A general framework is developed for separating classical and quantum correlations in a multipartite system. Entanglement is defined as the difference in the correlation information encoded by the state of a system and a suitably defined separable state with the same marginals. A generalization of the Schmidt decomposition is developed to implement the separation of correlations for any pure, multipartite state. The measure based on this decomposition is a generalization of the entanglement of formation to multipartite systems, provides an upper bound for the relative entropy of entanglement, and is directly computable on pure states. The example of pure three-qubit states is analyzed in detail, and a classification based on minimal, four-term decompositions is developed.
The class of local invertible operations is defined and the invariance of entanglement under such operations is established. For the quantification of entanglement, universal entanglement measures are defined, which are invariant under local invertible transformations. They quantify entanglement in a very general sense. It is shown that the Schmidt number is a universal entanglement measure, which is most important for the general amount of entanglement. For special applications, pseudo-measures are defined to quantify the useful entanglement for a certain quantum task. The entanglement quanti…
We construct a Universal Quantum Entanglement Concentration Gate (QEC-Gate). Special times operations of QEC-Gate can transform a pure 2-level bipartite entangled state to nearly maximum entanglement. The transformation can attain any required fidelity with optimal probability by adjusting concentration step. We also generate QEC-Gate to the Schmidt decomposable multi-partite system.
We study broadcasting of entanglement where we use universal quantum cloners (in general less optimal) to perform local cloning operations. We show that there is a lower bound on the fidelity of the universal quantum cloners that can be used for broadcasting. We prove that an entanglement is optimally broadcast only when optimal quantum cloners are used for local copying. We also show that broadcasting of entanglement into more than two entangled pairs is forbidden using only local operations.
We give a universal recipe for constructing nonlinear entanglement witnesses able to detect non-classical correlations in arbitrary systems of distinguishable and/or identical particles for an arbitrary number of constituents. The constructed witnesses are expressed in terms of expectation values of observables. As such they are, at least in principle, measurable in experiments
We consider two-qubit undergoing local dissipation and subject to local driving. We then determine the optimal Markovian feedback action to preserve initial entanglement as well as to create stationary entanglement with the help of an XY interaction Hamiltonian. Such feedback actions are worked out in a way not depending on the initial two-qubit state, whence called universal.
We consider the propagation of two photonic qubits, initially maximally entangled in their orbital angular momenta (OAM), across a turbulent atmosphere. By introducing the {\it phase correlation length} of an OAM beam, we show that the entanglement of OAM photons exhibits a universal exponential decay under turbulence.
We show that universal quantum computation can be achieved in the standard pure-state circuit model while, at any time, the entanglement entropy of all bipartitions is small---even tending to zero with growing system size. The result is obtained by showing that a quantum computer operating within a small region around the set of unentangled states still has universal computational power, and by using continuity of entanglement entropy. In fact an analogous conclusion applies to every entanglement measure which is continuous in a certain natural sense, which amounts to a large class. Other exam…
We show that the entanglement evolution of an open quantum system is the same for the vast majority of initial pure states, in the limit of large Hilbert space dimensions.
We consider the entanglement manipulation capabilities of the universal covariant quantum cloner or quantum processor circuit for quantum bits. We investigate its use for cloning a member of a bipartite or a genuine tripartite entangled state of quantum bits. We find that for bipartite pure entangled states a nontrivial behavior of concurrence appears, while for GHZ entangled states a possibility of the partial extraction of bipartite entanglement can be achieved.
Entanglement concentration from many copies of unknown pure states is discussed, and we propose the protocol which not only achieves entropy rate, but also produces the perfect maximally entangled state. Our protocol is induced naturally from symmetry of $n$-tensored pure state, and is optimal for all the protocols which concentrates entanglement from unknown pure states, in the sense of failure probability. In the proof of optimality, the statistical estimation theory plays a key role, for concentrated entanglement gives a natural estimate of the entropy of entanglement.
We study the entanglement evolution of photonic orbital angular momentum qubit states with opposite azimuthal indices $l_0$, in a weakly turbulent atmosphere. Using asymptotic methods, we deduce analytical expressions for the amplitude of turbulence-induced crosstalk between the modes $l_0$ and $-l_0$. Furthermore, we analytically establish distinct, universal entanglement decay laws for Kolmogorov's turbulence model and for two approximations thereof.
We propose and examine several candidates for universal multipartite entanglement measures. The most promising candidate for applications needing entanglement in the full Hilbert space is the ent-concurrence, which detects all entanglement correlations while distinguishing between different types of distinctly multipartite entanglement, and simplifies to the concurrence for two-qubit mixed states. For applications where subsystems need internal entanglement, we develop the absolute ent-concurrence which detects the entanglement in the reduced states as well as the full state.
Universal two-particle entanglement processes are analyzed in arbitrary dimensional Hilbert spaces. On the basis of this analysis the class of possible optimal universal entanglement processes is determined whose resulting output states do not contain any separable states. It is shown that these processes form a one-parameter family. For all Hilbert space dimensions larger than two the resulting optimally entangled output states are mixtures of anti-symmetric states which are freely entangled and which also preserve information about input states. Within this one-parameter family there is only…
Using invariance of the $n$-th tensored state w.r.t. the $n$-th symmetric group, we propose a 'variable length' universal entanglement concentration without classical communication. Like variable length data compression, arbitrary unknown states are concentrated into perfect Bell states and not approximate Bell states and the number of Bell states obtained is equal to the optimal rate asymptotically with the probability 1. One of the point of our scheme is that we need no classical communication at all. Using this method, we can construct a universal teleportation and a universal dense coding.
Detecting and quantifying quantum entanglement of a given unknown state poses problems that are fundamentally important for quantum information processing. Surprisingly, no direct (i.e., without quantum tomography) universal experimental implementation of a necessary and sufficient test of entanglement has been designed even for a general two-qubit state. Here we propose an experimental method for detecting a collective universal witness, which is a necessary and sufficient test of two-photon polarization entanglement. It allows us to detect entanglement for any two-qubit mixed state and to es…
We study the diffraction of two maximally entangled qubits encoded into spatial modes of light with orbital angular momenta $\pm \hbar l_0$, on angular apertures. Exploiting the uncertainty relation between angular position and momentum, we quantify the diffraction-induced entanglement loss, for different $l_0$, as a function of the angular uncertainty $Δϕ$ induced by the apertures. We demonstrate that the degradation of entanglement follows a universal law parametrised by the product $l_0Δϕ$ alone.} {We study the diffraction of two photonic qubits, which are initially maximally entangled in t…
We report a novel Bell-state synthesizer in which an interferometric entanglement concentration scheme is used. An initially mixed polarization state from type-II spontaneous parametric down-conversion becomes entangled after the interferometric entanglement concentrator. This Bell-state synthesizer is universal in the sense that the output polarization state is not affected by spectral filtering, crystal thickness, and, most importantly, the choice of pump source. It is also robust against environmental disturbance and a more general state, partially mixed$-$partially entangled state, can be …
We construct a single observable measurement of which mean value on four copies of an {\it unknown} two-qubit state is sufficient for unambiguous decision whether the state is separable or entangled. In other words, there exists a universal collective entanglement witness detecting all two-qubit entanglement. The test is directly linked to a function which characterizes to some extent the entanglement quantitatively. This function is an entanglement monotone under so--called local pure operations and classical communication (pLOCC) which preserve local dimensions. Moreover it provides tight up…
We investigate measurement-based entanglement purification protocols (EPP) in the presence of local noise and imperfections. We derive a universal, protocol-independent threshold for the required quality of the local resource states, where we show that local noise per particle of up to 24% is tolerable. This corresponds to an increase of the noise threshold by almost an order of magnitude, based on the joint measurement-based implementation of sequential rounds of few-particle EPP. We generalize our results to multipartite EPP, where we encounter similarly high error thresholds.
Quantum entanglement has been actively sought for in optomechanical and electromechanical systems. The simplest such system is a mechanical oscillator interacting with a coherent beam, while the oscillator also suffers from thermal decoherence. For this system, we show that quantum entanglement is always present between the oscillator and continuous outgoing fields -- even when the environmental temperature is high and the oscillator is highly classical. Such universal entanglement is also shown to be able to survive more than one oscillation cycle if characteristic frequency of the optomechan…
Entanglement is a key resource in many quantum information applications and achieving high values independently of the initial conditions is an important task. Here we address the problem of generating highly entangled states in a discrete time quantum walk irrespective of the initial state using two different approaches. First, we present and analyze a deterministic sequence of coin operators which produces high values of entanglement in a universal manner for a class of localized initial states. In a second approach, we directly optimize the sequence of coin operators using a reinforcement l…
We introduce the notion of entanglement measure for the universal classes of fractons as an entanglement between ocuppation-numbers of fractons in the lowest Landau levels and the rest of the many-body system of particles. This definition came as an entropy of the probability distribution {\it à la} Shannon. Fractons are charge-flux systems classified in universal classes of particles or quasiparticles labelled by a fractal or Hausdorff dimension defined within the interval $1 < h < 2$ and associated with the fractal quantum curves of such objects. They carry rational or irrational values of s…
Any bipartite entanglement witness $W$ can be written as $W=c_σI-σ$, where $σ$ is a quantum state, $I$ is the identity matrix, and $c_σ$ is a non-negative number. We present a general method to extend the given entanglement witness to multipartite cases via purification, partial purification, and direct tensor of the quantum state $σ$. Our methods extend $σ$ but leave the parameter $c_σ$ untouched. This is very valuable since the parameter is generally not easy to compute.
Entanglement is known to be a relative notion, defined with respect to the choice of physical observables to be measured (i.e., the measurement setup used). This implies that, in general, the same state can be both separable and entangled for different measurement setups, but this does not exclude the existence of states which are separable (or entangled) for all possible setups. We show that for systems of bosonic particles there indeed exist such universally separable states: they are i.i.d. pure states. In contrast, there is no such state for fermionic systems with a few exceptional cases. …
Entanglement is the defining characteristic of quantum mechanics. Bipartite entanglement is characterized by the von Neumann entropy. Entanglement is not just described by a number, however; it is also characterized by its level of complexity. The complexity of entanglement is at the root of the onset of quantum chaos, universal distribution of entanglement spectrum statistics, hardness of a disentangling algorithm and of the quantum machine learning of an unknown random circuit, and universal temporal entanglement fluctuations. In this paper, we numerically show how a crossover from a simple …
Fernando Galve \emph{et al.} $[Phys. Rev. Lett. \textbf{110}, 010501 (2013)]$ introduced discording power for a two-qubit unitary gate to evaluate its capability to produce quantum discord, and found that a $π/8$ gate has maximal discording power. This work analyzes the entangling power of a two-qubit unitary gate, which reflects its ability to generate quantum entanglement in another way. Based on the renowned Cartan decomposition of two-qubit unitary gates, we show that the magic power of the $π/8$ gate produces maximal entanglement for a general value of purities for two-qubit states.
Recently, a framework was established to systematically construct novel universal resource states for measurement-based quantum computation using techniques involving finitely correlated states. With these methods, universal states were found which are in certain ways much less entangled than the original cluster state model, and it was hence believed that with this approach many of the extremal entanglement features of the cluster states could be relaxed. The new resources were constructed as "computationally universal" states--i.e. they allow one to efficiently reproduce the classical output…
We study an initially inverted three-level atom in the lambda configuration embedded in a waveguide, interacting with a propagating single-photon pulse. Depending on the temporal shape of the pulse, the system behaves either as an optimal universal cloning machine, or as a highly efficient deterministic source of maximally entangled photon pairs. This quantum transistor operates over a wide range of frequencies, and can be implemented with today's solid-state technologies.
We prove that all states (mixed or pure) of qubit-qutrit ($2\times 3$) systems have entanglement-preserving unitary (EPU) equivalence to a compact subset of true-generalized X (TGX) states called EPU-minimal TGX states which we give explicitly. Thus, for any spectrum-entanglement combination achievable by general states, there exists an EPU-minimal TGX state of the same spectrum and entanglement. We use I-concurrence to measure entanglement and give an explicit formula for it for all $2\times 3$ minimal TGX states (a more general set than EPU-minimal TGX states) whether mixed or pure, yielding…
Universal entanglement detection with geometrically optimal witnesses
Southern University of Science and Technology（南方科技大学）；The Hong Kong University of Science and Technology（香港科技大学）；Quantum Science Center of Guangdong-HongKong-Macao Greater Bay Area（粤港澳大湾区量子科学中心）；The University of Hong Kong（香港大学）
We investigate which entanglement resources allow universal measurement-based quantum computation via single-qubit operations. We find that any entanglement feature exhibited by the 2D cluster state must also be present in any other universal resource. We obtain a powerful criterion to assess universality of graph states, by introducing an entanglement measure which necessarily grows unboundedly with the system size for all universal resource states. Furthermore, we prove that graph states associated with 2D lattices such as the hexagonal and triangular lattice are universal, and obtain the fi…
Entanglement, one of the central mysteries of quantum mechanics, plays an essential role in numerous applications of quantum information theory. A natural question of both theoretical and experimental importance is whether universal entanglement detection is possible without full state tomography. In this work, we prove a no-go theorem that rules out this possibility for any non-adaptive schemes that employ single-copy measurements only. We also examine in detail a previously implemented experiment, which claimed to detect entanglement of two-qubit states via adaptive single-copy measurements …
In the presence of boundaries, the entanglement entropy in lattice models is known to exhibit oscillations with the (parity of the) length of the subsystem, which however decay to zero with increasing distance from the edge. We point out in this article that, when the subsystem starts at the boundary and ends at an impurity, oscillations of the entanglement (as well as of charge fluctuations) appear which do not decay with distance, and which exhibit universal features. We study these oscillations in detail for the case of the XX chain with one modified link (a conformal defect) or two success…
We propose a new protocol of \textit{universal} entanglement concentration, which converts many copies of an \textit{unknown} pure state to an \textit{% exact} maximally entangled state. The yield of the protocol, which is outputted as a classical information, is probabilistic, and achives the entropy rate with high probability, just as non-universal entanglement concentration protocols do. Our protocol is optimal among all similar protocols in terms of wide varieties of measures either up to higher orders or non-asymptotically, depending on the choice of the measure. The key of the proof of o…
The University of Tokyo（东京大学）；RIKEN Center for Quantum Computing (RQC)（理化学研究所量子计算中心）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；International Quantum Academy（国际量子学院）；Nagoya University（名古屋大学）
Gapped two-dimensional topological phases can feature ungappable edge states which are robust even in the absence of protecting symmetries. In this work we show that a multipartite entanglement measure recently proposed in the context of holography, the Markov gap, provides a universal diagnostic of ungappable edge states. Defined as a difference of the reflected entropy and mutual information $h(A:B) = S_R(A:B) - I(A:B)$ between two parties, we argue that for $A,B$ being adjacent subregions in the bulk, $h=\frac{c_+}{3}\log 2$, where $c_+$ is the minimal total central charge of the boundary t…
Recently it was shown that the growth of entanglement in an initially separable state, as measured by the purity of subsystems, can be characterized by a timescale that takes a universal form for any Hamiltonian. We show that the same timescale governs the growth of entanglement for all Rényi entropies. Since the family of Rényi entropies completely characterizes the entanglement of a pure bipartite state, our timescale is a universal feature of bipartite entanglement. The timescale depends only on the interaction Hamiltonian and the initial state.
The ground state entanglement entropy is studied in a many-body bipartite quantum system with either a single or multiple conserved quantities. It is shown that the entanglement entropy exhibits a universal power-law behaviour at large $R$ -- the occupancy ratio between the two subsystems. Single and multiple conserved quantities lead to different power-law exponents, suggesting the entanglement entropy can serve to detect hidden conserved quantities. Moreover, occupancy measurements allow to infer the bipartite entanglement entropy. All the above results are generalized for the Rényi entropy.
Optimality of universal conclusive entanglement concentration protocols
Institute of Fundamental Technological Research, Polish Academy of Sciences（波兰科学院基础技术研究所）；Faculty of Mathematics, Informatics and Mechanics, University of Warsaw（华沙大学数学、信息学与力学系）；Institute for Theoretical Physics III, Heinrich Heine University Düsseldorf（杜塞尔多夫海因里希·海涅大学理论物理第三研究所）
In order to engineer an open quantum system and its evolution, it is essential to identify and control the memory effects. These are formally attributed to the non-Markovianity of dynamics that manifests itself by the evolution being indivisible in time, a property which can be witnessed by a non-monotonic behavior of contractive functions or correlation measures. We show that by monitoring directly the entanglement behavior of a system in a tripartite setting it is possible to witness all invertible non-Markovian dynamics, as well as all (also non-invertible) qubit evolutions. This is achieve…
No-cloning theorem forbids perfect cloning of an unknown quantum state. A universal quantum cloning machine (UQCM), capable of producing two copies of any input qubit with the optimal fidelity, is of fundamental interest and has applications in quantum information processing. This is enabled by delicately tailored nonclassical correlations between the input qubit and the copying qubits, which distinguish the UQCM from a classical counterpart, but whose experimental demonstrations are still lacking. We here implement the UQCM in a superconducting circuit, and investigate these correlations. The…
We revisit the problem of detection of entanglement of an unknown two-qubit state using minimal resources. Using weak values and just two copies of an arbitrary two-qubit state, we present a protocol where a post selection measurement in the computational basis provides enough information to identify if the state is entangled or not. Our protocol enables complete state identification with a single-setting post selection measurement on two copies of the state. It follows that by restricting to pure states, the global interaction required for determining the weak values can be realized by local …
Using the geometric entanglement measure, we study the scaling of multipartite entanglement in several 1D models at criticality, specifically the linear harmonic chain and the XY spin chain encompassing both the Ising and XX critical models. Our results provide convincing evidence that 1D models at criticality exhibit a universal logarithmic scaling behavior ~(c/12)log l in the multipartite entanglement per region for a partition of the system into regions of size l, where c is the central charge of the corresponding universality class in conformal field theory.
We present several criteria for genuine multipartite entanglement from universal uncertainty relations based on majorization theory. Under non-negative Schur-concave functions, the vector-type uncertainty relation generates a family of infinitely many detectors to check genuine multipartite entanglement. We also introduce the concept of $k$-separable circles via geometric distance for probability vectors, which include at most $(k-1)$-separable states. The entanglement witness is also generalized to a universal entanglement witness which is able to detect the $k$-separable states more accurate…
This paper presents a streaming (sequential) protocol for universal entanglement concentration at the Shannon bound. Alice and Bob begin with N identical (but unknown) two-qubit pure states, each containing E ebits of entanglement. They each run a reversible algorithm on their qubits, and end up with Y perfect EPR pairs, where Y = NE +- O(\sqrt N). Our protocol is streaming, so the N input systems are fed in one at a time, and perfect EPR pairs start popping out almost immediately. It matches the optimal block protocol exactly at each stage, so the average yield after n inputs is <Y> = nE - O(…
Clifford circuits are insufficient for universal quantum computation or creating $t$-designs with $t\ge 4$. While the entanglement entropy is not a telltale of this insufficiency, the entanglement spectrum is: the entanglement levels are Poisson-distributed for circuits restricted to the Clifford gate-set, while the levels follow Wigner-Dyson statistics when universal gates are used. In this paper we show, using finite-size scaling analysis of different measures of level spacing statistics, that in the thermodynamic limit, inserting a single T $(π/8)$ gate in the middle of a random Clifford ci…
Quantum telecloning is a multiparty quantum communication protocol which allows quantum information broadcasting. It can be, therefore, seen as a generalization of quantum teleportation. However, in contrast to quantum teleportation, it requires the resource of multipartite entanglement. Here we present an experimental demonstration of universal symmetric 1->2 quantum telecloning of qubits via four-photon polarisation entanglement.
We propose a multipartite extension of Matsumoto and Hayashi's distortion-free entanglement concentration protocol, which takes $n$ copies of a general multipartite state and, via local measurements, produces a maximally-entangled multipartite state between local spaces of dimensions $\sim 2^{n E_i}$, where $E_i$ are the local entropies of the input state. However, the extended protocol is generally not universal in the sense that for the same measurement outcomes, the output state will still depend on the input state. Our main result is that when specialized to any state in the multiqubit W c…
Using relative entropy, we derive bounds on the time rate of change of geometric entanglement entropy for any relativistic quantum field theory in any dimension. The bounds apply to both mixed and pure states, and may be extended to curved space. We illustrate the bounds in a few examples and comment on potential applications and future extensions.
