Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.
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.
The multipartite entangled state $|p,ξ_2,ξ_2,...,ξ_n>$ for the total momentum and relative coordinates of n particles is constructed. The corresponding quantum mechanical operator with respect to the classical transformation $p\to e^{λ_1} p$, $ξ_i\to e^{λ_i} ξ_i$, $i=1,...,n$, in the state $|p,ξ_2,ξ_2,...,ξ_n>$ is investigated.
In this note we generalize Nielsen's marjoization criterion for the convertibility of bipartite pure states [Phys. Rev. Lett \textbf{83}, 436(1999)] to a special class of multipartite pure states which have generalized Schmidt decompositions.
Transformations involving only local operations assisted with classical communication are investigated for multipartite entangled pure states having tensor rank 2. All necessary and sufficient conditions for the possibility of deterministically converting truly multipartite, rank-2 states into each other are given. Furthermore, a chain of local operations that successfully achieves the transformation has been identified for all allowed transformations. The identified chains have two nice features: (1) each party needs to carry out at most one local operation and (2) all of these local operatio…
Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure multipartite multilevel states. Combined with the analogous results for qubit states shown by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a characterization of almost all local transformations among multipartite pure states. We show that nontrivial LOCC transformations among generic, fully entangled…
Multipartite quantum correlations, in spite of years of intensive research, still leave many questions unanswered. While bipartite entanglement is relatively well understood for Gaussian states, the complexity of mere qualitative characterization grows rapidly with increasing number of parties. Here, we present two schemes for transformations of multipartite permutation invariant Gaussian states by Gaussian local operations and classical communication. To this end, we use a scheme for possible experimental realization, making use of the fact, that in this picture, the whole N - partite state c…
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.
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.
Characterizing the transformation and classification of multipartite entangled states is a basic problem in quantum information. We study the problem under two most common environments, local operations and classical communications (LOCC), stochastic LOCC and two more general environments, multi-copy LOCC (MCLOCC) and multi-copy SLOCC (MCSLOCC). We show that two transformable multipartite states under LOCC or SLOCC are also transformable under MCLOCC and MCSLOCC. What's more, these two environments are equivalent in the sense that two transformable states under MCLOCC are also transformable un…
Reversible state transformations under entanglement non-increasing operations give rise to entanglement measures. It is well known that asymptotic local operations and classical communication (LOCC) are required to get a simple operational measure of bipartite pure state entanglement. For bipartite mixed states and multipartite pure states it is likely that a more powerful class of operations will be needed. To this end \cite{BPRST01} have defined more powerful versions of state transformations (or reducibilities), namely LOCCq (asymptotic LOCC with a sublinear amount of quantum communication)…
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.
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 studied pure state transformations using local operations assisted by finitely many rounds of classical communication ($LOCC_{\mathbb{N}}$) in C. Spee, J.I. de Vicente, D. Sauerwein, B. Kraus, arXiv:1606.04418 (2016). Here, we first of all present the details of some of the proofs and generalize the construction of examples of state transformations via $LOCC_{\mathbb{N}}$ which require a probabilistic step. However, we also present explicit examples of SLOCC classes where any separable transformation can be realized by a protocol in which each step is deterministic (all-det-$LOCC_{\mathbb{N…
Recently, it was demonstrated by Son et al., Phys. Rev. Lett. \textbf{102}, 110404 (2009), that a separable bipartite continuous variable quantum system can violate the Clauser-Horne-Shimony-Holt (CHSH) inequality via operationally local transformations. Operationally local transformations are parameterized only by local variables, however in order to allow violation of the CHSH inequality a maximally entangled ancilla was necessary. The use of the entangled ancilla in this scheme caused the state under test to become dependent on the measurement choice one uses to calculate the CHSH inequalit…
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 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.
The ability to generate and verify multipartite entanglement is an important benchmark for near-term quantum devices devices. We develop a scalable entanglement metric based on multiple quantum coherences, and demonstrate experimentally on a 20-qubit superconducting device - the IBM Q System One. We report a state fidelity of 0.5165$\pm$0.0036 for an 18-qubit GHZ state, indicating multipartite entanglement across all 18 qubits. Our entanglement metric is robust to noise and only requires measuring the population in the ground state; it can be readily applied to other quantum devices to verify …
We consider a partial trace transformation which maps a multipartite quantum state to collection of local density matrices. We call this collection a mean field state. The necessary and sufficient conditions under which a mean field state is compatible with at least one multipartite pure state are found for the system of $n$ qubits and for the tripartite system with the Hilbert space of dimension 2x2x4. Compatibility of mean field states with more general classes of multipartite quantum states is discussed.
The tensor rank (also known as generalized Schmidt rank) of multipartite pure states plays an important role in the study of entanglement classifications and transformations. We employ powerful tools from the theory of homogeneous polynomials to investigate the tensor rank of symmetric states such as the tripartite state $\ket{W_3}=\tfrac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001})$ and its $N$-partite generalization $\ket{W_N}$. Previous tensor rank estimates are dramatically improved and we show that (i) three copies of $\ket{W_3}$ has rank either 15 or 16, (ii) two copies of $\ket{W_N}$ has…
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 develop graph theoretic methods for analysing maximally entangled pure states distributed between a number of different parties. We introduce a technique called {\it bicolored merging}, based on the monotonicity feature of entanglement measures, for determining combinatorial conditions that must be satisfied for any two distinct multiparticle states to be comparable under local operations and classical communication (LOCC). We present several results based on the possibility or impossibility of comparability of pure multipartite states. We show that there are exponentially many such entangl…
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 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.
Topos quantum theory provides representations of quantum states as direct generalizations of the probability distribution, namely probability valuation. In this article, we consider extensions of a known bijective correspondence between quantum states and probability valuations to composite systems and to state transformations. We show that multipartite probability valuations on composite systems have a bijective correspondence to positive over pure tensor states, according to a candidate definition of the composite systems in topos quantum theory. Among the multipartite probability valuations…
We propose a multi party quantum clock synchronization protocol that makes optimal use of the maximal multipartite entanglement of GHZ-type states. To realize the protocol, different versions of maximally entangled eigenstates of collective energy are generated by local transformations that distinguish between different groupings of the parties. The maximal sensitivity of the entangled states to time differences between the local clocks can then be accessed if all parties share the results of their local time dependent measurements. The efficiency of the protocol is evaluated in terms of the s…
Suppose several parties jointly possess a pure multipartite state, |ψ>. Using local operations on their respective systems and classical communication (i.e. LOCC) it may be possible for the parties to transform deterministically |ψ> into another joint state |ϕ>. In the bipartite case, Nielsen majorization theorem gives the necessary and sufficient conditions for this process of entanglement transformation to be possible. In the multipartite case, such a deterministic local transformation is possible only if both the states in the same stochastic LOCC (SLOCC) class. Here we generalize Nielsen m…
Entanglement shared among multiple parties presents complex challenges for the characterisation of different types of entanglement. One of the most fundamental insights is the fact that some mixed states can feature entanglement across every possible cut of a multipartite system yet can be produced via a mixture of states separable with respect to different partitions. To distinguish states that genuinely cannot be produced from mixing such partition-separable states, the term genuine multipartite entanglement was coined. All these considerations originate in a paradigm where only a single cop…
We demonstrate the presence of genuine multipartite entanglement between the modes of quantum fields in non-uniformly moving cavities. The transformations generated by the cavity motion can be considered as multipartite quantum gates. We present two setups for which multi-mode entanglement can be generated for bosons and fermions. As a highlight we show that the bosonic genuine multipartite correlations can be resonantly enhanced. Our results provide fundamental insights into the structure of Bogoliubov transformations and suggest strong links between quantum information, quantum fields in cur…
Using graphs and hypergraphs to systematically model collections of arbitrary subsets of parties representing {\it ensembles (or collections)} of shared multipartite CAT states, we study transformations between such {\it ensembles} under {\it local operations and classical communication (LOCC)}. We show using partial entropic criteria, that any two such distinct ensembles represented by {\it $r$-uniform hypergraphs} with the same number of hyperedges (CAT states), are LOCC incomparable for even integers $r\geq 2$, generalizing results in \cite{mscthesis,sin:pal:kum:sri}. We show that the cardi…
We propose a new scheme for realizing nonlocal nondegenerate optical parametric amplifier by means of genuine four-mode entanglement. The nondegenerate optical parametric amplifier is regarded as a unitary transformation from two input fields to two outputs. Two inputs and two outputs are located at the remote stations respectively. Employing off-line four-mode entanglement, homodyne detection, classical communication and local displacement, the nonlocal nondegenerate optical parametric amplifier can be achieved. This scheme can be implemented with a setup that is experimentally accessible at …
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…
Genuine multipartite entanglement (GME) is considered a powerful form of entanglement since it corresponds to those states that are not biseparable, i.e.\ a mixture of partially separable states across different bipartitions of the parties. In this work we study this phenomenon in the multiple-copy regime, where many perfect copies of a given state can be produced and controlled. In this scenario the above definition leads to subtle intricacies as biseparable states can be GME-activatable, i.e.\ several copies of a biseparable state can display GME. We show that the set of GME-activatable stat…
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 explore the question of using an entangled state as a universal resource for implementing quantum measurements by local operations and classical communication (LOCC). We show that for most systems consisting of three or more subsystems, there is no entangled state from the same space that can enable all measurements by LOCC. This is in direct contrast to the bipartite case, where a maximally entangled state is an universal resource. Our results are obtained showing an equivalence between the problem of local state transformation and that of entanglement-assisted local unambiguous state disc…
A new entanglement measure, the multiple entropy measures (MEMS), is proposed to quantify quantum entanglement of multi-partite quantum state. The MEMS is vector-like with $m=[N/2]$, the integer part of $N/2$, components: $[S_1, S_2,..., S_m]$, and the $i$-th component $S_i$ is the geometric mean of $i$-body partial entropy of the system. The $S_i$ measures how strong an arbitrary $i$ bodies from the system are entangled with the rest of the system. The MEMS is not only transparent in physical picture, but also simple to calculate. It satisfies the conditions for a good entanglement measure. W…
Understanding the nature of multipartite entanglement is a central mission of quantum information theory. To this end, we investigate the question of tripartite entanglement convertibility. We find that there exists no easy criterion to determine whether a general tripartite transformation can be performed with a nonzero success probability and in fact, the problem is NP-hard. Our results are based on the connections between multipartite entanglement and tensor rank (also called Schmidt rank), a key concept in algebraic complexity theory. Not only does this relationship allow us to characteriz…
Characterizing genuine quantum resources and determining operational rules for their manipulation are crucial steps to appraise possibilities and limitations of quantum technologies. Two such key resources are nonclassicality, manifested as quantum superposition between reference states of a single system, and entanglement, capturing quantum correlations among two or more subsystems. Here we present a general formalism for the conversion of nonclassicality into multipartite entanglement, showing that a faithful reversible transformation between the two resources is always possible within a pre…
In an effort to simplify the classification of pure entangled states of multi (m) -partite quantum systems, we study exactly and asymptotically (in n) reversible transformations among n'th tensor powers of such states (ie n copies of the state shared among the same m parties) under local quantum operations and classical communication (LOCC). With regard to exact transformations, we show that two states whose 1-party entropies agree are either locally-unitarily (LU) equivalent or else LOCC-incomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger (GHZ) states are LOCC-…
We explore the structure of multipartite quantum systems which are entangled in multiple degrees of freedom. We find necessary and sufficient conditions for the characterization of tripartite systems and necessary conditions for any number number of parties. Furthermore we develop a framework of multi-level witnesses for efficient discrimination and quantification of multidimensional entanglement that is applicable for an arbitrary number of systems and dimensions.
Encoding and decoding quantum information in a multipartite quantum system are indispensable for quantum error correction and also play crucial roles in multiparty tasks in distributed quantum information processing such as quantum secret sharing. To quantitatively characterize nonlocal properties of multipartite quantum transformations for encoding and decoding, we analyze entanglement costs of encoding and decoding quantum information in a multipartite quantum system distributed among spatially separated parties connected by a network. This analysis generalizes previous studies of entangleme…
Testing Genuine Multipartite Nonlocality via an Inflated Network with Multi-copy Entangled States
International Quantum Academy（国际量子科学院）；Southern University of Science and Technology（南方科技大学）；School of Information Science and Technology, Southwest Jiaotong University（西南交通大学信息科学与技术学院）；Hefei National Laboratory（合肥国家实验室）；School of Physics, Nankai University（南开大学物理学院）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）；Max-Planck-Institute for Mathematics in the Sciences（马克斯·普朗克科学研究所）
Distributed quantum sensing uses quantum correlations between multiple sensors to enhance the measurement of unknown parameters beyond the limits of unentangled systems. We describe a sensing scheme that uses continuous-variable multipartite entanglement to enhance distributed sensing of field-quadrature displacement. By dividing a squeezed-vacuum state between multiple homodyne-sensor nodes using a lossless beam-splitter array, we obtain a root-mean-square (rms) estimation error that scales inversely with the number of nodes (Heisenberg scaling), whereas the rms error of a distributed sensor …
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…
Several entanglement measures are used to define equivalence classes in the set of hypergraph states of three qubits. Our classifications reveal that (i) under local unitary transformations, hypergraph states of three qubits are split into six classes and only one class of them is not equivalent to any graph state; (ii) under stochastic local operations with classical communication, for the single copy case hypergraph states of three qubits, partitioned into five classes which can not be converted into a W state, are equivalent to graph states; and (iii) when bipartite entanglement in three qu…
Quantum coherence has received significant attention in recent years, but its study is mostly conducted in single party settings. In this paper, we generalize important results in multipartite entanglement theory to their counterparts in quantum coherence theory. First, we give a necessary and sufficient condition for when two pure multipartite states are equivalent under local quantum incoherent operations and classical communication (LICC), i.e., two states can be deterministically transformed to each other under LICC operations. Next, we investigate and give the conditions in which such a t…
The generation of genuine multipartite entangled states is challenging in practice. Here we explore a new route to this task, via autonomous entanglement engines which use only incoherent coupling to thermal baths and time-independent interactions. We present a general machine architecture, which allows for the generation of a broad range of multipartite entangled states in a heralded manner. Specifically, given a target multiple-qubit state, we give a sufficient condition ensuring that it can be generated by our machine. We discuss the cases of Greenberger-Horne-Zeilinger, Dicke and cluster s…
We introduce and analyze graph-associated entanglement cost, a generalization of the entanglement cost of quantum states to multipartite settings. We identify a necessary and sufficient condition for any multipartite entangled state to be constructible when quantum communication between the multiple parties is restricted to a quantum network represented by a tree. The condition for exact state construction is expressed in terms of the Schmidt ranks of the state defined with respect to edges of the tree. We also study approximate state construction and provide a second-order asymptotic analysis…
Entanglement is a resource in quantum information theory when state manipulation is restricted to Local Operations assisted by Classical Communication (LOCC). It is therefore of paramount importance to decide which LOCC transformations are possible and, particularly, which states are maximally useful under this restriction. While the bipartite maximally entangled state is well known (it is the only state that cannot be obtained from any other and, at the same time, it can be transformed to any other by LOCC), no such state exists in the multipartite case. In order to cope with this fact, we in…
Various parameterizations for the orbits under local unitary transformations of three-qubit pure states are analyzed. The interconvertibility, symmetry properties, parameter ranges, calculability and behavior under measurement are looked at. It is shown that the entanglement monotones of any multipartite pure state uniquely determine the orbit of that state under local unitary transformations. It follows that there must be an entanglement monotone for three-qubit pure states which depends on the Kempe invariant defined in [Phys. Rev. A 60, 910 (1999)]. A form for such an entanglement monotone …
