Quite recently, Cai et al. [arXiv:2006.07165v1] proposed a new concept "absolutely entangled set" for bipartite quantum systems: for any possible choice of global basis, at least one state of the set is entangled. There they presented a minimum example with a set of four states in two qubit systems and they proposed a quantitative measure for the absolute set entanglement. In this work, we derive two necessity conditions for a set of states to be an absolutely entangled set. In addition, we give a series constructions of absolutely entangled bases on $\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}$ for any nonprime dimension $d=d_1\times d_2$. Moreover, based on the structure of the orthogonal product basis in $\mathbb{C}^2\otimes \mathbb{C}^n$, we obtain another construction of absolutely entangled set with $2n+1$ elements in $\mathbb{C}^2\otimes \mathbb{C}^n$.
We develop a new technique to construct mutually unbiased tripartite absolutely maximally entangled bases. We first explore the tripartite absolutely maximally entangled bases and mutually unbiased bases in $\mathbb{C}^{d} \otimes \mathbb{C}^{d} \otimes \mathbb{C}^{d}$ based on mutually orthogonal Latin squares. Then we generalize the approach to the case of $\mathbb{C}^{d_{1}} \otimes \mathbb{C}^{d_{2}} \otimes \mathbb{C}^{d_{1}d_{2}}$ by mutually weak orthogonal Latin squares. The concise direct constructions of mutually unbiased tripartite absolutely maximally entangled bases are remarkably…
Absolutely maximally entangled (AME) states are multipartite entangled states that are maximally entangled for any possible bipartition. In this paper, we study the description of AME states within the graph state formalism. The graphical representation provides an intuitive framework to visualize the entanglement in graph states, which makes them a natural candidate to describe AME states. We show two different methods of determining bipartite entanglement in graph states and use them to define various AME graph states. We further show that AME graph states exist for all number of parties, an…
Absolutely Maximally Entangled (AME) states are maximally entangled for every bipartition of the system. They are crucial resources for various quantum information protocols. We present techniques for verifying either two AME states are equivalent concerning Stochastic Local Operations and Classical Communication (SLOCC). The conjecture that for a given multipartite quantum system all AME states are SLOCC-equivalent is falsified. We also show that the existence of AME states with minimal support of 6 or more particles results in the existence of infinitely many such non-SLOCC-equivalent states…
The absolutely maximally entangled (AME) states play key roles in quantum information processing. We provide an explicit expression of the generalized Bloch representation of AME states for general dimension $d$ of individual subsystems and arbitrary number of partite $n$. Based on this analytic formula, we prove that the trace of the squared support for any given weight is given by the so-called hyper-geometric function and is irrelevant with the choices of the subsystems. The optimal point for the existence of AME states is obtained.
For every stabiliser $N$-qudit absolutely maximally entangled state, we present a method for determining the stabiliser generators and logical operators of a corresponding quantum error correction code. These codes encode $k$ qudits into $N-k$ qudits, with $k\leq \left \lfloor{N/2} \right \rfloor$, where the local dimension $d$ is prime. We use these methods to analyse the concatenation of such quantum codes and link this procedure to entanglement swapping. Using our techniques, we investigate the spread of quantum information on a tensor network code formerly used as a toy model for the AdS/C…
Pure multiparticle quantum states are called absolutely maximally entangled if all reduced states obtained by tracing out at least half of the particles are maximally mixed. We provide a method to characterize these states for a general multiparticle system. With that, we prove that a seven-qubit state whose three-body marginals are all maximally mixed, or equivalently, a pure $((7,1,4))_2$ quantum error correcting code, does not exist. Furthermore, we obtain an upper limit on the possible number of maximally mixed three-body marginals and identify the state saturating the bound. This solves t…
We investigate absolutely maximally entangled (AME) states, which are multipartite quantum states that are maximally entangled with respect to any possible bipartition. These strong entanglement properties make them a powerful resource for a variety of quantum information protocols. In this paper, we show the existence of AME states for any number of parties, given that the dimension of the involved systems is chosen appropriately. We prove the equivalence of AME states shared between an even number of parties and pure state threshold quantum secret sharing (QSS) schemes, and prove necessary a…
We extend the relation between absolutely maximally entangled (AME) states and quantum maximum distance separable (QMDS) codes by constructing whole families of QMDS codes from their parent AME states. We introduce a reduction-friendly form for the generator set of the stabilizer representation of an AME state, from which the stabilizer form for children codes, all QMDS, can be obtained. We then relate this to optimal codes for one-way quantum repeaters, by minimizing the short-term infrastructure cost as well as the long-term running cost of such quantum repeaters. We establish that AME state…
In this paper we prove that no absolutely maximally entangled, AME, state with minimal support exists with 7 sites and 5 levels. General AME states are pure multipartite states that, when reduced to half or less of the sites, the maximum entropy mixed state is obtained. They have found applications in teleportation and quantum secret sharing, and finding conditions for their existence is a well known open problem. We consider the version of this problem for minimally supported AME states. We single out known both sufficient and necessary conditions in that case. From our negative result, we sh…
Quantum Circuits for High-Dimensional Absolutely Maximally Entangled States
Barcelona Supercomputing Center（巴塞罗那超级计算中心）；Universitat de Barcelona（巴塞罗那大学）；BEIT；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO-光子科学研究所，巴塞罗那科学技术研究院）；Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）；Institute of Theoretical Physics, Jagiellonian University（雅盖隆大学理论物理研究所）
A set of quantum states is said to be absolutely entangled, when at least one state in the set remains entangled for any definition of subsystems, i.e. for any choice of the global reference frame. In this work we investigate the properties of absolutey entangled sets (AES) of pure quantum states. For the case of a two-qubit system, we present a sufficient condition to detect an AES, and use it to construct families of $N$ states such that $N-3$ (the maximal possible number) remain entangled for any definition of subsystems. For a general bipartition $d=d_1d_2$, we prove that sets of $N>\left\…
Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed. These states are relevant for our understanding of multipartite entanglement, quantum information protocols, and the construction of quantum error correction codes. To our knowledge, the only known systematic construction of these quantum states is based on classical error correction codes. We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-un…
Absolutely maximally entangled, AME, states are pure multipartite states that give rise to the maximally mixed states when half or more of the parties are traced out. AME states find applications in fields like teleportation or quantum secret sharing, and the problem of finding conditions on their existence has been considered in a number of papers. We consider here AME states of minimal support, that are simpler to analyse. An equivalence with coding theory gives a sufficient condition for their existence, that the number of sites be equal to the local dimension plus one, when the local dimen…
We characterize the boundary of the convex compact set of absolutely separable states, referred as {\bf AS}, that cannot be transformed to entangled states by global unitary operators, in $2\otimes d$ Hilbert space. However, we show that the absolutely separable states of rank-$(2d-1)$ are extreme points of such sets. We then discuss conditions to examine if a given full-rank absolutely separable state is an interior point or a boundary point of {\bf AS}. Moreover, we construct two-qubit absolutely separable states which are boundary points but not extreme points of {\bf AS} and prove the exis…
Absolutely maximally entangled (AME) states are typically defined in homogeneous systems. However, the quantum system is more likely to be heterogeneous in a practical setup. In this work we pay attention to the construction of AME states in tripartite heterogeneous systems. We first introduce the concept of irreducible AME states as the basic elements to construct AME states with high local dimensions. Then we investigate the tripartite heterogeneous systems whose local dimensions are $l,m,n$, with $3\leq l<m<n\leq m+l-1$. We show the existence of AME states in such heterogeneous systems is r…
Absolutely maximally entangled pure states of multipartite quantum systems
BEIT sp. z o.o.（BEIT有限责任公司）；Jagiellonian University（雅盖隆大学）；Universität Siegen（锡根大学）；Indian Institute of Technology Madras（马德拉斯印度理工学院）；Polish Academy of Sciences（波兰科学院）
Entangled states are undoubtedly an integral part of various quantum information processing tasks. On the other hand, absolutely separable states which cannot be made entangled under any global unitary operations are useless from the resource theoretic perspective, and hence identifying non-absolutely separable states can be an important issue for designing quantum technologies. Here we report that nonlinear witness operators provide significant improvements in detecting non-absolutely separable states over their linear analogs, by invoking examples of states in various dimensions. We also add…
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…
Complete Existence Classification of Seven-Partite Absolutely Maximally Entangled States
School of Computer Science and Engineering, Sun Yat-sen University（中山大学计算机学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）；QICI Quantum Information and Computation Initiative, School of Computing and Data Science, The University of Hong Kong（香港大学计算与数据科学学院量子信息与计算倡议）
A pure multipartite quantum state is called absolutely maximally entangled (AME), if all reductions obtained by tracing out at least half of its parties are maximally mixed. Maximal entanglement is then present across every bipartition. The existence of such states is in many cases unclear. With the help of the weight enumerator machinery known from quantum error correction and the generalized shadow inequalities, we obtain new bounds on the existence of AME states in dimensions larger than two. To complete the treatment on the weight enumerator machinery, the quantum MacWilliams identity is d…
We study the conditions when mixtures of entangled pure states with maximally mixed one-qudit reduced density matrices remain entangled. We found that the resulting mixed state remains entangled when the number of entangled pure states to be mixed is less than or equal to the dimension of the pure states. For the latter case of mixing a number of pure states equal to their dimension, we found that the mixed state is entangled provided that the entangled pure states to be mixed are not equally weighted. We also found that one can restrict the set of pure states that one can mix from in order to…
We present two optimal methods of teleporting an unknown qubit using any pure entangled state. We also discuss how such methods can also have succesful application in quantum secret sharing with pure multipartite entangled states.
We propose a scheme to produce the pure entangled states through type II downconversion. In the scheme, the vacuum states are excluded and the a prori pure entangled states are produced and verified without destroying the state itself. This can help to carry out the many unditional experiments related to quantum entanglement.
A complete analysis of entangled bipartite qutrit pure states is carried out based on a simple entanglement measure. An analysis of all possible extremally entangled pure bipartite qutrit states is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis and the results should be helpful for finding different entanglement types in multipartite pure state systems.
I show that two distant parties can transform pure entangled states to arbitrary pure states by stochastic local operations and classical communication (SLOCC) at the single copy level, if they share bound entangled states. This is the effect of bound entanglement since this entanglement processing is impossible by SLOCC alone. Similar effect of bound entanglement occurs in three qubits where two incomparable entangled states of GHZ and W can be inter-converted. In general multipartite settings composed by $N$ distant parties, all $N$-partite pure entangled states are inter-convertible by SLOC…
We present an entanglement purification protocol for a mixture of a pure entangled state and a pure product state, which are orthogonal to each other. The protocol is combination of bisection method and one-way hashing protocol. We give recursive formula for the rate of the protocol for different states, i.e. the number of maximally entangled two-qubit pairs obtained with the protocol per a single copy of the initial state. We also calculate numerically the rate for some states.
Any pure entangled state of two particles violates a Bell inequality for two-particle correlation functions (Gisin's theorem). We show that there exist pure entangled N>2 qubit states that do not violate any Bell inequality for N particle correlation functions for experiments involving two dichotomic observables per local measuring station. We also find that Mermin-Ardehali-Belinskii-Klyshko inequalities may not always be optimal for refutation of local realistic description.
We derive a sufficient condition for a set of pure states, each entangled in two remote $N$-dimensional systems, to be transformable to $k$-dimensional-subspace equivalent entangled states ($k\leq N$) by same local operations and classical communication. If $k=N$, the condition is also necessary. This condition reveals the function of the relative marginal density operators of the entangled states in the entanglement manipulation without sufficient information of the initial states.
We show that all pure entangled states of two $d$-dimensional quantum systems (i.e., two qudits) can be generated from an initial separable state via a universal Yang--Baxter matrix if one is assisted by local unitary transformations.
We examine dense coding with an arbitrary pure entangled state sharing between the sender and the receiver. Upper bounds on the average success probability in approximate dense coding and on the probability of conclusive results in unambiguous dense coding are derived. We also construct the optimal protocol which saturates the upper bound in each case.
We construct a large family of Planar Maximally Entangled (PME) states which are a wider class of multi-partite entangled states than Absolutely Maximally Entangled (AME) states. These are states in which any half of the qudits are in a maximally mixed state, provided that they form a connected subset. We show that in contrast to AMEs, PMEs are easier to find and there are various PMEs for any even number of qudits. In particular, while it is known that no AME state of four qubits exists, we show that there are two distinct multi-parameter classes of four qubit PMEs. We also give explicit fami…
We prove that a pure entangled state of two subsystems with equal spin is equivalent to a two-mode spin-squeezed state under local operations except for a set of bipartite states with measure zero, and we provide a counterexample to the generalization of this result to two subsystems of unequal spin.
Geometric properties of the set of quantum entangled states are investigated. We propose an explicit method to compute the dimension of local orbits for any mixed state of the general K x M problem and characterize the set of effectively different states (which cannot be related by local transformations). Thus we generalize earlier results obtained for the simplest 2 x 2 system, which lead to a stratification of the 6D set of N=4 pure states. We define the concept of absolutely separable states, for which all globally equivalent states are separable.
Entangled pure-states, Werner-states and generalized mixed-states of any structure, spanning a 2x2 Hilbert space are created by a novel high-brilliance universal source of polarization-entangled photon pairs. The violation of a Bell inequality has been tested for the first time with a pure-state, indeed a conceptually relevant ideal condition, and with Werner-states. The generalized ''maximally entangled mixed states'' (MEMS) were also synthetized for the first time and their exotic properties investigated by means of a quantum tomographic technique.
The spectral decomposition of all (n,2,2) Bell operators ($2^{2^n}$ in number, $n \ge 2$), as introduced by Werner and Wolf, is done. Its implications on the characterization of Bell operators as probes of entanglement are considered in detail.
We report an exhaustive numerical analysis of violations of local realism by two qutrits in all possible pure entangled states. In Bell type experiments we allow any pairs of local unitary U(3) transformations to define the measurement bases. Surprisingly, Schmidt rank-2 states, resembling pairs of maximally entangled qubits, lead to the most noise-robust violations of local realism. The phenomenon seems to be even more pronounced for four and five dimensional systems, for which we tested a few interesting examples.
We demonstrate one-sided device-independent self-testing of any pure entangled two-qubit state based on a fine-grained steering inequality. The maximum violation of a fine-grained steering inequality can be used to witness certain steerable correlations, which certify all pure two-qubit entangled states. Our experimental results identify which particular pure two-qubit entangled state has been self-tested and which measurement operators are used on the untrusted side. Furthermore, we analytically derive the robustness bound of our protocol, enabling our subsequent experimental verification of …
We present a scheme for quantum secure direct communication with quantum encryption. The two authorized users use repeatedly a sequence of the pure entangled pairs (quantum key) shared for encrypting and decrypting the secret message carried by the traveling photons directly. For checking eavesdropping, the two parties perform the single-photon measurements on some decoy particles before each round. This scheme has the advantage that the pure entangled quantum signal source is feasible at present and any eavesdropper cannot steal the message.
We look for local unitary operators $W_1 \otimes W_2$ which would rotate all equally entangled two-qubit pure states by the same but arbitrary amount. It is shown that all two-qubit maximally entangled states can be rotated through the same but arbitrary amount by local unitary operators. But there is no local unitary operator which can rotate all equally entangled non-maximally entangled states by the same amount, unless it is unity. We have found the optimal sets of equally entangled non-maximally entangled states which can be rotated by the same but arbitrary amount via local unitary operat…
Nonlocal correlations as revealed by the violations to Bell inequalities are incompatible with local models without any nonlocal correlations. However some tripartite entangled states, e.g., symmetric pure states, exhibit a stronger nonlocality even incompatible with hybrid local/nonlocal models allowing nonlocal correlations between any two parties. Here we propose a three-particle Hardy-type test without inequality, as illustrated by a gedanken experiment, that is failed by all non-signaling local models while is passed by all tripartite entangled asymmetric pure states. Our result implies t…
In this paper, we construct a measure of entanglement by generalizing the quadric polynomial of the Segre variety for general multipartite states. We give explicit expressions for general pure three-partite and four-partite states. Moreover, we will discuss and compare this measure of entanglement with the generalized concurrence.
We report on the experimental generation of an entangled state with a spectrally pure heralded single-photon state and a weak coherent state. By choosing group-velocity matching in the nonlinear crystal, our system for producing entangled photons was 60 times brighter than that in the earlier experiment [Phys. Rev. Lett. 90, 240401 (2003)], with no need of bandpass filters. This entanglement system is useful for quantum information protocols that require indistinguishable photons from independent sources.
