Highly entangled multipartite states such as k-uniform (k-UNI) and absolutely maximally entangled (AME) states serve as critical resources in quantum networking and other quantum information applications. However, there does not yet exist a complete classification of such states, and much remains unknown about their entanglement structure. Here, we substantially broaden the class of known k-UNI and AME states by introducing a method for explicitly constructing such states that combines classical error correcting codes and qudit graph states. This method in fact constitutes a general recipe for obtaining multipartitite entangled states from classical codes. Furthermore, we show that at least for a large subset of this new class of k-UNI states, the states are inequivalent under stochastic local operations and classical communication. This subset is defined by an iterative procedure for constructing a hierarchy of k-UNI graph states.
We examine the existence and structure of particular sets of mutually unbiased bases (MUBs) in bipartite qudit systems. In contrast to well-known power-of-prime MUB constructions, we restrict ourselves to using maximally entangled stabilizer states as MUB vectors. Consequently, these bipartite entangled stabilizer MUBs (BES MUBs) provide no local information, but are sufficient and minimal for decomposing a wide variety of interesting operators including (mixtures of) Jamiolkowski states, entanglement witnesses and more. The problem of finding such BES MUBs can be mapped, in a natural way, to …
Self-testing was originally introduced as a device-independent method of certification of entangled quantum states and local measurements performed on them. Recently, in [F. Baccari \textit{et al.}, arXiv:2003.02285] the notion of state self-testing has been generalized to entangled subspaces and the first self-testing strategies for exemplary genuinely entangled subspaces have been given. The main aim of our work is to pursue this line of research and to address the question how "large" (in terms of dimension) are genuinely entangled subspaces that can be self-tested, concentrating on the mul…
Genuinely entangled subspaces are a class of subspaces in the multipartite Hilbert spaces that are composed of only genuinely entangled states. They are thus an interesting object of study in the context of multipartite entanglement. Here we provide a construction of multipartite subspaces that are not only genuinely entangled but also fully non-positive-partial-transpose (NPT) in the sense that any mixed state supported on them has non-positive partial transpose across any bipartition. Our construction originates from the stabilizer formalism known for its use in quantum error correction. To …
We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes ar…
There is a connection between classical codes, highly entangled pure states (called k-uniform or absolutely maximally entangled (AME) states), and quantum error correcting codes (QECCs). This leads to a systematic method to construct stabilizer QECCs by starting from a k-uniform state or the corresponding classical code and tracing out one party at each step. We provide explicit constructions for codewords, encoding procedure and stabilizer formalism of the QECCs by describing the changes that partial traces cause on the corresponding generator matrix of the classical codes. We then modify the…
How entangled is a randomly chosen bipartite stabilizer state? We show that if the number of qubits each party holds is large the state will be close to maximally entangled with probability exponentially close to one. We provide a similar tight characterization of the entanglement present in the maximally mixed state of a randomly chosen stabilizer code. Finally, we show that typically very few GHZ states can be extracted from a random multipartite stabilizer state via local unitary operations. Our main tool is a new concentration inequality which bounds deviations from the mean of random vari…
Bell inequality with self-testing property has played an important role in quantum information field with both fundamental and practical applications. However, it is generally challenging to find Bell inequalities with self-testing property for multipartite states and actually there are not many known candidates. In this work, we propose a systematical framework to construct Bell inequalities from stabilizers which are maximally violated by general stabilizer states, with two observables for each local party. We show that the constructed Bell inequalities can self-test any stabilizer state whi…
We find an interesting relationship between multipartite bound entangled states and the stabilizer formalism. We prove that if a set of commuting operators from the generalized Pauli group on $n$ qudits satisfy certain constraints, then the maximally mixed state over the subspace stabilized by them is an unlockable bound entangled state. Moreover, the properties of this state, such as symmetry under permutations of parties, undistillability and unlockability, can be easily explained from the stabilizer formalism without tedious calculation. In particular, the four-qubit Smolin state and its re…
In the seminal paper [Metger and Vidick, Quantum '21], they proposed a computational self-testing protocol for Bell states in a single quantum device. Their protocol relies on the fact that the target states are stabilizer states, and hence it is highly non-trivial to reveal whether the other class of quantum states, non-stabilizer states, can be self-tested within their framework. Among non-stabilizer states, magic states are indispensable resources for universal quantum computation. In this letter, we show that a magic state for the CCZ gate can be self-tested while that for the T gate canno…
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…
Entanglement of superpositions in zero-energy subspace: insights from stabilizer theory
Faculty of Physics, M.V. Lomonosov Moscow State University（莫斯科国立大学物理系）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Artificial Intelligence Research Institute (AIRI)（人工智能研究院）
Though the no-cloning theorem [1] prohibits exact replication of arbitrary quantum states, there are many instances in quantum information processing and entanglement measurement in which a weaker form of cloning may be useful. Here, I provide a construction for generating an "entangled clone" for a particular but rather expansive and rich class of states. Given a stabilizer code or free fermion Hamiltonian, this construction generates an exact entangled clone of the original ground state, in the sense that the entanglement between the original and the exact copy can be tuned to be arbitrarily…
Quantum error correction is an essential technique for constructing a scalable quantum computer. In order to implement quantum error correction with near-term quantum devices, a fast and near-optimal decoding method is demanded. A decoder based on machine learning is considered as one of the most viable solutions for this purpose, since its prediction is fast once training has been done, and it is applicable to any quantum error correcting code and any noise model. So far, various formulations of the decoding problem as the task of machine learning have been proposed. Here, we discuss general …
Controlling operational errors and decoherence is one of the major challenges facing the field of quantum computation and other attempts to create specified many-particle entangled states. The field of quantum error correction has developed to meet this challenge. A group-theoretical structure and associated subclass of quantum codes, the stabilizer codes, has proved particularly fruitful in producing codes and in understanding the structure of both specific codes and classes of codes. I will give an overview of the field of quantum error correction and the formalism of stabilizer codes. In th…
Suppose in a quantum network, there are $n$ qubits hold by Alice, Bob and Charlie, denoted by systems $A$, $B$ and $C$, respectively. We require the qubits to be described by a stabilizer state and assume the system $A$ is entangled with the combined system $BC$. An interesting question to ask is when it is possible to transfer all the entanglement to system $A$ and $B$ by local operation on $C$ and classical communication to $AB$, namely \textit{entanglement trimming}. We find a necessary and sufficient condition and prove constructively for this entanglement trimming, which we name it as "th…
Consider a stabilizer state on $n$ qudits, each of dimension $D$ with $D$ being a prime or a squarefree integer, divided into three mutually disjoint sets or parts. Generalizing a result of Bravyi et al. [J. Math. Phys. \textbf{47}, 062106 (2006)] for qubits (D=2), we show that up to local unitaries on the three parts the state can be written as a tensor product of unentangled single-qudit states, maximally entangled EPR pairs, and tripartite GHZ states. We employ this result to obtain a complete characterization of the properties of a class of channels associated with stabilizer error-correct…
We investigate stabilizer codes with carrier qudits of equal dimension $D$, an arbitrary integer greater than 1. We prove that there is a direct relation between the dimension of a qudit stabilizer code and the size of its corresponding stabilizer, and this implies that the code and its stabilizer are dual to each other. We also show that any qudit stabilizer can be put in a standard, or canonical, form using a series of Clifford gates, and we provide an explicit efficient algorithm for doing this. Our work generalizes known results that were valid only for prime dimensional systems and may be…
We study mutually unbiased bases (MUBs) in which all the bases are unextendible maximally entangled ones. We first present a necessary and sufficient condition of constructing a pair of MUBs in $C^2 \otimes C^4$. Based on this condition, an analytical and necessary condition for constructing MUBs is given. Moreover we illustrate our approach by some detailed examples in $C^2 \otimes C^4$. The results are generalized to $C^2 \otimes C^d$ $(d\geq 3)$ and a concrete example in $C^2 \otimes C^8$ is given.
We introduce a graphical representation of stabilizer states and translate the action of Clifford operators on stabilizer states into graph operations on the corresponding stabilizer-state graphs. Our stabilizer graphs are constructed of solid and hollow nodes, with (undirected) edges between nodes and with loops and signs attached to individual nodes. We find that local Clifford transformations are completely described in terms of local complementation on nodes and along edges, loop complementation, and change of node type or sign. Additionally, we show that a small set of equivalence rules g…
We propose a randomized variant of the stabilizer rewiring algorithm (SRA), a method for constructing a transversal circuit mapping between any pair of stabilizer codes. As gates along this circuit are applied, the initial code is deformed through a series of intermediate codes before reaching the final code. With this randomized variant, we show that there always exists a path of deformations which preserves the code distance throughout the circuit, while using at most linear overhead in the distance. Furthermore, we show that a random path will almost always suffice, and discuss prospects fo…
Based on the conception of quantum entanglement of Einstein-Podolsky-Rosen we construct generalized phase space representation associated with the entangled state |Gamma>_{e}, which is endowed with definite physical meaning. The set of states make up a complete and non-orthogonal representation. The Weyl ordered form of |Gamma>_{ee}_<Gamma| is derived which clearly exhibit the statistical behavior of marginal distribution of |Gamma>_{ee}_<Gamma|.The minimum uncertainty relation obeyed by |Gamma>_{e} is also demonstrated.
A new way of constructing unextendible maximally entangled basis (UMEB) from maximally entangled basis (MEB) is proposed. Consequently, it is shown that if there is an $N$-member UMEB in $\mathbb{C}^d\otimes \mathbb{C}^d$, then there exists a $(qd)^2-q(d^2-N)$-member UMEB in $\mathbb{C}^{qd}\otimes \mathbb{C}^{qd}$ for any $q\in\mathbb{N}$. This improves the results in [Phys. Rev. A 90, 034301(2014)], which shows that there exists a $(qd)^2-(d^2-N)$-member UMEB in $\mathbb{C}^{qd}\otimes \mathbb{C}^{qd}$ provided that an $N$-member UMEB exsits in $\mathbb{C}^d\otimes \mathbb{C}^d$. In addition…
Numerical optimization methods such as hillclimbing and simulated annealing have been applied to search for highly entangled multi-qubit states. Here the genetic algorithm is applied to this optimization problem -- to search not only for highly entangled states, but also for the corresponding quantum circuits creating these states. Simple quantum circuits for maximally (highly) entangled states are discovered for 3, 4, 5, and 6-qubit systems; and extension of the method to systems with more qubits is discussed. Among other results we have found explicit quantum circuits for maximally entangled…
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…
Vast developments in quantum technology have enabled the preparation of quantum states with more than a dozen entangled qubits. The full characterization of such systems demands distinct constructions depending on their specific type and the purpose of their use. Here we present a method that scales linearly with the number of qubits for characterizing stabilizer states. Our approach allows simultaneous extraction of information about the fidelity, the entanglement, and the nonlocality of the state and thus is of high practical relevance. We demonstrate the efficient applicability of our metho…
The 4-qubit unextendible product basis (UPB) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. From this result we show that there is only one UPB of size $6$ and six UPBs of size $9$ in $\cH=\bbC^2\ox\bbC^2\ox\bbC^4$, three UPBs of size $9$ in $\cK=\bbC^4\ox\bbC^4$, and no UPB of size $7$ in $\cH$ and $\cK$. Furthermore we construct a 4-qubit positive-partial-transpose (PPT) entangled state $\r$ of rank seven, and show that it is also a PPT entangled state in $\cH$ and $\cK$, respectively. We analytically derive the geometric measure of entanglement of a spec…
Asymptotic Entanglement Hiding under Stabilizer Restrictions
School of Computer Science and Technology, University of Science and Technology of China（中国科学技术大学计算机科学与技术学院）；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）；School of Software Engineering, University of Science and Technology of China（中国科学技术大学软件工程学院）；Suzhou Institute for Advanced Research, University of Science and Technology of China（中国科学技术大学苏州高等研究院）
We introduce a simple protocol for verifiable measurement-only blind quantum computing. Alice, a client, can perform only single-qubit measurements, whereas Bob, a server, can generate and store entangled many-qubit states. Bob generates copies of a graph state, which is a universal resource state for measurement-based quantum computing, and sends Alice each qubit of them one by one. Alice adaptively measures each qubit according to her program. If Bob is honest, he generates the correct graph state, and therefore Alice can obtain the correct computation result. Regarding the security, whateve…
Neural Quantum States in Non-Stabilizer Regimes: Benchmarks with Atomic Nuclei
Universität Bielefeld（比勒费尔德大学）；Argonne National Laboratory（阿贡国家实验室）；INFN-TIFPA Trento Institute for Fundamental Physics and Applications（特伦托基础物理与应用研究所）；GSI Helmholtzzentrum für Schwerionenforschung（重离子研究中心）
Finding ways to test the behaviour of quantum devices is a timely enterprise, especially in the light of the rapid development of quantum technologies. Device-independent self-testing is one desirable approach, as it makes minimal assumptions on the devices being tested. In this work, we address the question of which states can be self-tested. This has been answered recently in the bipartite case [Nat. Comm. 8, 15485 (2017)], while it is largely unexplored in the multipartite case, with only a few scattered results, using a variety of different methods: maximal violation of a Bell inequality, …
A Bacon-Shor code is a subsystem quantum error-correcting code on an $L \times L$ lattice where the $2(L-1)$ weight-$2L$ stabilizers are usually inferred from the measurements of $(L-1)^2$ weight-2 gauge operators. Here we show that the stabilizers can be measured directly and fault tolerantly with bare ancillary qubits by constructing circuits that follow the pattern of gauge operators. We then examine the implications of this method for small quantum error-correcting codes by comparing distance 3 versions of the rotated surface code and the Bacon-Shor code with the standard depolarizing mode…
Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and,…
We show how to explicitly construct an $O(nd)$ size and constant quantum depth circuit which encodes any given $n$-qubit stabilizer code with $d$ generators. Our construction is derived using the graphic description for stabilizer codes and the one-way quantum computation model. Our result demonstrates how to use cluster states as scalable resources for many multi-qubit entangled states and how to use the one-way quantum computation model to improve design of quantum algorithms.
Measurements play an important role in quantum computing (QC), by either providing the nonlinearity required for two-qubit gates (linear optics QC), or by implementing a quantum algorithm using single-qubit measurements on a highly entangled initial state (cluster state QC). Parity measurements can be used as building blocks for preparing arbitrary stabilizer states, and, together with 1-qubit gates are universal for quantum computing. Here we generalize parity gates by using a higher dimensional (qudit) ancilla. This enables us to go beyond the stabilizer/graph state formalism and prepare oth…
For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement. This paper is a sequel to arXiv:1708.01645, which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions $(d_1, d_2, \dots, d_n)$, and computes the dimension of the space ${\cal H}_{LME}/K$ of LME states up to local unitary transformations for all non-empty cases. In this paper, we provide a pedagogical overview and physical interpretation of the the underlying mathematics that leads…
Entanglement allows for the nonlocality of quantum theory, which is the resource behind device-independent quantum information protocols. However, not all entangled quantum states display nonlocality, and a central question is to determine the precise relation between entanglement and nonlocality. Here we present the first general test to decide whether a quantum state is local, and that can be implemented by semidefinite programming. This method can be applied to any given state and for the construction of new examples of states with local hidden-variable models for both projective and genera…
To build a fault-tolerant quantum computer, it is necessary to implement a quantum error correcting code. Such codes rely on the ability to extract information about the quantum error syndrome while not destroying the quantum information encoded in the system. Stabilizer codes are attractive solutions to this problem, as they are analogous to classical linear codes, have simple and easily computed encoding networks, and allow efficient syndrome extraction. In these codes, syndrome extraction is performed via multi-qubit stabilizer measurements, which are bit and phase parity checks up to local…
We present a general method for finding loss-tolerant teleportation on large, entangled stabilizer states using only single-qubit measurements, known as \emph{stabilizer pathfinding} (SPF). For heralded loss, SPF is shown to generate optimally loss-tolerant measurement patterns on any given stabilizer state. Furthermore, SPF also provides highly loss-tolerant teleportation strategies when qubit loss is unheralded. We provide a fast algorithm for SPF that updates continuously as a state is generated and measured, which is therefore suitable for real-time implementation on a quantum-computing de…
A five-qubit 1-resistant graph state and stabilizer marginal certificates
University of Science and Technology of China（中国科学技术大学）；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）
We introduce orthogonal quantum Latin squares, which restrict to traditional orthogonal Latin squares, and investigate their application in quantum information science. We use quantum Latin squares to build maximally entangled bases, and show how a pair of mutually unbiased maximally entangled bases can be constructed in square dimension from orthogonal quantum Latin squares. We also compare our construction to an existing construction due to Beth and Wocjan and show that ours is strictly more general.
