully insep…
Description of periodically and resonantly driven quantum systems can lead to solid state models where condensed matter phenomena can be investigated in time lattices formed by periodically evolving Wannier-like states. Here, we show that inseparable two-dimensional time lattices with the Möbius strip geometry can be realized for ultra-cold atoms bouncing between two periodically oscillating mirrors. Effective interactions between atoms loaded to a lattice can be long-ranged and can be controlled experimentally. As a specific example we show how to realize a Lieb lattice model with a flat band…
We propose local strategies to protect global quantum information. The protocols, which are quantum error correcting codes for dissipative systems, are based on environment measurements, direct feedback control and simple encoding of the logical qubits into physical qutrits whose decaying transitions are indistinguishable and equally probable. The simple addition of one extra level in the description of the subsystems allows for local actions to fully and deterministically protect global resources, such as entanglement. We present codes for both quantum jump and quantum state diffusion measure…
Information-theoretic aspects of quantum inseparability of mixed states are investigated in terms of the $α$-entropy inequalities and teleportation fidelity. Inseparability of mixed states is defined and a complete characterization of the inseparable $2\times2$ systems with maximally disordered subsystems is presented within the Hilbert-Schmidt space formalism. A connection between teleportation and negative conditional $α$-entropy is also emphasized.
We show how to decompose any density matrix of the simplest binary composite systems, whether separable or not, in terms of only product vectors. We determine for all cases the minimal number of product vectors needed for such a decomposition. Separable states correspond to mixing from one to four pure product states. Inseparable states can be described as pseudomixtures of four or five pure product states, and can be made separable by mixing them with one or two pure product states.
We show that the classification of bi-partite pure entangled states when local quantum operations are restricted yields a structure that is analogous in many respects to that of mixed-state entanglement. Specifically, we develop this analogy by restricting operations through local superselection rules, and show that such exotic phenomena as bound entanglement and activation arise using pure states in this setting. This analogy aids in resolving several conceptual puzzles in the study of entanglement under restricted operations. In particular, we demonstrate that several types of quantum optica…
We report our theoretical and experimental investigations into errors in quantum state estimation, putting a special emphasis on their asymptotic behavior. Tomographic measurements and maximum likelihood estimation are used for estimating several kinds of identically prepared quantum states (bi-photon polarization states) produced via spontaneous parametric down-conversion. Excess errors in the estimation procedures are eliminated by introducing a new estimation strategy utilizing Akaike's information criterion. We make a quantitative comparision between the errors of the experimentally estima…
Ying conceived of using two or more small-capacity quantum computers to produce a larger-capacity quantum computing system by quantum parallel programming ([M. S. Ying, Morgan-Kaufmann, 2016]). In doing so, the main obstacle is separating the quantum gates in the whole circuit to produce a tensor product of the local gates. It has been showed that there are few separable multipartite quantum gates, so the approximate separation problem involves finding local quantum gates that approximate a given inseparable gate. We propose and study a problem involving the approximate separation of multipart…
Distilling highly entangled quantum states from weaker ones is a process that is crucial for efficient and long-distance quantum communication, and has implications for several other quantum information protocols. We introduce the notion of distillation under limited resources, and specifically focus on the energy constraint. The corresponding protocol, which we call the canonical distillation of entanglement, naturally leads to the set of canonically distillable states. We show that for non-interacting Hamiltonians, almost no states are canonically distillable, while the situation can be dras…
Coherence analysis of local randomness and nonlocal correlation through polarization-basis projections of entangled photon pairs
本文通过偏振基投影对纠缠光子对的局域随机性和非局域关联进行相干性分析，揭示相位信息在量子特性中的确定性作用。
The homodyne detection is one of the most basic tools for identifying the quantum state of light. It has been used to detect useful non-local properties, such as entanglement for the quantum teleportation and distillability of a secret key in quantum key distribution. In so doing, the detection scheme employs a bright optical pulse, called the local oscillator (LO) pulse, and the LO pulse is usually transmitted along with the signal pulses. The LO pulse is presumed to be a coherent state with an infinite intensity. However, it is difficult in practice to hold this presumption owing to noise in…
Reliable high-fidelity quantum state transformation has always been considered as an inseparable part of quantum information processing. In this regard, Pontryagin maximum principle has proved to play an important role to achieve the maximum fidelity in an optimum time or energy. Motivated by this, in this work, we formulate a control constrained optimal control problem where we aim to minimize time and also energy subjected to a quantum system satisfying the bilinear Schrodinger equation. We derive the first order optimality conditions through the application of Pontryagin Maximum (minimum) P…
The fidelity and local unitary transformation are two widely useful notions in quantum physics. We study two constrained optimization problems in terms of the maximal and minimal fidelity between two bipartite quantum states undergoing local unitary dynamics. The problems are related to the geometric measure of entanglement and the distillability problem. We show that the problems can be reduced to semi-definite programming optimization problems. We give close-form formulae of the fidelity when the two states are both pure states, or a pure product state and the Werner state. We explain from t…
We study quantum information aspects of the fermionic entanglement negativity recently introduced in [Phys. Rev. B 95, 165101 (2017)] based on the fermionic partial transpose. In particular, we show that it is an entanglement monotone under the action of local quantum operations and classical communications--which preserves the local fermion-number parity-- and satisfies other common properties expected for an entanglement measure of mixed states. We present fermionic analogs of tripartite entangled states such as W and Greenberger-Horne-Zeilinger states and compare the results of bosonic and …
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