Peter Vrana
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.
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We give an universal algorithm for testing the local unitary equivalence of states for multipartite system with arbitrary dimensions.
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}.
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 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.
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 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.
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 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.
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.
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…
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 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.
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…
Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are responsible for many nonintuitive physical phenomena. The entanglement property of normalized Laplacian matrices is in general not invariant under graph isomorphism. In recent papers, graphs were identified wh…
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…
Multi-Unitary Complex Hadamard Matrices
本文研究受附加对称性约束的复 Hadamard 矩阵，通过重排与部分转置定义对偶、强对偶及 $k$-酉结构，并建立其与多体最大纠缠态及可解量子模型的联系。
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。
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…
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 investigate genuine multipartite nonlocality of pure permutationally invariant multimode Gaussian states of continuous variable systems, as detected by the violation of Svetlichny inequality. We identify the phase space settings leading to the largest violation of the inequality when using displaced parity measurements, distinguishing our results between the cases of even and odd total number of modes. We further consider pseudospin measurements and show that, for three-mode states with asymptotically large squeezing degree, particular settings of these measurements allow one to approach th…
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…
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.
For two symmetric quantum states one may be interested in maximizing the overlap under local operations applied to one of them. The question arises whether the maximal overlap can be obtained by applying the same local operation to each party. We show that for two symmetric multiqubit states and local unitary transformations this is the case; the maximal overlap can be reached by applying the same unitary matrix everywhere. For local invertible operations (stochastic local operations assisted by classical communication equivalence), however, we present counterexamples, demonstrating that consi…
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
Tensor invariants for multipartite entanglement classification
本文探讨了多粒子纠缠结构的空间组织，通过trace-invariants与着色图的对应关系，提出了一种分类多粒子纯态LU轨道的方法，并分析了局部操作对纠缠资源理论的影响。
Quantum Coherence: A Fundamental Resource for Establishing Genuine Multipartite Correlations
本文建立了真正多体纠缠度量与相干性度量间的等价关系，证明通过幺正非相干操作可从初始态相干性产生多体纠缠，并揭示了相干性与多体导引、非定域性间的依存关系。
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…
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。
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…
Quantum thermodynamics can be cast as a resource theory by considering free access to a heat bath, thereby viewing the Gibbs state at a fixed temperature as a free state and hence any other state as a resource. Here, we consider a multipartite scenario where several parties attempt at extracting work locally, each having access to a local heat bath (possibly with a different temperature), assisted with an energy-preserving global unitary. As a specific model, we analyze a collection of harmonic oscillators or a multimode bosonic system. Focusing on the Gaussian paradigm, we construct a reasona…
Constructing Multipartite Planar Maximally Entangled States from Phase States and Quantum Secret Sharing Protocol
本文研究从相位态经幺正动力学构造平面最大纠缠（PME）态的方法，给出二、三、四及K量子比特系统的构造方案，克服AME态在低维系统中的限制，并推动其在量子秘密共享等协议中的应用。
Absolutely Maximally Entangled (AME) states are those multipartite quantum states that carry absolute maximum entanglement in all possible partitions. AME states are known to play a relevant role in multipartite teleportation, in quantum secret sharing and they provide the basis novel tensor networks related to holography. We present alternative constructions of AME states and show their link with combinatorial designs. We also analyze a key property of AME, namely their relation to tensors that can be understood as unitary transformations in every of its bi-partitions. We call this property m…
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. $…
Recently, the problem of discriminating multipartite unitary operations by local operations and classical communication (LOCC) has attracted significant attention. The latest work in the literature on this problem showed that two multipartite unitary operations can always be perfectly distinguished by LOCC when a finite number of runs are allowable. However, in these schemes, local entanglement (an entangled state holden by one party) was required, which seems to imply that local entanglement is necessary for perfect discrimination between unitary operations by LOCC. In this article, we show t…
We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such an SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Our construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, we provide an elegant formula for the…
The stabilizer group of an n-qubit state ψis the set of all matrices of the form g=g_1\otimes\cdots\otimes g_n, with g_1,...,g_n being any 2x2 invertible complex matrices, that satisfy gψ=ψ. We show that for 5 or more qubits, except for a set of states of zero measure, the stabilizer group of multipartite entangled states is trivial; that is, containing only the identity element. We use this result to show that for 5 or more qubits, the action of deterministic local operations and classical communication (LOCC) can almost always be simulated simply by local unitary (LU) operations. This proves…
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…
On properties of Schmidt Decomposition
本文回顾二部施密特分解性质并研究其向多部态的扩展，指出施密特数通过可分幺正变换定义等价类，证明最高施密特数多部态划分是NP完全问题，且复合系统密度矩阵纯化保持施密特可分解性。
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