We develop a general theory of the relation between quantum phase transitions (QPTs) characterized by nonanalyticities in the energy and bipartite entanglement. We derive a functional relation between the matrix elements of two-particle reduced density matrices and the eigenvalues of general two-body Hamiltonians of $d$-level systems. The ground state energy eigenvalue and its derivatives, whose non-analyticity characterizes a QPT, are directly tied to bipartite entanglement measures. We show that first-order QPTs are signalled by density matrix elements themselves and second-order QPTs by the first derivative of density matrix elements. Our general conclusions are illustrated via several quantum spin models.
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.
We analyze the bipartite and multipartite entanglement for the ground state of the one-dimensional XY model in a transverse magnetic field in the thermodynamical limit. We explicitly take into account the spontaneous symmetry breaking in order to explore the relation between entanglement and quantum phase transitions. As a result we show that while both bipartite and multipartite entanglement can be enhanced by spontaneous symmetry breaking deep into the ferromagnetic phase, only the latter is affected by it in the vicinity of the critical point. This result adds to the evidence that multipart…
We calculate the geometric phase of a bipartite two-level system coupled to an external environment. We analyze the reduced density matrix for an arbitrary initial state of the composite system and compute the correction to the unitary geometric phase through a kinematic approach. In all cases considered, we observe a similar structure as a function of the degree of the entanglement of the initial state. Further, we compute the entanglement entropy and concurrence of the bipartite state and analyze if there is any relation among these quantities and the geometric phase acquired during the nonu…
We present an inequality for detecting entanglement and distillability of arbitrary dimensional bipartite systems. This inequality provides a sufficient condition of entanglement for bipartite mixed states, and a necessary and sufficient condition of entanglement for bipartite pure states. Moreover, the inequality also gives a necessary and sufficient condition for distillability.
Entanglement in a class of bipartite generalized coherent states is discussed. It is shown that a positive parameter can be associated with the bipartite generalized coherent states so that the states with equal value for the parameter are of equal entanglement. It is shown that the maximum possible entanglement of 1 bit is attained if the positive parameter equals $\sqrt{2}$. The result that the entanglement is one bit when the relative phase between the composing states is $π$ in bipartite coherent states is shown to be true for the class of bipartite generalized coherent states considered.
Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed. Using single particle feasible region, a way of constructing effective entanglement witnesses for bipartite systems is provided by exact convex optimization. Examples for some well known two qutrit quantum systems show these entanglement witnesses in most cases, provide necessary and sufficient conditions for separability of given bipartite system. Also this method is applied to a class of bipartite qudit quantum systems with details for d=3, 4 and 5. Keywords: non-linear and linear entanglement …
Based on the generators of $SU(n)$ we present inequalities for detecting quantum entanglement for $2 \otimes d$ and $M \otimes N$ systems. These inequalities provide a sufficient condition of entanglement for bipartite mixed states and give rise to an experimental way of entanglement detection.
We study bipartite entanglement in a general one-particle state, and find that the linear entropy, quantifying the bipartite entanglement, is directly connected to the paricitpation ratio, charaterizing the state localization. The more extended the state is, the more entangled the state. We apply the general formalism to investigate ground-state and dynamical properties of entanglement in the one-dimensional Harper model.
We give an explicit tight lower bound for the entanglement of formation for arbitrary bipartite mixed states by using the convex hull construction of a certain function. This is achieved by revealing a novel connection among the entanglement of formation, the well-known Peres-Horodecki and realignment criteria. The bound gives a quite simple and efficiently computable way to evaluate quantitatively the degree of entanglement for any bipartite quantum state.
We present an alternative definition of quantum entanglement for bipartite system based on Bell inequality and operators' noncommutativity. A state is said to be entangled, if the maximum of CHSH expectation value $F_{\max}$ is obtain by noncommutative measures on each particle of the bipartite system; otherwise, the state is a disentangled state. A uniform measure quantifying the degree of entanglement for any state of the bipartite system is also proposed.
Monogamy of bipartite correlations leads, for arbitrary pure multi-qubit states, to simple conditions able to indicate various types of multipartite entanglement by being capable to exclude the possibility of k-separability.
We investigate the generation of multipartite entangled SU(k+1) coherent states using a quantum network involving a sequence of k beam splitters. We particularly investigate the entanglement in multipartite SU(2) coherent states (k = 1). We employ the concurrence as measure of the degree of bipartite entanglement.
We derive exact formulas for bipartite von Neumann entanglement entropy after partial projective local measurement in $1+1$ dimensional conformal field theories with periodic and open boundary conditions. After defining the set up we will check numerically the validity of our results in the case of Klein-Gordon field theory (coupled harmonic oscillators) and spin-$1/2$ XX chain in a magnetic field. The agreement between analytical results and the numerical calculations is very good. We also find a lower bound for localizable entanglement in coupled harmonic oscillators.
The problem of the experimental determination of the amount of entanglement of a bipartite pure state is addressed. We show that measuring a single observable does not suffice to determine the entanglement of a given unknown pure state of two particles. Possible minimal local measuring strategies are discussed and a comparison is made on the basis of their best achievable precision.
We propose a novel parameterized entanglement measure $α$-concurrence for bipartite systems. By employing positive partial transposition and realignment criteria, we derive analytical lower bounds for the $α$-concurrence. Moreover, we calculate explicitly the analytic expressions of the $α$-concurrence for isotropic states and Werner states.
We construct a class of algebraic invariants for N-qubit pure states based on bipartite decompositions of the system. We show that they are entanglement monotones, and that they differ from the well know linear entropies of the sub-systems. They therefore capture new information on the non-local properties of multipartite systems.
A new criterion necessary and sufficient for the separability of pure bipartite systems for arbitrary finite dimensions is demonstrated; and the corresponding finer quantitative measures or characterizations of entanglement (beyond mere separability or non-separability determination) are discussed. Based on this criterion, we proved that the well-known Peres-Horodecki positivity-of-partial-transform criterion is also necessary and sufficient for separability in the case of pure bipartite systems. The maximum value of entanglement, and the corresponding maximally-entangled states are also worke…
A bipartite quantum channel represents the interaction between systems, generally allowing for exchange of information. A special class of bipartite channels are the no-signaling ones, which do not allow communication. In Ref. [1] it has been conjectured that all no-signaling channels are mixtures of entanglement-breaking and localizable channels, which require only local operations and entanglement. Here we provide the general realization scheme, giving a counterexample to the conjecture.
There has been spectacular progress in the field of quantum information in recent decades. The development of this field highlights the importance of the role of entanglement in quantum computing, quantum teleportation and quantum cryptography. These notes serve to provide a gentle introduction to the entanglement of bipartite states. In these notes, we introduce Bell's theorem in the form derived by Clauser, Horne, Shimony and Holt. We discuss the Schmidt decomposition and the Peres-Horodecki criterion in the entanglement of pure and mixed bipartite states. Finally, we describe a teleportatio…
This is a thesis submitted to university of Pune, India, for the Ph.D. degree. This work deals with entanglement production in two qubit, two qutrit and three qubit systems, entanglement in indistinguishable fermionic systems, quantum discord in a Heisenberg chain and geometric measure of quantum discord in an arbitrary state of a bipartite quantum system.
We derive a separability criterion for bipartite quantum systems which generalizes the already known criteria. It is based on observables having generic commutation relations. We then discuss in detail the relation among these criteria.
We present a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via local operations and classical communication. This framework unifies and greatly simplifies previous works. A necessary condition for ``pure contraction'' transformations is given. Finally, constructive protocols to achieve both probabilistic and deterministic entanglement transformations are presented.
We investigate the stationary bipartite entanglement is a useful hybrid optomechanical system, which is constituted of two coupled-cavity optomechanics through a photon hopping process and both are driven by squeezed light. The transfer of correlations from an entangled light source to optomechanical cavities is explored. It is found that the generation of bipartite entanglement and entanglement transfer depend strongly on photon hopping strength and the matching of the input squeezed modes to the cavity modes. It is revealed that the generated stationary bipartite entanglement due to squeezed…
We provide upper bound on the maximal rate at which irreversible quantum dynamics can generate entanglement in a bipartite system. The generator of irreversible dynamics consists of a Hamiltonian and dissipative terms in Lindblad form. The relative entropy of entanglement is chosen as a measure of entanglement in an ancilla-free system. We provide an upper bound on the entangling rate which has a logarithmic dependence on a dimension of a smaller system in a bipartite cut. We also investigate the rate of change of quantum mutual information in an ancilla-assisted system and provide an upper bo…
Entanglement of pure states of bipartite quantum systems has been shown to have a unique measure in terms of the von Neumann entropy of the reduced states of either of its subsystems. The measure is established under entanglement manipulation of an asymptotically large number of copies of the states. In this paper, different asymptotic measures of entanglement assigned to arbitrary sequences of bipartite pure states are shown to coincide only when the sequence is information stable, in terms of the quantum spectral information rates of the sequence of subsystem states. Additional bounds on the…
We theoretically study bipartite entanglement purification with neutral atoms via cavity-assistant interaction and linear optical elements. We focus on entanglement distillation and the recurrence protocol, whose performances under idealized and realistic conditions are discussed. The implementation of these purification protocols has been tested with numerical simulations. We analyze the performance and stability of all required operations and emphasize that all techniques are feasible with current experimental technology.
It is an interesting problem to construct genuine tripartite entangled states based on the collective use of two bipartite entangled states. We consider the case that the states are two-qubit Werner states, we construct the interval of parameter of Werner states such that the tripartite state is genuine entangled. Further, we present the way of detecting the tripartite genuine entanglement using current techniques in experiments. We also investigate the lower bound of genuine multipartite entanglement concurrence.
Multipartite quantum entanglement serves as a resource for spatially separated parties performing distributed quantum information processing. Any multipartite entangled state can be generated from appropriately distributed bipartite entangled states by local operations and classical communication (LOCC), and in this sense, any distributed process based on shared multipartite entanglement and LOCC is simulatable by using only bipartite entangled states and LOCC. We show here that this reduction scenario does not hold when there exists a limitation on the size of the local quantum system of each…
We explore how entanglement of a general bipartite system evolves when one subsystem undergoes the action of an arbitrary noisy channel. It is found that the dynamics of entanglement for general bipartite systems under the influence of such channel is determined by the channel's action on the maximally entangled state, which includes as a special case the results for two-qubit systems [Nature Physics 4, 99 (2008)]. In particular, for multi-qubit or qubit-qudit systems, we get a general factorization law for evolution equation of entanglement with one qubit being subject to a noisy channel. Our…
Based on the complementarity relation between entanglement of a composite system and the purity of a subsystem, we propose a simple method to measure the amount of entanglement. The method can be applied to a bipartite system in a pure state of any arbitrary dimension. It requires only single qudit rotations and straightforward probability measurements performed on one of the subsystems, and can thus be easily implemented experimentally using linear optical devices.
Recently, a sufficient condition on the structure of the Kossakowski-Lindblad master equation has been given such that the generated reduced dynamics of two qubits results entangling for at least one among their initial separable pure states. In this paper we study to which extent this condition is also necessary. Further, we find sufficient conditions for bath-mediated entanglement generation in higher dimensional bipartite open quantum systems.
We consider multipartite states of qubits and prove that their bipartite quantum entanglement, as quantified by the concurrence, satisfies a monogamy inequality conjectured by Coffman, Kundu, and Wootters. We relate this monogamy inequality to the concept of frustration of correlations in quantum spin systems.
We calculate analytic expressions for the distribution of bipartite entanglement for pure random quantum states. All moments of the purity distribution are derived and an asymptotic expansion for the purity distribution itself is deduced. An approximate expression for moments and distribution of Meyer-Wallach entanglement for random pure states is then obtained.
Photon subtraction and addition are experimental means of generating non-Gaussian states from Gaussian states. Coherent subtraction or addition is a combination of photon subtractions or additions. The resultant states are quite general non-Gaussian states. The states can be photon number entangled states with arbitrary coefficients. We derive the entanglement conditions for several classes of coherent subtraction or coherent addition bipartite continuous variable states. One of the entanglement conditions is necessary and sufficient.
Necessary and sufficient conditions for bipartite entanglement are derived, which apply to arbitrary Hilbert spaces. Motivated by the concept of witnesses, optimized entanglement inequalities are formulated solely in terms of arbitrary Hermitian operators, which makes them useful for applications in experiments. The needed optimization procedure is based on a separability eigenvalue problem, whose analytical solutions are derived for a special class of projection operators. For general Hermitian operators, a numerical implementation of entanglement tests is proposed. It is also shown how to id…
We {characterize the multipartite entanglement in a quantum system by the quantity} which vanishes if only the quantum system may be decomposed into two weakly entangled subsystems, unlike measures of multipartite entanglement introduced before. We refer to this {quantity} as the minimal entanglement of bipartite decompositions (MEBD). Big MEBD means that the system may not be decomposed into two weakly entangled subsystems. MEBD allows one to define, for instance, whether the given quantum system may be a candidate for a quantum register, where the above decomposition is undesirable. A method…
Controllable multipartite entanglement is a crucial element in quantum information processing. Here we present a scheme that generates switchable bipartite and genuine tripartite entanglement between microwave and optical photons via an optoelectromechanical interface, where microwave and optical cavities are coupled to a mechanical mode with controllable coupling constants. We show that by tuning an effective gauge phase between the coupling constants to the "sweet spots", bipartite entanglement can be generated and switched between designated output photons. The bipartite entanglement is rob…
An experimental scheme is suggested that permits a direct measure of entanglement in a two-qubit cavity system. It is realized in the cavity-QED technology utilizing atoms as flying qubits. With this scheme we generate two different measures of entanglement, namely logarithmic negativity and concurrence. The phenomenon of sudden death entanglement (ESD) in a bipartite system subjected to dissipative environment is examined. We show that the sudden death time of the entangled states depends on the initial preparation of the entangled state and the temperature of the reservoir.
A $q$-deformed Weyl-Heisenberg algebra is used to define a deformed displacement operator giving rise to a naturally normalized nonlinear coherent states type. Robust maximally entangled deformed coherent states are studied and the effect of such a deformation on the amount of the entanglement is discussed. The analogy between environment decoherence and algebra deformation is made through the deformation parameter.
We investigate the state space of bipartite qutrits. We construct an analog to the "magic" tetrahedron for bipartite qubits--a magic simplex W. It is formed by all convex combination of nine Bell states which are constructed using the Weyl operators. Due to the high symmetry it is enough to consider certain typical slices through W. Via optimal entanglement witnesses we find regions of bound entangled states inside W.
It has been observed that the reduced density matrices of bipartite qudit pure states possess a Gram matrix structure. This observation has opened a possibility of analysing the entanglement in such systems from the purely geometrical point of view. In particular, a new quantitative measure of an entanglement of the geometrical nature, has been proposed. Using the invented Gram matrix approach, a version of a non-linear purification of mixed states describing the system analysed has been presented.
We introduce a new class of unitary transformations based on the su(1,1) Lie algebra that generalizes, for certain particular representations of its generators, well-known squeezing transformations in quantum optics. To illustrate our results, we focus on the two-mode bosonic representation and show how the parametric amplifier model can be modified in order to generate such a generalized squeezing operator. Furthermore, we obtain a general expression for the bipartite Wigner function which allows us to identify two distinct sources of entanglement, here labelled by dynamical and kinematical e…
We extend the concept of the negativity, a good measure of entanglement for bipartite pure states, to mixed states by means of the convex-roof extension. We show that the measure does not increase under local quantum operations and classical communication, and derive explicit formulae for the entanglement measure of isotropic states and Werner states, applying the formalism presented by Vollbrecht and Werner [Phys. Rev. A {\bf 64}, 062307 (2001)].
We examine a class of bipartite mixed states which we call X states and report several analytic results related to the occurrence of so-called entanglement sudden death (ESD) under time evolution in the presence of common types of environmental noise. Avoidance of sudden death by application of purely local operations is shown to be feasible in some cases.
Measuring entanglement is a demanding task in the field of quantum computation and quantum information theory. Recently, some authors experimentally demonstrated an embedding quantum simulator, using it to efficiently measure two-qubit entanglement. Here, we are reviewing some measures of entanglement which are used for pure and mixed states. Furthermore, we have reported the efficient bipartite and multipartite entanglement measures.
The recognition of entanglement states is a notoriously difficult problem when no prior information is available. Here, we propose an efficient quantum adversarial bipartite entanglement detection scheme to address this issue. Our proposal reformulates the bipartite entanglement detection as a two-player zero-sum game completed by parameterized quantum circuits, where a two-outcome measurement can be used to query a classical binary result about whether the input state is bipartite entangled or not. In principle, for an $N$-qubit quantum state, the runtime complexity of our proposal is $O(\tex…
We investigate the properties and relations of two classes of operational bipartite and multipartite entanglement measures, the so-called source and the accessible entanglement. The former measures how easy it is to generate a given state via local operations and classical communication (LOCC) from some other state, whereas the latter measures the potentiality of a state to be convertible to other states via LOCC. Main emphasis is put on the bipartite pure states, single copy regime. We investigate which parameter regime is physically available, i.e for which values of these measures does ther…
Bound entanglement with a nonpositive partial transposition (NPT) does not exist. For any NPT entangled state a distillation procedure can be based on a certain number of copies. This number is the minimal Schmidt rank of a pure state needed to witness the NPT entanglement under study.
We study a random matrix model for the statistical properties of the purity of a bipartite quantum system at a finite (fictitious) temperature. This enables us to write the generating function for the cumulants, for both balanced and unbalanced bipartitions. It also unveils an unexpected feature of the system, namely the existence of two phase transitions, characterized by different spectra of the density matrices. One of the critical phases is described by the statistical mechanics of random surfaces, the other is a second-order phase transition.
We provide a graphical description of the entanglement of pure-state multiparty systems based on an analogy between a bipartite purity analysis and the centroid of a collection of point masses. This description applies to quantum systems with $N$ parties, each with an arbitrary number of (discrete) states. The case of $N$ qubits is highlighted for simplicity. This geometric description illustrates some of the restrictions in the form of inequalities that apply to entanglement in multiparty systems.
We propose a entanglement measure for pure $M \otimes N$ bipartite quantum states. We obtain the measure by generalizing the equivalent measure for a $2 \otimes 2$ system, via a $2 \otimes 3$ system, to the general bipartite case. The measure emphasizes the role Bell states have, both for forming the measure, and for experimentally measuring the entanglement. The form of the measure is similar to generalized concurrence. In the case of $2 \otimes 3$ systems, we prove that our measure, that is directly measurable, equals the concurrence. It is also shown that in order to measure the entanglemen…
In this work we propose the geometric mean of bipartite concurrences as a genuine multipartite entanglement measure. This measure achieves the maximum value for absolutely maximally entangled states and has desirable properties for quantifying potential quantum resources. The simplicity and symmetry in the definition facilitates its computation for various multipartite entangled states including the GHZ states and the $W$ states. With explicit examples we show that our measure results in distinct entanglement orderings from other measures, and can detect differences in certain types of genuine…
A review is given on the thermodynamical structure of bipartite entanglement. By comparing it to the axiomatic formulation of thermodynamics presented by Giles it is shown that for finite dimensional systems the two theories are formally inequivalent. The same approach is used to demonstrate the full equivalence in the asymptotic limit for pure quantum states. For mixed states a different method for obtaining the second law is described applied to two different classes of operations, PPT-preserving and asymptotically non-entangling operations.
We introduce a special class of truncated Weyl-Heisenberg algebra and discuss the corresponding Hilbertian and analytical representations. Subsequently, we study the effect of a quantum network of beam splitting on coherent states of this nonlinear class of harmonic oscillators. We particularly focus on quantum networks involving one and two beam splitters and examine the degree of bipartite as well as tripartite entanglement using the linear entropy.
