We present here an overview of our work concerning entanglement properties of composite quantum systems. The characterization of entanglement, i.e. the possibility to assert if a given quantum state is entangled with others and how much entangled it is, remains one of the most fundamental open questions in quantum information theory. We discuss our recent results related to the problem of separability and distillability for distinguishable particles, employing the tool of witness operators. Finally, we also state our results concerning quantum correlations for indistinguishable particles.
In our two preceding papers we studied bipartite composite boson (or quasiboson) systems through their realization in terms of deformed oscillators. Therein, the entanglement characteristics such as the entanglement entropy and purity were found and expressed, for both one-quasiboson and more complex states, through the parameter of deformation. In this work we initiate an analogous study of composite fermions for two major cases: (i) "boson + fermion" composites; (ii) "deformed-boson + fermion" composites. Both the entanglement entropy and purity of composite fermions are dealt with, their de…
A recent paper claimed to give an entanglement measure for composite systems, including Bose condensation and superconductivity. It is not an entanglement measure. It does not distinguish entanglement and non-factorization merely due to (anti)symmetrization for identical particles. In fact, the nature of entanglement, beyond the effect of (anti)symmetrization, in important states of many identical particles such as Bose condensation and superconductivity, had already been studied earlier.
We provide a constructive algorithm to find the best separable approximation to an arbitrary density matrix of a composite quantum system of finite dimensions. The method leads to a condition of separability and to a measure of entanglement.
Studying quantum entanglement in systems of indistinguishable particles, in particular anyons, poses subtle challenges. Here, we investigate a model of one-dimensional anyons defined by a generalized algebra. This algebra has the special property that fermions in this model are composites of anyons. A Hubbad-like Hamiltonian is considered that allows hopping between nearest neighbour sites not just for the fundamental anyons, but for the fermionic anyon composites. Some interesting results regarding the quantum entanglement of these particles are obtained.
We consider a composite particle formed by two fermions or two bosons. We discover that composite behavior is deeply related to the quantum entanglement between the constituent particles. By analyzing the properties of creation and annihilation operators, we show that bosonic character emerges if the constituent particles become strongly entangled. Such a connection is demonstrated explicitly in a class of two-particle wave functions.
The separability and entanglement of quantum mixed states in $\Cb^2 \otimes \Cb^3 \otimes \Cb^N$ composite quantum systems are investigated. It is shown that all quantum states $ρ$ with positive partial transposes and rank $r(ρ)\leq N$ are separable.
We analyse some compositeness effects and their relation with entanglement. We show that the purity of a composite system increases, in the sense of the expectation values of the deviation operators, with large values of the entanglement between the components of the system. We also study the validity of Pauli's principle in composite systems. It is valid within the limits of application of the approach presented here. We also present an example of two identical fermions, one of them entangled with a distinguishable particle, where the exclusion principle cannot be applied. This result can be …
We identify a general criterion for detecting entanglement of pure bipartite quantum states describing a system of two identical particles. Such a criterion is based both on the consideration of the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition and on the evaluation of the von Neumann entropy of the one-particle reduced statistical operators.
We show theoretically that according to the disentanglement behavior under composite noise environment, the Hilbert space of a two-qubit system can be divided into two separate parts: a 3-dimensional subspace in which all states disentangle asymptotically, and the rest in which all states disentangle abruptly. The violation of additivity for entanglement decay rates under weak noises [see, PRL 97, 140403 (2006)] therefore can be explained in terms of such division of the Hilbert space.
We build upon work by C. K. Law [Phys. Rev. A 71, 034306 (2005)] to show in general that the entanglement between two fermions largely determines the extent to which the pair behaves like an elementary boson. Specifically, we derive upper and lower bounds on a quantity that governs the bosonic character of a pair of fermions when N such pairs approximately share the same wavefunction. Our bounds depend on the purity of the single-particle density matrix, an indicator of entanglement, and demonstrate that if the entanglement is sufficiently strong, the quantity in question approaches its ideal …
We present a general description of separable states in Quantum Mechanics. In particular, our result gives an easy proof that inseparabitity (or entanglement) is a pure quantum (noncommutative) notion. This implies that distinction between separability and inseparabitity has sense only for composite systems consisting of pure quantum subsystems. Moreover, we provide the unified characterization of pure-state entanglement and mixed-state entanglement.
The finite time disentanglement or entanglement sudden death, when only one part of the composite system is subjected to a single noise, is examined. While it is shown that entanglement sudden death can occur when a part of the entangled mixed state is subjected to either amplitude noise or phase noise, local action of either of them does not cause entanglement sudden death in pure entangled states. In contrast, depolarizing noise is shown to have an abilitiy to cause sudden death of entanglement even in pure entangled states, when only one part of the state is exposed to it. The result is ill…
We show that all density operators of 2$\times N$--dimensional quantum systems that remain invariant after partial transposition with respect to the first system are separable. Based on this criterion, we derive a sufficient separability condition for general density operators in such quantum systems. We also give a simple proof of the separability criterion in $2\times 2$--dimensional systems [A. Peres, Phys. Rev. Lett {\bf 77}, 1413 (1996)]
Various topics concerning the entanglement of composite quantum systems are considered with particular emphasis concerning the strict relations of such a problem with the one of attributing objective properties to the constituents. Most of the paper deals with composite systems in pure states. After a detailed discussion and a precise formal analysis of the case of systems of distinguishable particles, the problems of entanglement and the one of the properties of subsystems of systems of identical particles are thourougly discussed. This part is the most interesting and new and it focuses vari…
We consider an evolution of two elementary quantum particles and ask the question: under what conditions such a system behaves as a single object? It is obvious that if the attraction between the particles is stronger than any other force acting on them the whole system behaves as one. However, recent insight from the quantum information theory suggests that in bipartite systems it is not attraction per se that is responsible for the composite nature, but the entanglement between the parts. Since entanglement can be present between the subsystems that interacted in the past, but do not interac…
We investigate separability and entanglement of mixed states in ${\cal C}^2\otimes{\cal C}^2\otimes{\cal C}^N$ three party quantum systems. We show that all states with positive partial transposes that have rank $\le N$ are separable. For the 3 qubit case (N=2) we prove that all states $ρ$ that have positive partial transposes and rank 3 are separable. We provide also constructive separability checks for the states $ρ$ that have the sum of the rank of $ρ$ and the ranks of partial transposes with respect to all subsystems smaller than 15N-1.
The Kochen--Specker (KS) theorem reveals the nonclassicality of single quantum systems. In contrast, Bell's theorem and entanglement concern the nonclassicality of composite quantum systems. Accordingly, unlike incompatibility, entanglement and Bell non-locality are not necessary to demonstrate KS-contextuality. However, here we find that for multiqubit systems, entanglement and non-locality are both essential to proofs of the Kochen--Specker theorem. Firstly, we show that unentangled measurements (a strict superset of local measurements) can never yield a logical (state-independent) proof of …
Entanglement transformation of composite quantum systems is investigated in the context of group representation theory. Representation of the direct product group $SL(2,C)\otimes SL(2,C)$, composed of local operators acting on the binary composite system, is realized in the four-dimensional complex space in terms of a set of novel bases that are pseudo orthonormalized. The two-to-one homomorphism is then established for the group $SL(2,C)\otimes SL(2,C)$ onto the $SO(4,C)$. It is shown that the resulting representation theory leads to the complete characterization for the entanglement transfor…
The correlation dynamics is investigated for various bi-partitions of a composite system consisting of two qubits, and two independent and non-identical noisy environments. The two qubits have no direct interaction with each other and locally interact with their environments. Classical and quantum correlations including entanglement are initially prepared only between the two qubits. We find that, contrary to the identical noisy environment case, the entanglement and quantum correlation transfer directions can be controlled by combining different noisy environments. The amplitude damping envir…
A general description of entanglement is suggested as an action realized by an arbitrary operator over given disentangled states. The related entanglement measure is defined. Because of its generality, this definition can be employed for any physical systems, pure or mixed, equilibrium or nonequilibrium, and characterized by any type of operators, whether these are statistical operators, field operators, spin operators, or anything else. Entanglement of any number of parts from their total ensemble forming a multiparticle composite system can be determined. Interplay between entanglement and o…
We compute the geometric phase for a spin-1/2 particle under the presence of a composite environment, composed of an external bath (modeled by an infinite set of harmonic oscillators) and another spin-1/2 particle. We consider both cases: an initial entanglement between the spin-1/2 particles and an initial product state in order to see if the initial entanglement has an enhancement effect on the geometric phase of one of the spins. We follow the nonunitary evolution of the reduced density matrix and evaluate the geometric phase for a single two-level system. We also show that the initial enta…
Both direct and indirect weak nonresonant interactions are shown to produce entanglement between two initially disentangled systems prepared as a tensor product of thermal states, provided the initial temperature is sufficiently low. Entanglement is determined by the Peres-Horodeckii criterion, which establishes that a composite state is entangled if its partial transpose is not positive. If the initial temperature of the thermal states is higher than an upper critical value $T_{uc}$ the minimal eigenvalue of the partially transposed density matrix of the composite state remains positive in th…
Three inter-related topics are discussed here. One, the Lindblad dynamics of quantum dissipative systems; two, quantum entanglement in composite systems and its quantification based on the Tsallis entropy; and three, robustness of entanglement under dissipation. After a brief review of the Lindblad theory of quantum dissipative systems and the idea of quantum entanglement in composite quantum systems illustrated by describing the three particle systems, the behavior of entanglement under the influence of dissipative processes is discussed. These issues are of importance in the discussion of qu…
Various topics concerning the entanglement of composite quantum systems are considered with particular emphasis concerning the strict relations of such a problem with the one of attributing objective properties to the constituents. In particular we will focus our attention to composite quantum systems composed of identical constituents, with the purpose of dealing with subtle issues, which have never been adequately discussed in the literature, originating from the true indistinguishability of the subsystems involved.
We study emerging notions of quantum correlations in compound systems. Based on different definitions of quantumness in individual subsystems, we investigate how they extend to the joint description of a composite system. Especially, we study the bipartite case and the connection of two typically applied and distinctively different concepts of nonclassicality in quantum optics and quantum information. Our investigation includes the representation of correlated states in terms of quasiprobability matrices, a comparative study of joint and conditional quantum correlations, and an entanglement ch…
In this contribution, we investigate the entanglement behavior of a composite system consists of two different dimensional subsystems in non-inertial frames. In particular, we consider a composite system of qubit(two-dimensional) subsystem, and qutrit, (three-dimensional) subsystem. The degree of entanglement is quantified for different cases, where it is assumed that the two-subsystems are simultaneously or non- simultaneously accelerated. The entanglement decays as the acceleration of any subsystem increases.In general, the decay rate of entanglement increases as the dimensional of the accel…
It was suggested that two entangled fermions can behave like a single boson and that the bosonic quality is proportional to the degree of entanglement between the two particles. The relation between bosonic quality and entanglement is quite natural if one takes into account the fact that entanglement appears in bound states of interacting systems. However, entanglement can still be present in spatially separated subsystems that do not interact anymore. These systems are often a subject of studies on quantum nonlocality and foundations of quantum physics. Here, we ask whether an entangled spati…
Composite structure of particles somewhat modifies their statistics, compared to the pure Bose- or Fermi-ones. The spin-statistics theorem, so, is not valid anymore. Say, $π$-mesons, excitons, Cooper pairs are not ideal bosons, and, likewise, baryons are not pure fermions. In our preceding papers, we studied bipartite composite boson (i.e. quasiboson) systems via a realization by deformed oscillators. Therein, the interconstituent entanglement characteristics such as entanglement entropy and purity were found in terms of the parameter of deformation. Herein, we perform an analogous study of co…
A classification of N-partite states, based on K-way negativities (K=2 to N), is proposed. The K-way partial transpose with respect to a subsystem is defined so as to shift the focus to K-way coherences instead of K subsystems of the composite system. For an N-partite system, the fraction of K-way negativity contributing to global negativity, is obtained. After minimizing K-way negativities through local unitary qubit rotations, a combined analysis of 2-way, 3-way and global negativities is shown to provide distinct measures of genuine tripartite, W-state like and bipartite entanglement, for t…
We study the entanglement structure, i.e., the structure of quantum composite system from operational aspects. The structure is not uniquely determined in General Probabilistic Theories (GPTs) even if we impose reasonable postulate about local systems. In this paper, we investigate the possibility that the standard entanglement structure can be determined uniquely by repeatability of measurement processing and its saturated situation called self-duality. Surprisingly, self-duality cannot determine the standard entanglement structure even if we additionally impose local unitary symmetry assumpt…
We address a problem of generating a robust entangling gate between electronic and nuclear spins in the system of a single nitrogen-vacany centre coupled to a nearest Carbon-13 atom in diamond against certain types of systematic errors such as pulse-length and off-resonance errors. We analyse the robustness of various control schemes: sequential pulses, composite pulses and numerically-optimised pulses. We find that numerically-optimised pulses, produced by the gradient ascent pulse engineering algorithm (GRAPE), are more robust than the composite pulses and the sequential pulses. The optimise…
We revisist the issue of entanglement of thermal equilibrium states in composite quantum systems. The possible scenarios are exemplified in bipartite qubit/qubit and qubit/qutrit systems.
We experimentally demonstrate the migration of coherence between composite quantum systems and their subsystems. The quantum systems are implemented using polarization states of photons in two experimental setups. The first setup is based on linear optical controlled-phase quantum gate and the second scheme is utilizing effects of nonlinear optics. Our experiment allows to verify the relation between correlations of the subsystems and the coherence of the composite system, which was given in terms of a conservation law for maximal accessible coherence by Svozilík \emph{et al.} [Phys. Rev. Lett…
We propose to quantify three qubit entanglement using global negativity along with K-way negativities, where K=2 and 3. K-way partial transpose with respect to a subsystem is defined so as to shift the focus to K-way coherences of the composite system instead of K subsystems of the composite system. While genuine tripartite entanglement of three qubit composite system is generated by 3-way coherences, tripartite entanglement can be present due to 2-way coherences as well.
Composite bosons (or quasibosons), as recently proven, are realizable by deformed oscillators and due to that can be effectively treated as particles of nonstandard statistics (deformed bosons). This enables us to study quasiboson states and their inter-component entanglement aspects using the well developed formalism of deformed oscillators. We prove that the internal entanglement characteristics for single two-component quasiboson are determined completely by the parameter(s) of deformation. The bipartite entanglement characteristics are generalized and calculated for arbitrary multi-quasibo…
Bipartite composite boson (quasiboson) systems, which admit realization in terms of deformed oscillators, were considered in our previous paper from the viewpoint of entanglement characteristics. These characteristics, including entanglement entropy and purity, were expressed through the relevant deformation parameter for different quasibosonic states. On the other hand, it is of interest to present the entanglement entropy and likewise the purity as function of energy for those states. In this work, the corresponding dependencies are found for different states of composite bosons realized by …
Entanglement within a given device provides a potential resource for quantum information processing. Entanglement between system and environment leads to decoherence (thus suppressing non-classical features within the system) but also opens up a route to robust and universal control. The latter is related to thermodynamic equilibrium, a generic behavior of bi-partite quantum systems. Fingerprints of this equilibrium behavior (including relaxation and stability) show up already far from the thermodynamic limit, where a complete solution of the underlying Schroedinger dynamics of the total syste…
We present a two-part program for state space decomposition. States are classified into entanglement classes based on local unitary transformations, and then characterized as elements of topological spaces using the language of fibre bundles.
Entanglement is closely related to some fundamental features of the dynamics of composite quantum systems: quantum entanglement enhances the "speed" of evolution of certain quantum states, as measured by the time required to reach an orthogonal state. The concept of "speed" of quantum evolution constitutes an important ingredient in any attempt to determine the fundamental limits that basic physical laws impose on how fast a physical system can process or transmit information. Here we explore the relationship between entanglement and the speed of quantum evolution in the context of the quantum…
This is an extended version of the previous paper arXiv:2103.05303 to study entanglement entropy (EE) of a half space in interacting field theories. In the previous paper, we have proposed a novel method to calculate EE based on the notion of $\mathbb{Z}_M$ gauge theory on Feynman diagrams, and shown that EE consists of two particular contributions, one from a renormalized two-point correlation function in the two-particle irreducible (2PI) formalism and another from interaction vertices. In this paper, we further investigate them in more general field theories and show that the non-Gaussian c…
We study geometrical aspects of entanglement, with the Hilbert--Schmidt norm defining the metric on the set of density matrices. We focus first on the simplest case of two two-level systems and show that a ``relativistic'' formulation leads to a complete analysis of the question of separability. Our approach is based on Schmidt decomposition of density matrices for a composite system and non-unitary transformations to a standard form. The positivity of the density matrices is crucial for the method to work. A similar approach works to some extent in higher dimensions, but is a less powerful to…
I investigate quantum decoherence in a one-body density matrix of a composite particle consisting of two correlated particles. Because of a two-body correlation in the composite particle, quantum decoherence occurs in the one-body density matrix that has been reduced from the two-body density matrix. As the delocalization of the distribution of the composite particle grows, the entanglement entropy increases, and the system can be well-described by a semi-classical approximation, wherein the center position of the composite particle can be regarded as a classical coordinate. I connect the quan…
We study the geometric Uhlmann phase of entangled mixed states in a composite system made of two coupled spin-$\frac 1 2$ particles with a magnetic field acting on one of them. Within a depolarizing channel setup, an exact analytical expression for such a phase in each subsystem is derived. We find an explicit connection to the concurrence of the depolarizing channel density matrix, which allows to characterize the features of the Uhlmann phase in terms of the degree of entanglement in the system. In the space of field direction and coupling parameter, it exhibits a phase singularity revealing…
Using well known duality between quantum maps and states of composite systems we introduce the notion of Schmidt number of a quantum channel. It enables one to define classes of quantum channels which partially break quantum entanglement. These classes generalize the well known class of entanglement breaking channels.
We begin with the simple model of phase sychronization in open classical nonlinear system which is represented in the language of angular momentum variables. After that we propose the relevant quantum counterpart of this system. Using the appropriate Lindblad master equation for the density matrix of two qubit realization of such system we have revealed that stationary state of this composite system is pure and entangled with small dispersion of phase observable. We believe that such curious connection between entangled stationary states of quantum composite system and phase sychronization bet…
The state of a quantum system, consisting of two distinct subsystems, is called separable if it can be prepared by two distant experimenters who receive instructions from a common source, via classical communication channels. A necessary condition is derived and is shown to be more sensitive than Bell's inequality for detecting quantum inseparability. Moreover, collective tests of Bell's inequality (namely, tests that involve several composite systems simultaneously) may sometimes lead to a violation of Bell's inequality, even if the latter is satisfied when each composite system is tested sep…
We extend study of the Jaynes-Cummings model involving a pair of identical two-level atoms (or qubits) interacting with a single mode quantized field. We investigate the effects of replacing the radiation field mode with a composite spin, comprising $N$ qubits, or spin-1/2 particles. This model is relevant for physical implementations in superconducting circuit QED, ion trap and molecular systems. For the case of the composite spin prepared in a spin coherent state, we demonstrate the similarities of this set-up to the qubits-field model in terms of the time evolution, attractor states and in …
The problem of defining quantum probabilities of composite events is considered. This problem is of high importance for the theory of quantum measurements and for quantum decision theory that is a part of measurement theory. We show that the Luders probability of consecutive measurements is a transition probability between two quantum states and that this probability cannot be treated as a quantum extension of the classical conditional probability. The Wigner distribution is shown to be a weighted transition probability that cannot be accepted as a quantum extension of the classical joint prob…
In this paper, I will derive a measure of entanglement that coincides with the generalized concurrence for a general pure bi-and three-partite state based on wedge product. I will show that a further generalization of this idea to a general pure multipartite state with more than three subsystems will fail to quantify entanglement, but it defines the set of separable state for such composite state.
Complementary correlations can reveal the genuine quantum correlations present in a composite quantum system. Here we investigate the relation between complementary correlations and other aspects of genuine quantum correlations. We show that for a certain class of states quantum correlations revealed through complementary correlations become equal to entanglement and discord. We also provide a necessary and sufficient condition for entanglement distribution with separable Bell diagonal states in terms of complementary correlations.
The understanding of the behaviour of systems of identical composite bosons has progressed significantly in connection with the analysis of the entanglement between constituents and the development of coboson theory. The basis of these treatments is a coboson ansatz for the ground state of a system of N pairs, stating that in appropriate limits this state is well approximated by the account of Pauli exclusion in what would otherwise be the product state of N independent pairs, each described by the single-pair ground state. In this work we study the validity of this ansatz for particularly sim…
We consider an arbitrary d_{1}\otimes d_{2}\otimes ... \otimes d_{N} composite quantum system and find necessary conditions for general m-party subsystem states to be the reduced states of a common N-party state. These conditions will lead to various monogamy inequalities for bipartite quantum entanglement and partial disorder in multipartite states. Our results are tightly connected with the measures of multipartite entanglement.
The concept of relative state is used to introduce geometric phases that originate from correlations in states of composite quantum systems. In particular, we identify an entanglement-induced geometric phase in terms of a weighted average of geometric phase factors associated with a decomposition that define the entanglement of formation. An explicit procedure to calculate the entanglement-induced geometric phase for qubit pairs is put forward. We illustrate it for maximally entangled mixed states (MEMS) of two qubits.
Some new identities for quantum variance and covariance involving commutators are presented, in which the density matrix and the operators are treated symmetrically. A measure of entanglement is proposed for bipartite systems, based on covariance. This works for two- and three-component systems but produces ambiguities for multicomponent systems of composite dimension. Its relationship to angular momentum dispersion for symmetric symmetric spin states is described.
We propose a fault-tolerant implementation of the quantum Householder reflection, which is a key operation in various quantum algorithms, quantum-state engineering, generation of arbitrary unitaries, and entanglement characterization. We construct this operation using the modular approach of composite pulses and a relation between the Householder reflection and the quantum phase gate. The proposed implementation is highly insensitive to variations in the experimental parameters, which makes it suitable for high-fidelity quantum information processing.
