We seek to derive the probability--expressed in terms of the Hilbert-Schmidt (Euclidean or flat) metric--that a generic (nine-dimensional) real two-qubit system is separable, by implementing the well-known Peres-Horodecki test on the partial transposes (PTs) of the associated 4 x 4 density matrices. But the full implementation of the test--requiring that the determinant of the PT be nonnegative for separability to hold--appears to be, at least presently, computationally intractable. So, we have previously implemented--using the auxiliary concept of a diagonal-entry-parameterized separability function (DESF)--the weaker implied test of nonnegativity of the six 2 x 2 principal minors of the PT. This yielded an exact upper bound on the separability probability of 1024/(135 Pi^2) = 0.76854. Here, we extend this line of work by requiring that the four 3 x 3 principal minors of the PT be nonnegative, giving us an improved/reduced upper bound of 22/35 = 0.628571. Numerical simulations--as opposed to exact symbolic calculations--indicate, on the other hand, that the true probability is certainly less than 1/2. Our combined analyses lead us to suggest a possible form for the true DESF, yielding a separability probability of 29/64 = 0.453125, while the best exact lower bound established so far is (6928-2205 Pi)/2^(9/2) = 0.0348338.
Separability and entanglement for n-qubits systems are quantified by using Hilbert-Schmidt (HS) decompositions in which the density matrices are decomposed into various terms representing certain one qubit, two-qubits,and larger qubits measurements. The present method is more general than previous methods for bipartite systems as it can be used for quantification of entanglement for large n-qubits systems.We demonstrate the use of such method by analyzing three qubits GHZ states, and three qubits general Bell states produced by a certain multiplications of Braid operators, operating on the com…
The Horodecki family employed the Jaynes maximum-entropy principle, fitting the mean (b_{1}) of the Bell-CHSH observable (B). This model was extended by Rajagopal by incorporating the dispersion (σ_{1}^2) of the observable, and by Canosa and Rossignoli, by generalizing the observable (B_α). We further extend the Horodecki one-parameter model in both these manners, obtaining a three-parameter (b_{1},σ_{1}^2,α) two-qubit model, for which we find a highly interesting/intricate continuum (-\infty < α< \infty) of Hilbert-Schmidt (HS) separability probabilities -- in which, the golden ratio is featu…
A confluence of numerical and theoretical results leads us to conjecture that the Hilbert-Schmidt separability probabilities of the 15- and 9-dimensional convex sets of complex and real two-qubit states (representable by 4 x 4 density matrices rho) are 8/33 and 8/17, respectively. Central to our reasoning are the modifications of two ansatze, recently advanced (quant-ph/0609006), involving incomplete beta functions B_{nu}(a,b), where nu= (rho_{11} rho_{44})/(rho_{22} rho_{33}). We, now, set the separability function S_{real}(nu) propto B_{nu}(nu,1/2},2) =(2/3) (3-nu) sqrt{nu}. Then, in the com…
Hilbert-Schmidt (HS) decompositions are employed for analyzing systems of n-qubits, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary transformations (PTU) for one qubit from the whole system, are used for indicating inseparability. A sufficient criterion for full separability of the n-qubits and qubit-qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and n-qubits systems, …
We seek to derive the probability--expressed in terms of the Hilbert-Schmidt (Euclidean or flat) metric--that a generic (nine-dimensional) real two-qubit system is separable, by implementing the well-known Peres-Horodecki test on the partial transposes (PT's) of the associated 4 x 4 density matrices). But the full implementation of the test--requiring that the determinant of the PT be nonnegative for separability to hold--appears to be, at least presently, computationally intractable. So, we have previously implemented--using the auxiliary concept of a diagonal-entry-parameterized separability…
We consider the Schmidt decomposition of a bipartite density operator induced by the Hilbert-Schmidt scalar product, and we study the relation between the Schmidt coefficients and entanglement. First, we define the Schmidt equivalence classes of bipartite states. Each class consists of all the density operators (in a given bipartite Hilbert space) sharing the same set of Schmidt coefficients. Next, we review the role played by the Schmidt coefficients with respect to the separability criterion known as the `realignment' or `computable cross norm' criterion; in particular, we highlight the fact…
We study the mathematical structures and relations among some quantities in the theory of quantum entanglement, such as separability, weak Schmidt decompositions, Hadamard matrices etc.. We provide an operational method to identify the Schmidt-correlated states by using weak Schmidt decomposition. We show that a mixed state is Schmidt-correlated if and only if its spectral decomposition consists of a set of pure eigenstates which can be simultaneously diagonalized in weak Schmidt decomposition, i.e. allowing for complex-valued diagonal entries. For such states, the separability is reduced to t…
We treat 3-qubits states with maximally disordered subsystems, by using Hilbert-Schmidt decompositions.By using unfolding methods, the tensors are converted into matrices and by applying singular values decompositions to these matrices the number 27 of the Hilbert Schmidt parameters, is reduced to 9 and under the condition that the sum of absolute values of these parameters is not larger than 1, we conclude that the density matrix is fully separable . In another method we divide the 27 Hilbert Schmidt parameters into 9 triads where for each triad we calculate the Frobenius norm of 3 Hilbert-Sc…
Paralleling our recent computationally-intensive (quasi-Monte Carlo) work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric -- and also their analogous formulas (quant-ph/0302197) for the (non-monotone) Hilbert-Schmidt metric. With the same seven billion well-distributed (``low-discrepancy'') sample points, w…
Jakobczyk and Siennicki studied two-dimensional sections of a set of (generalized) Bloch vectors corresponding to n x n density matrices of two-qubit systems (that is, the case n = 4). They found essentially five different types of (nontrivial) separability regimes. We compute the Euclidean/Hilbert-Schmidt (HS) separability probabilities assigned to these regimes, and conduct parallel two-dimensional sectional analyses for the higher-level cases n=6,8,9 and 10. Making use of the newly-introduced capability for integration over implicitly defined regions of version 5.1 of Mathematica -- also fr…
Firstly, we reduce the long-standing problem of ascertaining the Hilbert-Schmidt probability that a generic pair of qubits is separable to that of determining the specific nature of a one-dimensional (separability) function of the radial coordinate (r) of the unit ball in 15-dimensional Euclidean space, and similarly for a generic pair of rebits, using the 9-dimensional unit ball. Separability probabilities, could, then, be directly obtained by integrating the products of these functions (which we numerically estimate and plot) with jacobian factors of r^m over r in [0,1], with m=17 for the tw…
Two-qubit X-matrices have been the subject of considerable recent attention, as they lend themselves more readily to analytical investigations than two-qubit density matrices of arbitrary nature. Here, we maximally exploit this relative ease of analysis to formally derive an exhaustive collection of results pertaining to the separability probabilities of generalized two-qubit X-matrices endowed with Hilbert-Schmidt and, more broadly, induced measures. Further, the analytical results obtained exhibit interesting parallels to corresponding earlier (but, contrastingly, not yet fully rigorous) res…
We show that the Hilbert-Schmidt distance, unlike the trace distance, between quantum states is generally not monotonic for open quantum systems subject to Lindblad semigroup dynamics. Sufficient conditions for contractivity of the Hilbert-Schmidt norm in terms of the dissipation generators are given. Although these conditions are not necessary, simulations suggest that non-contractivity is the typical case, i.e., that systems for which the Hilbert-Schmidt distance between quantum states is monotonically decreasing form only a small set of all possible dissipative systems for N>2, in contrast …
Using simple quantum analysis we describe the correlations of Greenberger-Horne-Zeilinger (GHZ) states by the use of Hilbert-Schmidt (HS) representation. Our conclusion is that while these states disprove local-realism they do not prove any nonlocality property.
In this paper, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let $M$ be the matrix of the amplitudes of $\ketψ$, we prove $M$ is a compact operator. We also prove $\ketψ$ is separable if and only if $M$ is a bounded linear operator with rank 1, that is the image of $M$ is a one dimensional Hilbert space. So we have related the separability for bipartite pure states in infinite dimensional Hilbert spaces to an important class of bounded linear operators in Functional analysis which has many interesting proper…
In this letter we discuss a new entanglement measure. It is based on the Hilbert-Schmidt norm of operators. We give an explicit formula for calculating the entanglement of a large set of states on C^2 \times C^2. Furthermore we find some relations between the entanglement of relative entropy and the Hilbert-Schmidt entanglement. A rigorous definition of partial transposition is given in the appendix.
In order to construct a measure of entanglement on the basis of a ``distance'' between two states, it is one of desirable properties that the ``distance'' is nonincreasing under every completely positive trace preserving map. Contrary to a recent claim, this letter shows that the Hilbert-Schmidt distance does not have this property.
We extend the findings and analyses of our two recent studies (Phys. Rev. A, 75, 032326 [2007] and arXiv:0704.3723) by, first, obtaining numerical estimates of the separability function based on the (Euclidean, flat) Hilbert-Schmidt (HS) metric for the 27-dimensional convex set of quaternionic two-qubit systems. The estimated function appears to be strongly consistent with our previously-formulated Dyson-index (beta = 1, 2, 4) ansatz, dictating that the quaternionic (beta=4) separability function should be exactly proportional to the square of the separability function for the 15-dimensional c…
Hilbert - Schmidt and trace norm geometric quantum discord are compared with regard to their behavior during local time evolution. We consider the system of independent two - level atoms with time evolution given by the dissipative process of spontaneous emission. It is explicitly shown that the Hilbert - Schmidt norm discord has nonphysical properties with respect to such local evolution and cannot serve as a reasonable measure of quantum correlations and the better choice is to use trace norm discord as such a measure.
We significantly advance the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A function P(alpha), incorporating a family of six hypergeometric functions, all with argument 27/64 =(3/4)^3, is obtained here. It reproduces a series, alpha = 1/2, 1, 3/2, 2,...,32$, of sixty-four Hilbert-Schmidt generic 2 x 2 generalized separability probabilities, advanced on the basis of systematic high-accuracy probability-distribution-reconstruction computations, employing 7,501 determinant…
Using the Hilbert-Schmidt (HS) decomposition we suggest new possible choices of Bell operators and entanglement witnesses (EW ) for n (>2) qubits systems for (full/bi) separability. The latter give upper bounds for (full/bi) separability. Also using the HS decomposition, we find explicitly (full/bi) separable forms for some qubits states which give lower bounds for (full/bi) separability. When the lower bounds and upper bounds coincide it means that the EW is optimal. In the case of full separability, the positive transpose method can sometimes give optimal results. As concrete examples, we gi…
Pati (Physics Letters A, 2000) derived a sufficient condition for the existence of Schmidt decomposition in tripartite Hilbert spaces. In this paper, we show that the condition is erroneous by demonstrating some counter-examples. Moreover, we suitably modify the condition and provide a correctness proof. We also show for the first time how this can be generalized to $n$-partite Hilbert spaces for any $n\geq 2$. Finally, we prove that this condition is also a necessary condition.
In this letter, we show that the measure of entanglement using the generalized Bell inequality and the distance between states coincide when we use the Hilbert-Schmidt norm. Our conclusions apply to the spin-1/2 Heisenberg chains with interaction between the first neighbors.
Hilbert-Schmidt distance reduces to Euclidean distance in Bell decomposable states. Based on this, entanglement of these states are obtained according to the protocol proposed in Ref. [V. Vedral et al, Phys. Rev. Lett. 78, 2275 (1995)] with Hilbert-Schmidt distance. It is shown that this measure is equal to the concurrence and thus can be used to generate entanglement of formation. We also introduce a new measure of distance and show that under the action of restricted LQCC operations, the associated measure of entanglement transforms in the same way as the concurrence transforms .
The question of the generation of random mixed states is discussed, aiming for the computation of probabilistic characteristics of composite finite dimensional quantum systems. Particularly, we consider the generation of the random Hilbert-Schmidt and Bures ensembles of qubit and qutrit pairs and compute the corresponding probabilities to find a separable state among the states of a fixed rank.
The (complex) two-qubit systems comprise a 15-dimensional convex set and the real two-qubit systems, a 9-dimensional convex set. While formulas for the Hilbert-Schmidt volumes of these two sets are known -- owing to recent important work of Sommers and Zyczkowski (J. Phys. A {36}, 10115 [2003]) -- formulas have not been so far obtained for the volumes of the separable subsets. We reduce these two problems to the determination of certain functions of a single variable.
We report major advances in the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A highly succinct separability probability function P(alpha) is put forth, yielding for generic (9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional) two-qubit systems, P(1) = 8/33 and (27-dimensional) two-quater(nionic)bit systems, P(2)=26/323. This particular form of P(alpha) was obtained by Qing-Hu Hou and colleagues by applying Zeilberger's algorithm ("creative telescoping") t…
A generic model of measurement device which is able to directly measure commonly used quantum-state characteristics such as fidelity, overlap, purity and Hilbert-Schmidt distance for two general uncorrelated mixed states is proposed. In addition, for two correlated mixed states, the measurement realizes an entanglement witness for Werner's separability criterion. To determine these observables, the estimation only one parameter - the visibility of interference, is needed. The implementations in cavity QED, trapped ion and electromagnetically induced transparency experiments are discussed.
The Hilbert-Schmidt distance between a mixed three-qubit state and its closest state is used to quantify the amount of pairwise quantum correlations in a tripartite system. Analytical expressions of geometric quantum discord are derived. A particular attention is devoted to two special classes of three-qubit $X$ states. They include three-qubit states of ${\rm W}$, ${\rm GHZ}$ and Bell type. We also discuss the monogamy property of geometric quantum discord in some mixed three-qubit systems.
Quantum correlations in the state of four-level atom are investigated by using generic unitary transforms of the classical (diagonal) density matrix. Partial cases of pure state, $X$-state, Werner state are studied in details. The geometrical meaning of unitary Hilbert reference-frame rotations generating entanglement in the initially separable state is discussed. Characteristics of the entanglement in terms of concurrence, entropy and negativity are obtained as functions of the unitary matrix rotating the reference frame.
Gilbert proposed an algorithm for bounding the distance between a given point and a convex set. In this article we apply the Gilbert's algorithm to get an upper bound on the Hilbert-Schmidt distance between a given state and the set of separable states. While Hilbert Schmidt Distance does not form a proper entanglement measure, it can nevertheless be useful for witnessing entanglement. We provide here a few methods based on the Gilbert's algorithm that can reliably qualify a given state as strongly entangled or practically separable, while being computationally efficient. The method also outpu…
We compute the volume of the convex N^2-1 dimensional set M_N of density matrices of size N with respect to the Hilbert-Schmidt measure. The hyper--area of the boundary of this set is also found and its ratio to the volume provides an information about the complex structure of M_N. Similar investigations are also performed for the smaller set of all real density matrices. As an intermediate step we analyze volumes of the unitary and orthogonal groups and of the flag manifolds.
Hilbert-Schmidt (HS) decompositions and Frobenius norms are used to analyze biseparability of 3-qubit systems, with particular emphasis on density matrices with maximally disordered subsystems (MDS) and on the W state mixed with white noise. The biseparable form of a MDS density matrix is obtained by using the Bell states of a 2-qubit subsystem, multiplied by density matrices of the third qubit, which include the relevant HS parameters. Using our methods a sufficient condition and explicit biseparability of the W state mixed with white noise are given. They are compared with the sufficient con…
We use classical results from the theory of linear preserver problems to characterize operators that send the set of pure states with Schmidt rank no greater than k back into itself, extending known results characterizing operators that send separable pure states to separable pure states. We also provide a new proof of an analogous statement in the multipartite setting. We use these results to develop a bipartite version of a classical result about the structure of maps that preserve rank-1 operators and then characterize the isometries for two families of norms that have recently been studied…
We apply the generalised concept of witness operators to arbitrary convex sets, and review the criteria for the optimisation of these general witnesses. We then define an embedding of state vectors and operators into a higher-dimensional Hilbert space. This embedding leads to a connection between any Schmidt number witness in the original Hilbert space and a witness for Schmidt number two (i.e. the most general entanglement witness) in the appropriate enlarged Hilbert space. Using this relation we arrive at a conceptually simple method for the construction of Schmidt number witnesses in bipart…
Compelling evidence-though yet no formal proof-has been adduced that the probability that a generic (standard) two-qubit state ($ρ$) is separable/disentangled is $\frac{8}{33}$ (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal $4F3$-hypergeometric moment formula (Appendix A), we reach, {\it via} density-approximation procedures, the conclusion that one-half ($\frac{4}{33}$) of this probability arises when the determinantal inequality $|ρ^{PT}|>|ρ|$, where $PT$ denotes the partial transpose, is satisfied, and, the oth…
Decomposition of (finite-dimensional) operators in terms of orthogonal bases of matrices has been a standard method in quantum physics for decades. In recent years, it has become increasingly popular because of various methodologies applied in quantum information, such as the graph state formalism and the theory of quantum error correcting codes, but also due to the intensified research on the Bloch representation of quantum states. In this contribution we collect various interesting facts and identities that hold for finite-dimensional orthogonal matrix bases.
We investigate how the Hilbert-Schmidt speed (HSS), a special type of quantum statistical speed, can be exploited as a powerful and easily computable tool for quantum phase estimation in a $n$-qubit system. We find that, when both the HSS and quantum Fisher information (QFI) are computed with respect to the phase parameter encoded into the initial state of the $n$-qubit register, the zeros of the HSS dynamics are essentially the same as those of the QFI dynamics. Moreover, the positivity (negativity) of the time-derivative of the HSS exactly coincides with the positivity (negativity) of the ti…
A witness of non-Markovianity based on the Hilbert-Schmidt speed (HSS), a special type of quantum statistical speed, has been recently introduced for low-dimensional quantum systems. Such a non-Markovianity witness is particularly useful, being easily computable since no diagonalization of the system density matrix is required. We investigate the sensitivity of this HSS-based witness to detect non-Markovianity in various high-dimensional and multipartite open quantum systems. We find that the time behaviors of the HSS-based witness are always in agreement with those of quantum negativity or qu…
We reexamine a recent analysis in which, using the volume of the associated quantum steering ellipsoid (QES) as a measure, we sought to estimate the probability that a two-qubit state is separable. In the estimation process, we, in effect, sought to attach to states random with respect to Hilbert-Schmidt (HS) measure, the corresponding QES volumes. However, a study of the relations between HS and Bures ensembles and their well-supported separability probabilities of $\frac{8}{33}$ and $\frac{25}{341}$, respectively, now lead us to explore as a possible alternative measure, the QES volume divid…
The operator Schmidt rank is the minimum number of terms required to express a state as a sum of elementary tensor factors. Here we provide a new proof of the fact that any bipartite mixed state with operator Schmidt rank two is separable, and can be written as a sum of two positive semidefinite matrices per site. Our proof uses results from the theory of free spectrahedra and operator systems, and illustrates the use of a connection between decompositions of quantum states and decompositions of nonnegative matrices. In the multipartite case, we prove that any Hermitian Matrix Product Density …
Using Bloch's parametrization for qudits ($d$-level quantum systems), we write the Hilbert-Schmidt distance (HSD) between two generic $n$-qudit states as an Euclidean distance between two vectors of observables mean values in $\mathbb{R}^{Π_{s=1}^{n}d_{s}^{2}-1}$, where $d_{s}$ is the dimension for qudit $s$. Then, applying the generalized Gell Mann's matrices to generate $SU(d_{s})$, we use that result to obtain the Hilbert-Schmidt quantum coherence (HSC) of $n$-qudit systems. As examples, we consider in details one-qubit, one-qutrit, two-qubit, and two copies of one-qubit states. In this las…
Non-Markovian effects can speed up the dynamics of quantum systems while the limits of the evolution time can be derived by quantifiers of quantum statistical speed. We introduce a witness for characterizing the non-Markovianity of quantum evolutions through the Hilbert-Schmidt speed (HSS), which is a special type of quantum statistical speed. This witness has the advantage of not requiring diagonalization of evolved density matrix. Its sensitivity is investigated by considering several paradigmatic instances of open quantum systems, such as one qubit subject to phase-covariant noise and Pauli…
We begin by investigating relationships between two forms of Hilbert-Schmidt two-re[al]bit and two-qubit "separability functions"--those recently advanced by Lovas and Andai (J. Phys. A 50 [2017] 295303), and those earlier presented by Slater (J. Phys. A 40 [2007] 14279). In the Lovas-Andai framework, the independent variable $\varepsilon \in [0,1]$ is the ratio $σ(V)$ of the singular values of the $2 \times 2$ matrix $V=D_2^{1/2} D_1^{-1/2}$ formed from the two $2 \times 2$ diagonal blocks ($D_1, D_2$) of a $4 \times 4$ density matrix $D$. In the Slater setting, the independent variable $μ$ i…
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…
We employ a quasirandom methodology, recently developed by Martin Roberts, to estimate the separability probabilities, with respect to the Bures (minimal monotone/statistical distinguishability) measure, of generic two-qubit and two-rebit states. This procedure, based on generalized properties of the golden ratio, yielded, in the course of almost seventeen billion iterations (recorded at intervals of five million), two-qubit estimates repeatedly close to nine decimal places to $\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.073313783$. Howeer, despite the use of over twenty-three billion it…
The practically useful criteria of separable states $ρ=\sum_{k}w_{k}ρ_{k}$ in $d=2\times2$ are discussed. The equality $G({\bf a},{\bf b})= 4[\langle ψ|P({\bf a})\otimes P({\bf b})|ψ\rangle-\langle ψ|P({\bf a})\otimes{\bf 1}|ψ\rangle\langle ψ|{\bf 1}\otimes P({\bf b})|ψ\rangle]=0$ for any two projection operators $P({\bf a})$ and $P({\bf b})$ provides a necessary and sufficient separability criterion in the case of a separable pure state $ρ=|ψ\rangle\langleψ|$. We propose the separability criteria of mixed states, which are given by ${\rm Tr}ρ\{{\bf a}\cdot {\bf σ}\otimes {\bf b}\cdot {\bf σ}\…
The nonnegativity of the determinant of the partial transpose of a two-qubit (4 x 4) density matrix is both a necessary and sufficient condition for its separability. While the determinant is restricted to the interval [0,1/256], the determinant of the partial transpose can range over [-1/16,1/256], with negative values corresponding to entangled states. We report here the exact values of the first nine moments of the probability distribution of the partial transpose over this interval, with respect to the Hilbert-Schmidt (metric volume element) measure on the nine-dimensional convex set of re…
We prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kullback-Leibler divergence. Suppose Alice is sending classical information to Bob using a quantum channel, while Bob is performing some projective measurement. We bound the classical mutual information in terms of the Hilbert-Schmidt norm by its quantum Hilbert-Schmidt counterpart. This constitutes a Holevo-type upper bound on the classical information transmission rate via a quantum channel. The resulting inequality is rather natural and intuitive relating classical and quantum expressions using the …
Minimum error state discrimination between two mixed states ρand σcan be aided by the receipt of "classical side information" specifying which states from some convex decompositions of ρand σapply in each run. We quantify this phenomena by the average trace distance, and give lower and upper bounds on this quantity as functions of ρand σ. The lower bound is simply the trace distance between ρand σ, trivially seen to be tight. The upper bound is \sqrt{1 - tr(ρσ)}, and we conjecture that this is also tight. We reformulate this conjecture in terms of the existence of a pair of "unbiased decomposi…
Employing the volume of quantum steering ellipsoids (QSEs) as a measure on the fifteen-dimensional convex set of two-qubit states, we estimate the ratio of the integral of the measure over the separable states to its integral over all (separable and entangled) states to be 0.0288. This can be contrasted with the considerably larger separability ratios (probabilities) of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and $\frac{25}{341}=\frac{5^2}{11 \cdot 31} \approx 0.0733138$ that various forms of evidence point to with the use of the prominent Hilbert-Schmidt and Bures measures, r…
Employing Hilbert-Schmidt measure, we explicitly compute and analyze a number of determinantal product (bivariate) moments |rho|^k |rho^{PT}|^n, k,n=0,1,2,3,..., PT denoting partial transpose, for both generic (9-dimensional) two-rebit (alpha = 1/2) and generic (15-dimensional) two-qubit (alpha=1) density matrices rho. The results are, then, incorporated by Dunkl into a general formula (Appendix D6), parameterized by k, n and alpha, with the case alpha=2, presumptively corresponding to generic (27-dimensional) quaternionic systems. Holding the Dyson-index-like parameter alpha fixed, the induce…
Explicit separable density matrices, for mixed two qubits states, are derived by the use of Hilbert Schmidt decompositions and Peres Horodecki criterion. A strongly separable two qubits mixed state is defined by multiplications of two density matrices given with pure states while weakly separable two qubits state is defined by multiplications of two density matrices which includes non-pure states. We find the sufficient and necessary condition for separability of two-qubits density matrices and show that under this condition the two-qubit density matrices are strongly separable.
