Dariusz Chruscinski, Andrzej Kossakowski
We propose a natural generalization of bipartite Werner and isotropic states to multipartite systems consisting of an arbitrary even number of d-dimensional subsystems (qudits). These generalized states are invariant under the action of local unitary operations. We study basic properties of multipartite invariant states: separability criteria and multi-PPT conditions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The equivalence of multipartite quantum mixed states under local unitary transformations is studied. A criterion for the equivalence of non-degenerate mixed multipartite quantum states under local unitary transformations is presented.
We study the invariants of arbitrary dimensional multipartite quantum states under local unitary transformations. For multipartite pure states, we give a set of invariants in terms of singular values of coefficient matrices. For multipartite mixed states, we propose a set of invariants in terms of the trace of coefficient matrices. For full ranked mixed states with nondegenerate eigenvalues, this set of invariants is also the necessary and sufficient conditions for the local unitary equivalence of such two states.
A Comment on the Letter by B. Kraus {\it Phys. Rev. Lett.}{\bf 104}, 020504 (2010).
A general method for implementing weakly entangling multipartite unitary operations using a small amount of entanglement and classical communication is presented. For the simple Hamiltonian σ_z\otimesσ_z this method requires less entanglement than previously known methods. In addition, compression of multiple operations is applied to reduce the average communication required.
We study the local unitary equivalence of arbitrary dimensional multipartite quantum mixed states. We present a necessary and sufficient criterion of the local unitary equivalence for general multipartite states based on matrix realignment. The criterion is shown to be operational even for particularly degenerated states by detailed examples. Besides, explicit expressions of the local unitary operators are constructed for locally equivalent states. In complement to the criterion, an alternative approach based on partial transposition of matrices is also given, which makes the criterion more ef…
We present an approach of constructing invariants under local unitary transformations for multipartite quantum systems. The invariants constructed in this way can be complement to that in [Science 340 (2013) 1205-1208]. Detailed examples are given to compute such invariant in detail. It is shown that these invariants can be used to detect the local unitary equivalence of degenerated quantum states.
We show that any two different unitary operations acting on an arbitrary multipartite quantum system can be perfectly distinguishable by local operations and classical communication when a finite number of runs is allowed. We then directly extend this result into the case when the number of unitary operations to be discriminated is more than two. Intuitively, our result means that the lost identity of a nonlocal (entangled) unitary operation can be recovered locally, without any use of entanglement or joint quantum operations.
Necessary and sufficient conditions for the equivalence of arbitrary n-qubit pure quantum states under Local Unitary (LU) operations are derived. First, an easily computable standard form for multipartite states is introduced. Two generic states are shown to be LU-equivalent iff their standard forms coincide. The LU-equivalence problem for non--generic states is solved by presenting a systematic method to determine the LU operators (if they exist) which interconvert the two states.
Local unitary equivalence of arbitrary-dimensional multipartite quantum states
本文基于广义 Bloch 表示的系数矩阵构造任意维度两体及多体量子态的局域酉不变量，并将其用于验证局域酉等价和估计并发度。
Classification of Schmidt-rank-two multipartite unitary gates by singular number
该研究针对施密特秩为2的真多体幺正门，利用其施密特分解的唯一性提出奇异数概念完成分类，并拓展研究了三量子比特对角幺正门的施密特秩性质，给出了秩为3的充要条件。
Characterizing local unitary equivalence of multipartite states by local unitary Bargmann invariants
本文研究局部酉Bargmann不变量（LUBIs）能否充分判定多体态的局部酉（LU）等价性，证明对单方约化矩阵非简并的CLC态成立，并给出简并情形下反例。
We construct a new class of multipartite states possessing orthogonal symmetry. This new class defines a convex hull of multipartite states which are invariant under the action of local unitary operations introduced in our previous paper "On multipartite invariant states I. Unitary symmetry". We study basic properties of multipartite symmetric states: separability criteria and multi-PPT conditions.
The necessary and sufficient conditions for the equivalence of arbitrary n-qubit pure quantum states under Local Unitary (LU) operations derived in [B. Kraus Phys. Rev. Lett. 104, 020504 (2010)] are used to determine the different LU-equivalence classes of up to five-qubit states. Due to this classification new parameters characterizing multipartite entanglement are found and their physical interpretation is given. Moreover, the method is used to derive examples of two n-qubit states (with n>2 arbitrary) which have the properties that all the entropies of any subsystem coincide, however, the s…
In previous work the authors introduced a notion of generic states and obtained criteria for local equivalence of them. Here they introduce the concept of CHG states maintaining the criteria of local equivalence. This fact allows the authors to halve the number of invariants necessary to characterize the equivalence classes under local unitary transformations for the set of tripartite states whose partial trace with respect to one of the subsystems belongs to the class of CHG mixed states.
We study the entangling properties of multipartite unitary gates with respect to the measure of entanglement called one-tangle. Putting special emphasis on the case of three parties, we derive an analytical expression for the entangling power of an $n$-partite gate as an explicit function of the gate, linking the entangling power of gates acting on $n$-partite Hilbert space of dimension $d_1 \ldots d_n$ to the entanglement of pure states in the Hilbert space of dimension $(d_1 \ldots d_n)^2$. Furthermore, we evaluate its mean value averaged over the unitary and orthogonal groups, analyze the m…
The dynamics of a system composed by two pairs of dipolarly coupled two-level atoms is exactly studied. We show that the initial entanglement stored in a couple of atoms not directly interacting is fully transferred to the other pair in a periodic way. The observability of this phenomenon in laboratory is briefly discussed both in terms of its temporal scale and of its stability against uncertainties in the geometrical parameters defining the physical system.
The equivalence problem under local unitary transformation for $n$--partite pure states is reduced to the one for $(n-1)$--partite mixed states. In particular, a tripartite system $\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_C$, where $\mathcal{H}_j$ is a finite dimensional complex Hilbert space for $j=A,B,C$, is considered and a set of invariants under local transformations is introduced, which is complete for the set of states whose partial trace with respect to $\mathcal{H}_A$ belongs to the class of generic mixed states.
We analyze the entanglement evolution of two cavity photons being affected by the dissipation of two individual reservoirs. Under an arbitrary local unitary operation on the initial state, it is shown that there is only one parameter which changes the entanglement dynamics. For the bipartite subsystems, we show that the entanglement of the cavity photons is correlated with that of the reservoirs, although the local operation can delay the time at which the photon entanglement disappears and advance the time at which the reservoir entanglement appears. Furthermore, via a new defined four-qubit …
The scheme for entanglement teleportation is proposed to incorporate multipartite entanglement of four qubits as a quantum channel. Based on the invariance of entanglement teleportation under arbitrary two-qubit unitary transformation, we derive relations of separabilities for joint measurements at a sending station and for unitary operations at a receiving station. From the relations of separabilities it is found that an inseparable quantum channel always leads to a total teleportation of entanglement with an inseparable joint measurement and/or a nonlocal unitary operation.
We give an universal algorithm for testing the local unitary equivalence of states for multipartite system with arbitrary dimensions.
We revisit the genuine multipartite entanglement by a simplified method, which only involves the Schmidt decomposition and local unitary transformation. We construct a local unitary equivalent class of the tri-qubit quantum state, then use the trace norm of the whole correlation tensor as a measurement to detect genuine multipartite entanglement. By detailed examples, we show our result can detect more genuinely entangled states. Furthermore, we generalize the genuine multipartite entanglement criterion to tripartite higher-dimensional systems.
We study the equivalence of quantum states under local unitary transformations by using the singular value decomposition. A complete set of invariants under local unitary transformations is presented for several classes of tripartite mixed states in KxMxN composite systems. Two density matrices in the same class are equivalent under local unitary transformations if and only if all these invariants have equal values for these density matrices.
We propose a practical entanglement classification scheme for general multipartite pure states in arbitrary dimensions under local unitary equivalence by exploiting the high order singular value decomposition technique and local symmetries of the states. By virtue of this scheme, the method of determining the local unitary equivalence of $n$-qubit states proposed by Kraus is extended to the case for arbitrary dimensional multipartite states.
Within the unified framework of exploiting the relative entropy as a distance measure of quantum correlations, we make explicit the hierarchical structure of quantum coherence, quantum discord and quantum entanglement in multipartite systems. On this basis, we introduce a new measure of quantum coherence, the basis-free quantum coherence and prove that this quantity is exactly equivalent to quantum discord. Furthermore, since the original relative entropy of coherence is a basis-dependent quantity, we investigate the local and nonlocal unitary creation of quantum coherence, focusing on the two…
We show that any unitary operator on the $d_A\times d_B$ system ($d_A\ge 2$) can be decomposed into the product of at most $4d_A-5$ controlled unitary operators. The number can be reduced to $2d_A-1$ when $d_A$ is a power of two. We also prove that three controlled unitaries can implement a bipartite complex permutation operator, and discuss the connection to an analogous result on classical reversible circuits. We further show that any $n$-partite unitary on the space $\mathbb{C}^{d_1}\otimes...\otimes\mathbb{C}^{d_n}$ is the product of at most $[2\prod^{n-1}_{j=1}(2d_j-2)-1]$ controlled unit…
We review the generation of random pure states using a protocol of repeated two qubit gates. We study the dependence of the convergence to states with Haar multipartite entanglement distribution. We investigate the optimal generation of such states in terms of the physical (real) time needed to apply the protocol, instead of the gate complexity point of view used in other works. This physical time can be obtained, for a given Hamiltonian, within the theoretical framework offered by the quantum brachistochrone formalism. Using an anisotropic Heisenberg Hamiltonian as an example, we find that di…
A method is presented to obtain local unitary invariants for multipartite quantum systems consisting of fermions or distinguishable particles. The invariants are organized into infinite families, in particular, the generalization to higher dimensional single particle Hilbert spaces is straightforward. Many well-known invariants and their generalizations are also included.
The degree of the generators of invariant polynomial rings of is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary group---the tensor product of unitary group, and local general linear group---the tensor product of general linear group. For these two groups, we prove polynomial upper bounds on the degree of the generators of invariant polynomial rings. On the other hand, systematic methods are provided to to…
We study the growth of genuine multipartite entanglement in random quantum circuit models, which include random unitary circuit models and the random Clifford circuit. We find that for the random Clifford circuit, the growth of multipartite entanglement remains slower in comparison to the random unitary case. However, the final saturation value of multipartite entanglement is almost the same in both cases. The behavior is then compared to the genuine multipartite entanglement obtained in random matrix product states with a moderately high bond dimension. We then relate the behavior of multipar…
In this paper, we give a method for the local unitary equivalent problem which is more efficient than that was proposed by Bin Liu $et \ al$ \cite{bliu}.
We study necessary conditions for the efficient simulation of both bipartite and multipartite Hamiltonians, which are independent of the eigenvalues and based on the algebraic-geometric invariants introduced in [1-2]. Our results indicate that the problem of efficient simulation of Hamiltonians for arbitrary and multipartite quantum systems cannot be described by using only eigenvalues, unlike that in the two-qubit case.
Multipartite entangling power by von Neumann entropy
本文提出多体纠缠、辅助纠缠和解纠缠能力的概念，通过最小凸和计算多体单元操作的纠缠能力，并分析了多体门的纠缠能力。
Hierarchies among genuine multipartite entangling capabilities of quantum gates
本研究通过分析量子门生成真正多体纠缠的能力，揭示了不同单位ary算符在纠缠生成中的效率差异及复系数在纠缠生成中的作用。
We construct a class of multipartite states possessing rotational SO(3) symmetry -- these are states of K spin-j_A particles and K spin-j_B particles. The construction of symmetric states follows our two recent papers devoted to unitary and orthogonal multipartite symmetry. We study basic properties of multipartite SO(3) symmetric states: separability criteria and multi-PPT conditions.
We investigate the entanglement properties of pure quantum states describing $n$ qubits. We characterize all multipartite states which can be maximally entangled to local auxiliary systems using controlled operations. A state has this property iff one can construct out of it an orthonormal basis by applying independent local unitary operations. This implies that those states can be used to encode locally the maximum amount of $n$ bits. Examples of these states are the so--called stabilizer states, which are used for quantum error correction and one--way quantum computing. We give a simple char…
Unitary operations are physically implementable. We further the understanding of such operations by studying the possible forms of nonlocal unitary operators, which are bipartite or multipartite unitary operators that are not tensor product operators. They are of broad relevance in quantum information processing. We prove that any nonlocal unitary operator of Schmidt rank three on a $d_A\times d_B$ bipartite system is locally equivalent to a controlled unitary. This operator can be locally implemented assisted by a maximally entangled state of Schmidt rank $\min\{d_A^2,d_B\}$ when $d_A\le d_B$…
We construct a nontrivial set of invariants for any multipartite mixed states under the SLOCC symmetry. These invariants are given by hyperdeterminants and independent from basis change. In particular, a family of d^2 invariants for arbitrary d-dimensional even partite mixed states are explicitly given.
We propose a practical method for finding the canonical forms of arbitrary dimensional multipartite entangled states, either pure or mixed. By extending the technique developed in one of our recent works, the canonical forms for the mixed $N$-partite entangled states are constructed where they have inherited local unitary symmetries from their corresponding $N+1$ pure state counterparts. A systematic scheme to express the local symmetries of the canonical form is also presented, which provides a feasible way of verifying the local unitary equivalence for two multipartite entangled states.
We introduce algebraic sets in the products of complex projective spaces for the mixed states in multipartite quantum systems as their invariants under local unitary operations. The algebraic sets have to be the union of the linear subspaces if the state is separable. Some examples are studied based on our criterion
We construct quantum gate entangler for general multipartite states based on topological unitary operators. We show that these operators can entangle quantum states if they satisfy the separability condition that is given by the complex multi-projective Segre variety. We also in detail discuss the construction of quantum gate entangler for higher dimensional bipartite and three-partite quantum systems.
In this work we consider the permutational properties of multipartite entanglement monotones. Based on the fact that genuine multipartite entanglement is a property of the entire multi-qubit system, we argue that ideal definitions for its characterizing quantities must be permutation-invariant. Using this criterion, we examine the three 4-qubit entanglement monotones introduced by Osterloh and Siewert [Phys. Rev. A. 72, 012337]. By expressing them in terms of quantities whose permutational properties can be easily derived, we find that one of these monotones is not permutation-invariant. We pr…
We prove that every unitary acting on any multipartite system and having operator Schmidt rank equal to 2 can be diagonalized by local unitaries. This then implies that every such multipartite unitary is locally equivalent to a controlled unitary with every party but one controlling a set of unitaries on the last party. We also prove that any bipartite unitary of Schmidt rank 2 is locally equivalent to a controlled unitary where either party can be chosen as the control, and at least one party can control with two terms, which implies that each such unitary can be implemented using local opera…
A general mathematical framework is presented to describe local equivalence classes of multipartite quantum states under the action of local unitary and local filtering operations. This yields multipartite generalizations of the singular value decomposition. The analysis naturally leads to the introduction of entanglement measures quantifying the multipartite entanglement (as generalizations of the concurrence and the 3-tangle), and the optimal local filtering operations maximizing these entanglement monotones are obtained. Moreover a natural extension of the definition of GHZ-states to e.g. $…
Local unitary equivalence of absolutely maximally entangled states constructed from orthogonal arrays
本研究利用不变量分析由正交阵列构造的绝对最大纠缠态的局部酉等价性，证明存在无限多个局部酉不等价的三方和五方AME态。
Hypergraph states are multiqubit states whose combinatorial description and entanglement properties generalize the well-studied class of graph states. Graph states are important in applications such as measurement-based quantum computation and quantum error correction. The study of hypergraph states, with their richer multipartite entanglement and other nonlocal properties, has a promising outlook for new insight into multipartite entanglement. We present results analyzing local unitary symmetries of hypergraph states, including both continuous and discrete families of symmetries. In particula…
We consider the action of the group of local unitary transformations, U(m) x U(n), on the set of (mixed) states W of the bipartite m x n quantum system. We prove that the generic U(m) x U(n)--orbits in W have dimension m^2+n^2-2. This problem was mentioned (and left open) by Kus and Zyczkowski in their paper Geometry of entangled states. The proof can be extended to the case of arbitrary finite-dimensional multipartite quantum systems.
We study the equivalence of mixed states under local unitary transformations. First we express quantum states in Bloch representation. Then based on the coefficient matrices, some invariants are constructed. This method and results can be extended to multipartite high dimensional system.
Bell inequalities are natural tools that allow one to certify the presence of nonlocality in quantum systems. The known constructions of multipartite Bell inequalities contain, however, correlation functions involving all observers, making their experimental implementation difficult. The main purpose of this work is to explore the possibility of witnessing nonlocality in multipartite quantum states from the easiest-to-measure quantities, that is, the two-body correlations. In particular, we determine all three and four-partite Bell inequalities constructed from one and two-body expectation val…
We investigate the dynamics of entanglement and nonlocality for multipartite quantum systems under collective dephasing. Using an exact and computable measure for genuine entanglement, we demonstrate the possibility of a non trivial phenomenon of time-invariant entanglement for multipartite quantum systems. We find that for four qubits, there exist quantum states, which are changing continously nevertheless their genuine entanglement remains constant. Based on our numerical results, we conjecture that there is no evidence of time-invariant entanglement for quantum states of three qubits. We po…
We investigate the structural aspects of genuine multipartite entanglement in Gaussian states of continuous variable systems. Generalizing the results of [Adesso & Illuminati, Phys. Rev. Lett. 99, 150501 (2007)], we analyze whether the entanglement shared by blocks of modes distributes according to a strong monogamy law. This property, once established, allows to quantify genuine N-partite entanglement in terms of the "residual contangle" not encoded into 2,...,K,...,(N-1)-partite quantum correlations. The explicit expression of this entanglement measure is derived, by a recursive formula, for…
Weyl channels for multipartite systems
本文通过 Weyl 算符将 Pauli 映射推广至多体高维量子系统，以 Fourier 变换矩阵导出完全正性条件，求出极值点并发现代数结构，进而拓展了分量擦除信道，给出算法构造及生成全集的最小子集。
We consider the local unitary equivalence of a class of quantum states in bipartite case and multipartite case. The necessary and sufficient condition is presented. As special cases, the local unitary equivalent classes of isotropic state and werner state are provided. Then we study the local unitary similar equivalence of this class of quantum states and analyze the necessary and sufficient condition.
The invariant-comb approach is a method to construct entanglement measures for multipartite systems of qubits. The essential step is the construction of an antilinear operator that we call {\em comb} in reference to the {\em hairy-ball theorem}. An appealing feature of this approach is that for qubits (or spins 1/2) the combs are automatically invariant under $SL(2,\CC)$, which implies that the obtained invariants are entanglement monotones by construction. By asking which property of a state determines whether or not it is detected by a polynomial $SL(2,\CC)$ invariant we find that it is the …
We introduce algebriac sets in the products of complex projective spaces for multipartite mixed states, which are independent of their eigenvalues and only measure the "position" of their eigenvectors, as their non-local invariants (ie. remaining invariant after local untary transformations). These invariants are naturally arised from the physical consideration of checking multipartite mixed states by measuring them with multipartite separable pure states. The algebraic sets have to be the sum of linear subspaces if the multipartite mixed state is separable, and thus we give a new separability…
Local unitary invariants allow one to test whether multipartite states are equivalent up to local basis changes. Equivalently, they specify the geometry of the "orbit space" obtained by factoring out local unitary action from the state space. This space is of interest because of its intimate relationship to entanglement. Unfortunately, the dimension of the orbit space grows exponentially with the number of subsystems, and the number of invariants needed to characterise orbits grows at least as fast. This makes the study of entanglement via local unitary invariants seem very daunting. I point o…
Production and verification of multipartite quantum state are an essential step in quantum information processing. In this work, we propose an efficient method to decompose symmetric multipartite observables, which are invariant under permutations between parties, with only $(N+1)(N+2)/2$ local measurement settings, where $N$ is the number of qubits. We apply the decomposition technique to evaluate the fidelity between an unknown prepared state and any target permutation invariant state. In addition, for some typical permutation invariant states, such as the Dicke state with a constant number …
We show that a positive homogeneous function that is invariant under determinant-1 stochastic local operations and classical communication (SLOCC) transformations defines an N-qubit entanglement monotone if and only if the homogeneous degree is not larger than four. In particular this implies that any power larger than one of the well-known N-tangle (N > 2) is not an entanglement monotone anymore. We then describe a common basis and formalism for the N-tangle and other known invariant polynomials of degree four. This allows us to elucidate the relation of the four-qubit invariants defined by L…
Macroscopic theory of multipartite correlations in permutation-invariant open quantum systems
本文提出了一种方法，用于研究可交换开放量子系统中多重粒子关联的宏观行为，发现系统弛豫到时间依赖吸引子时，互信息的广泛缩放具有鲁棒性。
On genuine multipartite entanglement signals
本文提出了一种通用方法，通过Möbius反演构造真正的多粒子纠缠信号，并展示了其在多不变量中的应用。
It is a well-established fact in quantum information theory, that uniform averaging over the collective action of a unitary group on a multipartite quantum state projects the state to a form equivalent to a permutation operator of the subsystems. Hence states equivalent to permutation operators are untouched by collective unitary noise. A trivial observation shows that uniform averaging over permutation operators projects the state into a form with block-diagonal structure equivalent to the one of the collective action of the unitary group. We introduce a name for this property: duality of ave…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。