Preeti Parashar, Swapan Rana
We prove that the most general W class of N-qubit states are uniquely determined among arbitrary states (pure or mixed) by just their bipartite reduced density matrices. Moreover, if we consider only pure states, then (N-1) of them are shown to be sufficient.
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The partial separability of multipartite qubit density matrixes is strictly defined. We give a reduction way from N-partite qubit density matrixes to bipartite qubit density matrixes, and prove a necessary condition that a N-partite qubit density matrix to be partially separable is its reduced density matrix to satisfy PPT condition.
We discuss the problem of characterizing upper bounds on entanglement in a bipartite quantum system when only the reduced density matrices (marginals) are known. In particular, starting from the known two-qubit case, we propose a family of candidates for maximally entangled mixed states with respect to fixed marginals for two qudits. Interestingly, it turns out such states are always quasidistillable. Moreover, they are extremal in the convex set of two qudit states with fixed marginals. Our observations are supported by numerical analysis.
We consider a four-state pure bipartite system consisting of four qubits shared among two parties using the same Schmidt basis, two qubits per party. In some cases, transformation between two known pure states may not be possible using LOCC transformations but may be possible with the addition of a two-qubit catalyst. We provide a necessary and sufficient condition for this to occur.
In this work we study an inverse dynamical problem for a bipartite quantum system governed by the time local master equation: to find the class of generators which give rise to a certain time evolution with the constraint of fixed reduced states (marginals). The compatibility of such choice with a global unitary evolution is considered. For the non unitary case we propose a systematic method to reconstruct examples of master equations and address them to different physical scenarios.
We construct a class of algebraic invariants for N-qubit pure states based on bipartite decompositions of the system. We show that they are entanglement monotones, and that they differ from the well know linear entropies of the sub-systems. They therefore capture new information on the non-local properties of multipartite systems.
SU(N) algebras and new thumbrules for entanglement of bipartite qubit and qutrit systems
本文基于施密特分解提出检验二分量子比特与量子三态系统纠缠的新经验法则，并揭示其分别由SU(2)和SU(3)代数结构支撑。
Consider a qubit-qutrit ($2 \times 3$) composite state space. Let $C(\{1}{2}I_2, \{1}{3}I_3)$ be a convex set of all possible states of composite system whose marginals are given by $\{1}{2}I_2$ and $\{1}{3}I_3$ in two and three dimensional spaces respectively. We prove that there exists no pure state in $C(\{1}{2}I_2, \{1}{3}I_3)$. Further we generalize this result to an arbitrary $m \times n$ bipartite systems. We prove that for $m < n$, no pure state exists in the convex set $C(ρ_A,ρ_B)$, for an arbitrary $ρ_A$ and rank of $ρ_B >m$.
We numerically study protocols consisting of repeated applications of two qubit gates used for generating random pure states. A necessary number of steps needed in order to generate states displaying bipartite entanglement typical of random states is obtained. For generic two qubit entangling gate the decay rate of purity is found to scale as $\sim n$ and therefore of order $\sim n^2$ steps are necessary to reach random bipartite entanglement. We also numerically identify the optimal two qubit gate for which the convergence is the fastest. Perhaps surprisingly, applying the same good two qubit…
We consider multipartite states of qubits and prove that their bipartite quantum entanglement, as quantified by the concurrence, satisfies a monogamy inequality conjectured by Coffman, Kundu, and Wootters. We relate this monogamy inequality to the concept of frustration of correlations in quantum spin systems.
Quantum coherence and quantum entanglement are two strong pillars in quantum information theory. We study here for the possibility of any connection between these two important aspects of quantum mechanics while studying the entanglement detection problem for the detection of bipartite higher dimensional entangled states and multipartite entangled states. To achieve our goal, we derive the lower bound $L$ of $l_{1}$ norm of coherence of bipartite qubit-qudit system using the criterion that detect entanglement. Furthermore, we deduce the upper bound $U$ of $l_{1}$ norm of coherence of separable…
If we distribute n qubits between two parties, which quantum pure states and distributions of qubits would allow all-versus-nothing (or Greenberger-Horne-Zeilinger-like) proofs of Bell's theorem using only single-qubit measurements? We show a necessary and sufficient condition for the existence of these proofs for any number of qubits, and provide all distinct proofs up to n=7 qubits. Remarkably, there is only one distribution of a state of n=4 qubits, and six distributions, each for a different state of n=6 qubits, which allow these proofs.
Quantum secret sharing is one of the most important and interesting quantum information processing task. In quantum secret sharing, information is split among several parties such that only one of them is able to recover the qubit exactly provided all the other parties agree to cooperate. To achieve this task, all the parties need to share entangled state. As far as my knowledge, all the previous quantum secret sharing protocol used either pure tripartite or pure bipartite entangled state. In this work we use for the first time bipartite two qubit mixed state (formed due to noisy environment) …
One of the last open problems concerning two qubits in a pure state is to find the exact local content of their correlation, in the sense of Elitzur, Popescu and Rohrlich (EPR2) [Phys. Lett. A 162, 25 (1992)]. We propose a new EPR2 decomposition that allows us to prove, for a wide range of states psi(theta)=cos(theta)|00>+sin(theta)|11>, that their local content is pL(theta)=cos(2theta). We also share reflections on how to possibly extend our result to all two-qubit pure states.
We find a single parameter family of genuinely entangled three qubit pure states, called the maximally Bell inequality violating states (MBV), which exhibit maximum Bell inequality violation by the reduced bipartite system for a fixed amount of genuine tripartite entanglement quantified by the so called tangle measure. This in turn implies that there holds a complementary relation between the Bell inequality violation by the reduced bipartite systems and the tangle present in the three qubit states, not necessarily pure. The MBV states also exhibit maximum Bell inequality violation by the redu…
Toy model for the correlation of qudit bipartite states with maximally mixed marginals
针对高维 qudit 二部最大混合边缘态局域幺正分类难题，本文构造态族并给出不变量，还完整分类了一个双参数两 qutrit 族，将其与纯度和负性关联。
We advocate the step change in properties of discrete $d$-level quantum systems, between $d=2$ and $d\geq 3$. Qubit systems, or multipartite systems containing qubit subsystem, are exceptional in their relative simplicity. One faces a step in complexity in valuating measures of quantum correlations for qutrits and then other higher dimensional qudits. There is a growing number of arguments leading to such conclusion: recently found no-go theorem for generalization of the Peres-Horodecki's PPT criterion \cite{sko}, change in geometry of state spaces of qubit and higher degree qudits (the so cal…
We consider entanglement detection for quantum key distribution systems that use two signal states and continuous variable measurements. This problem can be formulated as a separability problem in a qubit-mode system. To verify entanglement, we introduce an object that combines the covariance matrix of the mode with the density matrix of the qubit. We derive necessary separability criteria for this scenario. These criteria can be readily evaluated using semidefinite programming and we apply them to the specific quantum key distribution protocol.
Quantum entanglement is an enigmatic and powerful property that has attracted much attention due to its usefulness in new ways of communications, like quantum teleportation and quantum key distribution. Much effort has been done to quantify entanglement. Indeed, there exist some well-established separability criterion and analytical formulas for the entanglement of bipartite systems. In some of these, the crucial elements are the eigenvalues of the partial transpose of the density matrix. In this paper, we show that one can also have information about the entanglement of bipartite state, in C2…
In this paper we consider feedback control algorithms for the rapid purification of a bipartite state consisting of two qubits, when the observer has access to only one of the qubits. We show 1) that the algorithm that maximizes the average purification rate is not the same as that that for a single qubit, and 2) that it is always possible to construct an algorithm that generates a deterministic rate of purification for {\em both} qubits. We also reveal a key difference between projective and continuous measurements with regard to state-purification.
Quantum entanglement in multipartite systems cannot be shared freely. In order to illuminate basic rules of entanglement sharing between qubits we introduce a concept of an entangled structure (graph) such that each qubit of a multipartite system is associated with a point (vertex) while a bi-partite entanglement between two specific qubits is represented by a connection (edge) between these points. We prove that any such entangled structure can be associated with a pure state of a multi-qubit system. Moreover, we show that a pure state corresponding to a given entangled structure is a superpo…
We explore how entanglement of a general bipartite system evolves when one subsystem undergoes the action of an arbitrary noisy channel. It is found that the dynamics of entanglement for general bipartite systems under the influence of such channel is determined by the channel's action on the maximally entangled state, which includes as a special case the results for two-qubit systems [Nature Physics 4, 99 (2008)]. In particular, for multi-qubit or qubit-qudit systems, we get a general factorization law for evolution equation of entanglement with one qubit being subject to a noisy channel. Our…
We investigate bipartite and tripartite entanglement in an open quantum system, specifically three qubits, all of which are damped, and one of which is driven. We adapt a systematic approach in calculating the entanglement of various bipartite splits usinga generalized concurrence as an indicator of entanglement. Our calculations are based on a direct detection scheme that is a particular unravelling of the density matrix. This system has a collective dipole-dipole energy shift that couples the atoms and the dissipation is via partially collective spontaneous emission described by the Lehmberg…
Reducing the Entanglement Cost of Distributed Bipartite Quantum Computation with Constant Qubit Overhead
本研究针对分布式量子计算中的通信成本问题，提出在恒定量子比特开销下逼近纠缠成本下界的方法，对克利福德酉算子可达下界，对非克利福德算子成本至多高出2t个贝尔对。
We consider multi-qubit systems and relate quantitatively the problems of generating cluster states with high value of concurrence of assistance, and that of generating states with maximal bipartite entanglement. We prove an upper bound for the concurrence of assistance. We consider dynamics of spin-1/2 systems that model qubits, with different couplings and possible presence of magnetic field to investigate the appearance of the discussed entanglement properties. We find that states with maximal bipartite entanglement can be generated by an XY Hamiltonian, and their generation can be controll…
The local and non-local contents of non-local probability distributions are studied using the approach of Elitzur, Popescu and Rohrlich [Phys. Lett. A \textbf{162}, 25 (1992)]. This work focuses on distributions that can be obtained by single-copy von Neumann measurements on bipartite quantum systems. For pure two-qubit states Psi(theta)=cos(theta)|00>+sin(theta)|11>, with cos(theta)>=sin(theta), the local content of the corresponding probability distribution is found to lie between 1-sin(2*theta) and cos(2*theta). For the family Psi(gamma)= (|00>+|11>+gamma*|22>)/sqrt(2+gamma^2) of two-qutrit…
We show that violation of variance based local sum uncertainty relation (LSUR) for angular momentum operators of a bipartite system, proposed by Hofmann and Takeuchi~[Phys.Rev.A {\bf 68}, 032103 (2003)], reflects entanglement in the equal bipartitions of an $N$-qubit symmetric state with even qubits. We establish the one-to-one connection with the violation of LSUR with negativity of covariance matrix [Phys. Lett. A, {\bf 364}, 203 (2007)] of the two-qubit reduced system of a permutation symmetric $N$-qubit state.
We model repetitive quantum error correction (QEC) with the single-error-correcting five-qubit code on a network of individually-controlled qubits with always-on Ising couplings, using our previously designed universal set of quantum gates based on sequences of shaped decoupling pulses. In addition to serving as accurate quantum gates, the sequences also provide dynamical decoupling (DD) of low-frequency phase noise. The simulation involves integrating unitary dynamics of six qubits over the duration of tens of thousands of control pulses, using classical stochastic phase noise as a source of …
A class of quantum protocols to teleport bipartite (entangled) states of two qubits is suggested. Our schemes require a single entangled pair shared by the two parties and the transmission of three bits of classical information, as well as a two-qubit gate with an additional qubit at the receiver' location. Noisy quantum channels are considered and the effects on both the teleportation fidelity and the entanglement of the replica are evaluated.
Quantum properties of correlations have a key role in disparate fields of physics, from quantum information processing, to quantum foundations, to strongly correlated systems. We tackle a specific aspect of the fundamental quantum marginal problem: we address the issue of deducing the global properties of correlations of tripartite quantum states based on the knowledge of their bipartite reductions, focusing on relating specific properties of bipartite correlations to global correlation properties. We prove that strictly classical bipartite correlations may still require global entanglement an…
We study the explicit relation between violation of Bell inequalities and bipartite distillability of multi-qubit states. It has been shown that even though for $N\ge 8$ there exist $N$-qubit bound entangled states which violates a Bell inequality [Phys. Rev. Lett. {\bf 87}, 230402 (2001)], for all the states violating the inequality there exists at least one splitting of the parties into two groups such that pure-state entanglement can be distilled [Phys. Rev. Lett. {\bf 88}, 027901 (2002)]. We here prove that for all $N$-qubit states violating the inequality the number of distillable biparti…
Controlling nonlocality of bipartite qubit states via quantum channels
本文分析了从X态导出的量子信道，刻画了破坏Bell CHSH条件的逆Choi-Jamiolkowski映射（非定域性生成），发现稳态偏离最大混合态时该信道区域收缩，并证明特殊情况仅需单参数即可简化非定域性分析。
We consider the entanglement manipulation capabilities of the universal covariant quantum cloner or quantum processor circuit for quantum bits. We investigate its use for cloning a member of a bipartite or a genuine tripartite entangled state of quantum bits. We find that for bipartite pure entangled states a nontrivial behavior of concurrence appears, while for GHZ entangled states a possibility of the partial extraction of bipartite entanglement can be achieved.
Ising interaction between qubits could produce distortion in entangled pairs generated for engineering purposes (as in quantum computation) in presence of parasite magnetic fields, destroying or altering the expected behavior of process in which is projected to be used. Quantum control could be used to correct that situation in several ways. Sometimes the user should be make some measurement upon the system to decide which is the best control scheme; other posibility is try to reconstruct the system using similar procedures without perturbate it. In the complete pictures both schemes are prese…
Aharonov, Jones, and Landau [Algorithmica 55, 395 (2009)] have presented a polynomial quantum algorithm for approximating the Jones polynomial. We investigate the bipartite entanglement properties in AJL's algorithm for three-strand braids. We re-describe AJL's algorithm as an equivalent algorithm which involves three work qubits in some mixed state coupled to a single control qubit. Furthermore, we use the Peres entanglement criterion to study the entanglement features of the state before measurements present in the re-described algorithm for all bipartitions. We show that the state is a prod…
We investigate the entanglement, CHSH nonlocality, fully entangled fraction and symmetric extendibility of two-qubit states that have a single maximally mixed marginal. Within this set of states, the steering ellipsoid formalism has recently highlighted an interesting family of so-called 'maximally obese' states. These are found to have extremal quantum correlation properties that are significant in the steering ellipsoid picture and for the study of two-qubit states in general.
For the first time, we compute the quantum discord in bipartite systems containing up to nine qubits. An analytical expression is obtained for the discord in a bipartite system with three qubits. The dependence of the discord on the temperature and the structural parameter of the model is studied.
Hamiltonian Switching Control of Noisy Bipartite Qubit Systems
本文提出一种受QAOA启发的哈密顿量切换控制方法，结合强化学习优化，有效抑制双量子比特系统中的环境噪声，实现高保真度量子门，并分析关键参数影响。
In this letter, a nonlocal effect for a bipartite system which is induced by a local cyclic evolution of one of its subsystem is suggested. This effect vanishes when the system is at a disentangled pure state but can be observed for some disentangled mixed states. As a paradigm, we study the effect for the system of two qubits in detail. It is interesting that the effect is directly related to the degree of entanglement for pure state of qubit pairs. Furthermore, we suggest a Bell-type experiment to measure this nonlocal effect for qubit pairs.
The dynamics of entanglement and quantum discord for qubit-qutrit systems are studied in the presence of phase damping and amplitude damping noises. Both one way and two couplings of the marginal systems with the environments are considered. Entanglement sudden death is unavoidable under any setup, however, the required time span depends on the way of coupling. On the other hand, the dynamics of quantum discord strongly depends both on the nature of environment and on the number of dimensions of the Hilbert space of the coupled marginal system. We show that freezing and invariance of quantum d…
This paper is to investigate the effects of quantum noises on entanglement localization by taking an example of reducing a three-qubit Greenberger-Horne-Zeilinger (GHZ) state to a two-qubit entangled state. We consider, respectively, two types of quantum decoherence, i.e., amplitude-damping and depolarizing decoherence, and explore the best von Neumann measurements on one of three qubits of the triple GHZ state for making the amount of entanglement of the collapsed bipartite state be as large as possible. The results indicate that different noises have different impacts on entanglement localiz…
We investigate some properties of multipartite entanglement of hypergraph states in purely hypergraph theoretical terms. We first introduce an approach for computing the concurrence between two specific qubits of a hypergraph state by using the so-called Hamming weights of several special subhypergraphs of the corresponding hypergraph. Then we quantify and characterize bipartite entanglement between each qubit pair of several special hypergraph states in terms of the concurrence obtained by using the above approach. Our main results include that (i) a graph g has a component with the vertex se…
Thermal effects on bipartite and multipartite correlations in fiber coupled cavity arrays
本文研究光纤耦合腔阵列中热涨落对极化激元量子比特两体及多体关联的影响，通过主方程数值分析并发度、量子失谐和自旋压缩，发现热效应可阻塞纠缠但保留失谐，且长链关联更鲁棒。
We analyze the effect of position dependent excitation phase on the properties of entanglement between two qubits formed in atomic systems. We show that the excitation phase induces a vacuum mediated quantum interference in the system that affects the dynamical behavior of entanglement between the qubits. It is also found that the quantum interference leads to a coherent population transfer between the symmetric and antisymmetric states which can considerably modify the dynamics of two-qubit entanglement and can even prevent finite time disentanglement (sudden death) under certain conditions.
Bipartite operations underpin both classical communication and entanglement generation. Using a superposition of classical messages, we show that the capacity of a two-qubit operation for error-free entanglement-assisted bidirectional classical communication can not exceed twice the entanglement capability. In addition we show that any bipartite two-qubit operation can increase the communication that may be performed using an ensemble by twice the entanglement capability.
We use singular value decomposition to derive a tight lower bound for geometric discord of arbitrary bipartite states. In a single shot this also leads to an upper bound of measurement-induced non locality which in turn yields that for Werner and isotropic states the two measures coincide. We also emphasize that our lower bound is saturated for all $2\otimes n$ states. Using this we show that both the generalized Greenberger-Horne-Zeilinger and $W$ states of $N$ qubits satisfy monogamy of geometric discord. Indeed, the same holds for all $N$-qubit pure states which are equivalent to $W$ states…
In the present work, we address the question of how bipartite steering violation takes place among multi-partite systems (where each sub-system have Hilbert space dimension restricted to two) based on the maximal violations of the bipartite steering inequality of the reduced pairwise qubit systems. We have derived a trade-off relation which is satisfied by those pairwise bipartite maximal steering violations, which physically can be understood as providing restrictions on the distribution of steering among subsystems. For a three-qubit system, it is impossible that all pairs of qubits violate …
In this thesis work, we have studied the role of positive and completely positive maps in detecting entanglement.
Teleportation with non-maximally entangled states and underlying unitary algebras of certain bipartite systems
本文针对双粒子量子比特和量子三态系统，提出基于矩阵行列式与迹的解析判据，揭示底层SU(2)与SU(3)代数，并利用非最大纠缠态实现增强密码学安全性的量子隐形传态。
Three-qubit W state tomography via full and marginal state reconstructions on ibm_osaka
通过全状态和边缘状态重建在ibm_osaka上实现三量子比特W态的高效成像，展示子系统方法在提高保真度方面的优势。
The recognition of entanglement states is a notoriously difficult problem when no prior information is available. Here, we propose an efficient quantum adversarial bipartite entanglement detection scheme to address this issue. Our proposal reformulates the bipartite entanglement detection as a two-player zero-sum game completed by parameterized quantum circuits, where a two-outcome measurement can be used to query a classical binary result about whether the input state is bipartite entangled or not. In principle, for an $N$-qubit quantum state, the runtime complexity of our proposal is $O(\tex…
We introduce the entanglement gauge describing the combined effects of local operations and nonlocal unitary transformations on bipartite quantum systems. The entanglement gauge exploits the invariance of nonlocal properties for bipartite systems under local (gauge) transformations. This new formalism yields observable effects arising from the gauge geometry of the bipartite system. In particular, we propose a non-Abelian gauge theory realized via two separated spatial modes of the quantized electromagnetic field manipulated by linear optics. In this linear optical realization, a bi-partite st…
Detection and Classification of Bipartite and Multipartite Entangled States
本文基于SPA-PT方法系统研究二分与多分量子纠缠态的检测、分类与量化，提出三量子比特SLOCC不等价类分类判据，并定义新纠缠度量“结构化负度”，且方法具备实验可实现性。
We characterize the positive maps detecting the entangled bipartite states of n x n qubits that are diagonal with respect to the orthonormal basis constructed by tensor products of Pauli matrices acting on the totally symmetric state. We then discuss the case n=2 for a class of states completely determined by the geometric patterns of subsets of a 16 point lattice.
One-qubit pure states, living on the surface of Bloch sphere, can be mapped onto the usual complex plane by using stereographic projection. In this paper, after reviewing the entanglement of two-qubit pure state, it is shown that the \emph{quaternionic} stereographic projection is related to concurrence measure. This is due to the fact that every two-qubit state, in ordinary complex field, corresponds to the one-qubit state in quaternionic skew field, called quaterbit. Like the one-qubit states in complex field, the stereographic projection maps every quaterbit onto a quaternion number whose c…
An experimental scheme is suggested that permits a direct measure of entanglement in a two-qubit cavity system. It is realized in the cavity-QED technology utilizing atoms as flying qubits. With this scheme we generate two different measures of entanglement, namely logarithmic negativity and concurrence. The phenomenon of sudden death entanglement (ESD) in a bipartite system subjected to dissipative environment is examined. We show that the sudden death time of the entangled states depends on the initial preparation of the entangled state and the temperature of the reservoir.
Entanglement generated by Ising model has been studied for several authors in order to understand the relation between it and magnetic properties of materials, principally using one or two dimensional models for two or more particles. In this work, Ising model evolution is solved in three dimensions for two parts including an inhomogeneous magnetic field, giving an detailed study of the entanglement properties derived from interaction. Some relations between entanglement and energy or spin are developed specifically to stablish candidates for entanglement witness. Finally, some basic quantum c…
A first approach to the open dynamics of bipartite systems
本文研究了双分系统（如qubit-qubit、oscillator-oscillator和qubit-oscillator系统）的开放量子动力学，比较了旋转波近似与非近似情况下的动力学影响，并分析了不同系统间的相似性和差异性。
This is a thesis submitted to university of Pune, India, for the Ph.D. degree. This work deals with entanglement production in two qubit, two qutrit and three qubit systems, entanglement in indistinguishable fermionic systems, quantum discord in a Heisenberg chain and geometric measure of quantum discord in an arbitrary state of a bipartite quantum system.
We study the nonlocality of arbitrary dimensional bipartite quantum states. By computing the maximal violation of a set of multi-setting Bell inequalities, an analytical and computable lower bound has been derived for general two-qubit states. This bound gives the necessary condition that a two-qubit state admits no local hidden variable models. The lower bound is shown to be better than that from the CHSH inequality in judging the nonlocality of some quantum states. The results are generalized to the case of high dimensional quantum states, and a sufficient condition for detecting the non-loc…
Monogamy of bipartite correlations leads, for arbitrary pure multi-qubit states, to simple conditions able to indicate various types of multipartite entanglement by being capable to exclude the possibility of k-separability.
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