We obtain a new lower bound on the radius of the largest ball of separable unnormalized states around the identity matrix for a 3-qubit system. This also enables us to improve the corresponding lower bounds for multi-qubit systems. These bounds are approximately 5% better than the previously known ones. As a by-product, we compute the radius of the largest ball that fits into the triple projective tensor product of the unit ball in R^3.
We have investigated the dynamics of quantum discord and entanglement for two qubits subject to independent global transverse and/or longitudinal memoryless noisy classical fields. Global transverse and/or longitudinal random fields are found to drive the system to maximally discordant mixed separable steady states for suitable initial conditions. Moreover, two independent noises in the system are found to enhance both the steady state randomness and quantum discord in the absence of entanglement for some initial states.
In a recent article S. Gharibian [\href{http://dx.doi.org/10.1103/PhysRevA.86.042106}{Phys. Rev. A {\bf 86}, 042106 (2012)}] has conjectured that no two qubit separable state of rank greater than two could be maximally non classical (defined to be those which have normalized geometric discord $1/4$) and asked for an analytic proof. In this work we prove analytically that among the subclass of $X$ states, there is a unique (up to local unitary equivalence) maximal separable state of rank two. Partial progress has been made towards the general problem and some necessary conditions have been deri…
We show that for an m-partite quantum system, there is a ball of radius 2^{-(m/2-1)} in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices. This can be used to derive an epsilon below which mixtures of epsilon of any density matrix with 1 - epsilon of the maximally mixed state will be separable. The epsilon thus obtained is exponentially better (in the number of systems) than existing results. This gives a number of qubits below which NMR with standard pseudopure-state preparation techniques can access only unentangled states; with parame…
We extend the relation between absolutely maximally entangled (AME) states and quantum maximum distance separable (QMDS) codes by constructing whole families of QMDS codes from their parent AME states. We introduce a reduction-friendly form for the generator set of the stabilizer representation of an AME state, from which the stabilizer form for children codes, all QMDS, can be obtained. We then relate this to optimal codes for one-way quantum repeaters, by minimizing the short-term infrastructure cost as well as the long-term running cost of such quantum repeaters. We establish that AME state…
In this paper, based on a matrix norm, we first present a ball of separable unnormalized states around the identity matrix for the bipartite quantum system, which is larger than the separable ball in Frobenius norm. Then the proposed ball is used to get not only simple sufficient conditions for the separability of pseudopure states and the states with strong positive partial transposes, but also a separable ball centered at the identity matrix for the multipartite quantum system.
A lower bound on the amount of noise that must be added to a GHZ-like entangled state to make it separable (also called the random robustness) is found using the transposition condition. The bound is applicable to arbitrary numbers of subsystems, and dimensions of Hilbert space, and is shown to be exact for qubits. The new bound is compared to previous such bounds on this quantity, and found to be stronger in all cases. It implies that increasing the number of subsystems, rather than increasing their Hilbert space dimension is a more effective way of increasing entanglement. An explicit decomp…
Modern applications in quantum computation and quantum communication require the precise characterization of quantum states and quantum channels. In practice, this means that one has to determine the quantum capacity of a physical system in terms of measurable quantities. Witnesses, if properly constructed, succeed in performing this task. We derive a method that is capable to compute witnesses for identifying deterministic evolutions and measurement-induced collapse processes. At the same time, applying the Choi-Jamiolkowski isomorphism, it uncovers the entanglement characteristics of biparti…
Let B_1,B_2 be balls in finite-dimensional real vector spaces E_1,E_2, centered around unit length vectors v_1,v_2 and not containing zero. An element in the tensor product space E_1 \otimes E_2 is called B_1 \otimes B_2-separable if it is contained in the convex conic hull of elements of the form w_1 \otimes w_2, where w_1 \in B_1, w_2 \in B_2. We study the cone formed by the separable elements in E_1 \otimes E_2. We determine the largest faces of this cone via a description of the extreme rays of the dual cone, i.e. the cone of the corresponding positive linear maps. We compute the radius of…
For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral $l_p$ norms for $1 \le p \le \infty$, of separable (unentangled) matrices around the identity matrix. This implies a simple and intutively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity $\tr ρ^2$ not be too large. Theoretical and experimental applications of these results include algorithmic problems such as computing whether or not a state is entangled, and practical ones such as obtaining information about the existence or n…
Absolute separable states is a kind of separable state that remain separable under the action of any global unitary transformation. These states may or may not have quantum correlation and these correlations can be measured by quantum discord. We find that the absolute separable states are useful in quantum computation even if it contains infinitesimal quantum correlation in it. Thus to search for the class of two-qubit absolute separable states with zero discord, we have derived an upper bound for $Tr(\varrho^{2})$, where $\varrho$ denoting all zero discord states. In general, the upper bound…
We give improved upper bounds on the radius of the largest ball of separable states of an m-qubit system around the maximally mixed state. The ratio between the upper bound and the best known lower bound (Hildebrand, quant.ph/0601201) thus shrinks to a constant c = \sqrt{34/27} ~ 1.122, as opposed to a term of order \sqrt{m\log m} for the best upper bound known previously (Aubrun and Szarek, quant.ph/0503221). We give concrete examples of separable states on the boundary to entanglement which realize these upper bounds. As a by-product, we compute the radii of the largest balls that fit into t…
As separable states are a convex combination of product states, the geometry of the manifold of product states is studied. Prior results by Sanpera, Vidal and Tarrach are extended. Furthermore, it is proven that states in the set tangent to the manifold of product states, at the maximally mixed state are separable; the set normal constains, among others, all maximally entangled states. A canonical decomposition is given. A surprising result is that for the case of two particles, the closest product state to the maximally entangled state is the maximally mixed state. An algorithm is provided to…
We present a general method for constructing pure-product-state representations for density operators of $N$ quantum bits. If such a representation has nonnegative expansion coefficients, it provides an explicit separable ensemble for the density operator. We derive the condition for separability of a mixture of the Greenberger-Horne-Zeilinger state with the maximally mixed state.
Every entangled state can be perturbed, for instance by decoherence, and stay entangled. For a large class of pure entangled states, we show how large the perturbation can be. Our class includes all pure bipartite and all maximally entangled states. For an entangled state, E, the constucted neighborhood of entangled states is the region outside two parallel hyperplanes, which sandwich the set of all separable states. The states for which these neighborhoods are largest are the maximally entangled ones. As the number of particles, or the dimensions of the Hilbert spaces for two of the particles…
We show that for an m-qubit quantum system, there is a ball of radius asymptotically approaching kappa 2^{-gamma m} in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent gamma = (1/2)((ln 3/ln 2) - 1), roughly .29248125. This is much smaller in magnitude than the best previously known exponent, from our earlier work, of 1/2. For normalized m-qubit states, we get a separable ball of radius sqrt(3^(m+1)/(3^m+3)) * 2^{-(1 + γ)m}, i.e. sqrt{3^{m+1}/(3^m+3)}\times 6^{-m/2} (note that κ= \sqrt{3}), compared to the previous 2 * …
We explore classical to quantum transition of correlations by studying the quantum states located just outside of the classically-correlated-states-only neighborhood of the maximally mixed state (the largest separable ball (LSB)). We show that a natural candidate for such states raises the possibility of a layered transition, i.e., an annular region comprising only classical and the classical-like bound entangled states, followed by free or distillable entanglement. Surprisingly, we find the transition to be abrupt for bipartite systems: distillable states emerge arbitrarily close to the LSB. …
We consider the separability of various joint states for N qutrits. We derive two results: (i) the separability condition for a two-qutrit state that is a mixture of the maximally mixed state and a maximally entangled state (such a state is a generalization of the Werner state for two qubits); (ii) upper and lower bounds on the size of the neighborhood of separable states surrounding the maximally mixed state for N qutrits.
The separable mixed 2-qubit X-states are classified in accordance with degeneracies in the spectrum of density matrices. It is shown that there are four classes of separable X-states, among them: one 4D family, a pair of 2D family and a single, zero-dimensional maximally mixed state.
We consider mixed states of two qubits and show under which global unitary operations their entanglement is maximized. This leads to a class of states that is a generalization of the Bell states. Three measures of entanglement are considered: entanglement of formation, negativity and relative entropy of entanglement. Surprisingly all states that maximize one measure also maximize the others. We will give a complete characterization of these generalized Bell states and prove that these states for fixed eigenvalues are all equivalent under local unitary transformations. We will furthermore chara…
We study the efficacy of two-qubit mixed entangled states as resources for quantum teleportation. We first consider two maximally entangled mixed states, viz., the Werner state\cite{werner}, and a class of states introduced by Munro {\it et al.} \cite{munro}. We show that the Werner state when used as teleportation channel, gives rise to better average teleportation fidelity compared to the latter class of states for any finite value of mixedness. We then introduce a non-maximally entangled mixed state obtained as a convex combination of a two-qubit entangled mixed state and a two-qubit separa…
In this paper we explore the possibility of performing Heisenberg limited quantum metrology of a phase, without any prior, by employing only maximally entangled states. Starting from the estimator introduced by Higgins et al. in New J. Phys. 11, 073023 (2009), the main result of this paper is to produce an analytical upper bound on the associated Mean Squared Error which is monotonically decreasing as a function of the square of the number of quantum probes used in the process. The analysed protocol is non-adaptive and requires in principle (for distinguishable probes) only separable measureme…
We study the relative strength of classical and quantum correlations, as measured by discord, for two-qubit states. Quantum correlations appear only in the presence of classical correlations, while the reverse is not always true. We identify the family of states that maximize the discord for a given value of the classical correlations and show that the largest attainable discord for mixed states is greater than for pure states. The difference between discord and entanglement is emphasized by the remarkable fact that these states do not maximize entanglement and are, in some cases, even separab…
We consider the separability of various joint states of D-dimensional quantum systems, which we call "qudits." We derive two main results: (i) the separability condition for a two-qudit state that is a mixture of the maximally mixed state and a maximally entangled state; (ii) lower and upper bounds on the size of the neighorhood of separable states surrounding the maximally mixed state for N qudits.
Using a geometric measure of entanglement quantification based on Euclidean distance of the Hermitian matrices, we obtain the minimum distance between a bipartite bound entangled $n$- qudit density matrix and the maximally mixed state.This minimum distance for which entangled density matrices necessarily have positive partial transpose (PPT) is obtained as $\frac{1}{\sqrt{\sqrt{d^n(d^n-1)}+1}}$, which is also a lower limit for the existence of 1-distillable entangled states. The separable states necessarily lie within a minimum distance of $\frac{R}{1+d^{n-1}}$ from the Identity,where R is the…
We investigate the asymptotic rates of entanglement transformations for bipartite mixed states by local operations and classical communication (LOCC). We analyse the relations between the rates for different transitions and obtain simple lower and upper bound for these transitions. In a transition from one mixed state to another and back, the amount of irreversibility can be different for different target states. Thus in a natural way, we get the concept of "amount" of irreversibility in asymptotic manipulations of entanglement. We investigate the behaviour of these transformation rates for di…
As an analogy of best separable approximation (BSA) in the framework of entanglement theory, here we concentrate on the notion of best incoherent approximation, with application to characterizing and quantifying quantum coherence. From both analytical and numerical perspectives, we have demonstrated that the weight-based coherence measure displays some unusual properties, in sharp contrast to other popular coherence quantifiers. First, by deriving a closed formula for qubit states, we have showed the weight-based coherence measure exhibits a rich (geometrical) structure even in this simplest c…
Geometric relations between separable and entangled two-qubit and two-qutrit quantum information states are studied. To characterize entanglement of two qubit states, we establish a relation between reduced density matrix and the concurrence. For the rebit states, the geometrical meaning of concurrence as double area of a parallelogram is found and for generic qubit states it is expressed by determinant of the complex Hermitian inner product metric, where reduced density matrix coincides with the inner product metric. In the case of generic two-qutrit state, for reduced density matrix we find …
We present a method to construct entanglement measures for pure states of multipartite qubit systems. The key element of our approach is an antilinear operator that we call {\em comb} in reference to the {\em hairy-ball theorem}. For qubits (or spin 1/2) the combs are automatically invariant under $SL(2,\CC)$. This implies that the {\em filters} obtained from the combs are entanglement monotones by construction. We give alternative formulae for the concurrence and the 3-tangle as expectation values of certain antilinear operators. As an application we discuss inequivalent types of genuine four…
There is growing belief that the next decade will see the emergence of sensing devices based on the laws of quantum physics that outperform some of our current sensing devices. For example, in frequency estimation, using a probe prepared in an entangled state can, in principle, lead to a precision gain compared to a probe prepared in a separable state. Even in the presence of some forms of decoherence, it has been shown that the precision gain can increase with the number of probe particles $N$. Usually, however, the entangled and separable state preparation and readout times are assumed to be…
Entanglement can be considered as a special quantum correlation, but not the only kind. Even for a separable quantum system, it is allowed to exist non-classical correlations. Here we propose two dissipative schemes for generating a maximally correlated state of two qubits in the absence of quantum entanglement, which was raised by [F. Galve, G. L. Giorgi, and R. Zambrini, {\color{blue}Phys. Rev. A {\bf 83}, 012102 (2011)}]. These protocols take full advantages of the interaction between four-level atoms and strongly lossy optical cavities. In the first scenario, we alternatively change the ph…
We present a proof-of-principle experiment demonstrating measurement of the collectibility, a nonlinear entanglement witness proposed by Rudnicki et al. [Phys. Rev. Lett. 107, 150502 (2011)]. This entanglement witness works for both mixed and pure two-qubit states. In the later case it can be used to measure entanglement in terms of the negativity. We measured the collectibility for three distinct classes of photonic polarization-encoded two-qubit states, i.e., maximally entangled, separable and maximally mixed states. We demonstrate that the measurement procedure is feasible and robust agains…
We propose a measurement-based method to produce a maximally-entangled state from a partially-entangled pure state. Our goal can be thought of as entanglement distillation from a single copy of a partially-entangled state. The present approach involves local two-outcome weak measurements. We show that application of these local weak measurements leads to a probabilistic amplification of entanglement. In addition, we examine how the probability to find the maximally-entangled state is related to the entanglement of the input state. We also study the application of our method to a mixed initial …
We obtain the necessary and sufficient separability and distillability conditions of mixtures of a maximally entangled state and the completely separable state in relativistic setting. In an inertial frame we study the entanglement under Wigner rotations induced by Lorentz transformations. We also investigate the mixed state entanglement of scalar and Dirac fields as seen by two relatively accelerated observers. For scalar field we show that in infinite acceleration limit the state has no longer distillable entanglement. For dirac field the entanglement in the infinite acceleration limit is fi…
Entanglement is a resource to overcome the natural restriction of operations used for state manipulation to Local Operations assisted by Classical Communication (LOCC). Hence, a bipartite maximally entangled state is a state which can be transformed deterministically into any other state via LOCC. In the multipartite setting no such state exists. There, rather a whole set, the Maximally Entangled Set of states (MES), which we recently introduced, is required. This set has on the one hand the property that any state outside of this set can be obtained via LOCC from one of the states within the …
We give a constructive proof that all mixed states of N qubits in a sufficiently small neighborhood of the maximally mixed state are separable. The construction provides an explicit representation of any such state as a mixture of product states. We give upper and lower bounds on the size of the neighborhood, which show that its extent decreases exponentially with the number of qubits. We also discuss the implications of the bounds for NMR quantum computing.
Our work addresses the problem of generating maximally entangled two spin-1/2 (qubit) symmetric states using NMR, NQR, Lipkin-Meshkov-Glick Hamiltonians. Time evolution of such Hamiltonians provides various logic gates which can be used for quantum processing tasks. Pairs of spin-1/2's have modeled a wide range of problems in physics. Here we are interested in two spin-1/2 symmetric states which belong to a subspace spanned by the angular momentum basis {|j = 1, μ>; μ = +1, 0,-1}. Our technique relies on the decomposition of a Hamiltonian in terms of SU(3) generators. In this context, we defin…
For a given pure state of a composite quantum system we analyze the product of its projections onto a set of locally orthogonal separable pure states. We derive a bound for this product analogous to the entropic uncertainty relations. For bipartite systems the bound is saturated for maximally entangled states and it allows us to construct a family of entanglement measures, we shall call collectibility. As these quantities are experimentally accessible, the approach advocated contributes to the task of experimental quantification of quantum entanglement, while for a three-qubit system it is cap…
Milz and Strunz recently reported substantial evidence to further support the previously conjectured separability probability of $\frac{8}{33}$ for two-qubit systems ($ρ$) endowed with Hilbert-Schmidt measure. Additionally, they found that along the radius ($r$) of the Bloch ball representing either of the two single-qubit subsystems, this value appeared constant (but jumping to unity at the locus of the pure states, $r=1$). Further, they also observed (personal communication) such separability probability $r$-invariance, when using, more broadly, random induced measure ($K=3,4,5,\ldots$), wit…
Quantities invariant under local unitary transformations are of natural interest in the study of entanglement. This paper deduces and studies a particularly simple quantity that is constructed from a combination of two standard permutations of the density matrix, namely realignment and partial transpose. This bipartite quantity, denoted here as $R_{12}$, vanishes on large classes of separable states including classical-quantum correlated states, while being maximum for only maximally entangled states. It is shown to be naturally related to the 3-tangle in three qubit states via their two-qubit…
