We develop a sufficient condition for the least-squares measurement (LSM), or the square-root measurement, to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states. Using this condition we derive the optimal measurement for state sets with a broad class of symmetries.
We first consider geometrically uniform (GU) state sets with a possibly nonabelian generating group, and show that if the generator satisfies a certain constraint, then the LSM is optimal. In particular, for pure-state GU ensembles the LSM is shown to be optimal. For arbitrary GU state sets we show that the optimal measurement operators are GU with generator that can be computed very efficiently in polynomial time, within any desired accuracy.
We then consider compound GU (CGU) state sets which consist of subsets that are GU. When the generators satisfy a certain constraint, the LSM is again optimal. For arbitrary CGU state sets the optimal measurement operators are shown to be CGU with generators that can be computed efficiently in polynomial time.
Majorana Representation (MR) of symmetric $ N $-qubit pure states has been used successfully in entanglement classification. Generalization of this has been a long standing open problem due to the difficulties faced in the construction of a Majorana like geometric representation for symmetric mixed state. We have overcome this problem by developing a method of classifying local unitary (LU) equivalent classes of symmetric $N$-qubit mixed states based on the geometrical Multiaxial Representation (MAR) of the density matrix. In addition to the two parameters defined for the entanglement classifi…
We classify, up to local unitary equivalence, local unitary stabilizer Lie algebras for symmetric mixed states into six classes. These include the stabilizer types of the Werner states, the GHZ state and its generalizations, and Dicke states. For all but the zero algebra, we classify entanglement types (local unitary equivalence classes) of symmetric mixed states that have those stabilizers. We make use of the identification of symmetric density matrices with polynomials in three variables with real coefficients and apply the representation theory of SO(3) on this space of polynomials.
Study of an N qubit mixed symmetric separable states is a long standing challenging problem as there exist no unique separability criterion. In this regard, we take up the N-qubit mixed symmetric separable states for a detailed study as these states are of experimental importance and offer elegant mathematical analysis since the dimension of the Hilbert space reduces from 2^N to N + 1. Since there exists a one to one correspondence between spin-j system and an N-qubit symmetric state, we employ Fano statistical tensor parameters for the parametrization of spin density matrix. Further, we use g…
We study an optimized measurement which discriminates N mixed quantum states occurring with given prior robabilities. The measurement yields the maximum achievable confidence for each of the N conclusive outcomes, thereby keeping the overall probability of inconclusive outcomes as small as possible. It corresponds to optimum unambiguous discrimination when for each outcome the confidence is equal to unity. Necessary and sufficient optimality conditions are derived and general properties of the optimum measurement are obtained. The results are applied to the optimized maximum-confidence discrim…
We introduce an operational entanglement classification of symmetric mixed states for an arbitrary number of qubits based on stochastic local operations assisted with classical communication (SLOCC operations). We define families of SLOCC entanglement classes successively embedded into each other, we prove that they are of non-zero measure, and we construct witness operators to distinguish them. Moreover, we discuss how arbitrary symmetric mixed states can be realized in the lab via a one-to-one correspondence between well-defined sets of controllable parameters and the corresponding entanglem…
We discuss how to calculate genuine multipartite quantum and classical correlations in symmetric, spatially invariant, mixed $n$-qubit density matrices. We show that the existence of symmetries greatly reduces the amount of free parameters to be optimized in order to find the optimal measurement that minimizes the conditional entropy in the discord calculation. We apply this approach to the states exhibited dynamically during a thermodynamic protocol to extract maximum work. We also apply the symmetry criterion to a wide class of physically relevant cases of spatially homogeneous noise over mu…
Induced random mixed states are symmetric Dirichlet mixtures
DEVCOM Army Research Laboratory（美国陆军研究与发展司令部陆军研究实验室）；Tulane University（杜兰大学）；University of Toronto（多伦多大学）；Southern Methodist University（南方卫理公会大学）；Purdue University（普渡大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem whether a complete set of five iso-entangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced density matrices of these $20$ pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius $\sqrt{3/20}$ located inside the Bloch ball of radius $1/2$. Such a set forms a mixed-state $2$-design --- a discrete set of quantum states with the property that the …
We present a comprehensive study of maximally entangled symmetric states of arbitrary numbers of qubits in the sense of the maximal mixedness of the one-qubit reduced density operator. A general criterion is provided to easily identify whether given symmetric states are maximally entangled in that respect or not. We show that these maximally entangled symmetric (MES) states are the only symmetric states for which the expectation value of the associated collective spin of the system vanishes, as well as in corollary the dipole moment of the Husimi function. We establish the link between this ki…
We consider the problem of minimum-error quantum state discrimination for single-qubit mixed states. We present a method which uses the Helstrom conditions constructively and analytically; this algebraic approach is complementary to existing geometric methods, and solves the problem for any number of arbitrary signal states with arbitrary prior probabilities.
We present theory and experiment for the task of discriminating two nonorthogonal states, given multiple copies. We implement several local measurement schemes, on both pure states and states mixed by depolarizing noise. We find that schemes which are optimal (or have optimal scaling) without noise perform worse with noise than simply repeating the optimal single-copy measurement. Applying optimal control theory, we derive the globally optimal local measurement strategy, which outperforms all other local schemes, and experimentally implement it for various levels of noise.
We develop an approach of quantifying entanglement in mixed quantum states by the optimal entanglement witness operator. We identify the convex set of mixed states for which a single witness provides the exact value of an entanglement measure, and show that the convexity, properties, and symmetries of entanglement or of a target state considerably fix the form of the optimal witness. This greatly reduces difficulty in computing and experimentally determining entanglement measures. As an example, we show how to experimentally quantify bound entanglement in four-qubit noisy Smolin states and thr…
We study quantum metrology for unitary dynamics. Analytic solutions are given for both the optimal unitary state preparation starting from an arbitrary mixed state and the corresponding optimal measurement precision. This represents a rigorous generalization of known results for optimal initial states and upper bounds on measurement precision which can only be saturated if pure states are available. In particular, we provide a generalization to mixed states of an upper bound on measurement precision for time-dependent Hamiltonians that can be saturated with optimal Hamiltonian control. These r…
Recently, it has been shown that the generalized symmetric Woods-Saxon potential energy, in which surface interaction terms are taken into account, describes the physical processes better than the standard form. Therefore in this study, we investigate the scattering of Klein-Gordon particles in the presence of both generalized symmetric Woods-Saxon vector and scalar potential. In one spatial dimension we obtain the solutions in terms of hypergeometric functions for spin symmetric or pseudo-spin symmetric cases. Finally, we plot transmission and reflection probabilities for incident particles w…
Mixed states appear naturally in experiment over pure states. So for studying different notions of nonlocality and their relation with entanglement in realistic scenarios, one needs to consider mixed states. In a recent article [Phys. Rev. Lett. \textbf{108}, 020502 (2012)], a complete characterization of entanglement of an entire class of mixed three qubit states with the same symmetry as Greenberger-Horne-Zeilinger state known as GHZ-symmetric states, has been achieved. In this paper we investigate different notions of nonlocality of the same class of states. By finding the analytical expres…
Mixed entangled states are generic resource for quantum teleportation. Optimal teleportation fidelity measures the success of quantum teleportation. The relevance of rank in the teleportation process is investigated by constructing three new maximally entangled mixed states (MEMS) of different ranks. Linear entropy, concurrence, optimal teleportation fidelity and Bell function are obtained for each of the state analytically. It is found that mixed states with higher rank are better resource for teleportation. In order to achieve a fixed value of optimal teleportation fidelity, we find that low…
In an earlier publication we had given an exhaustive analysis of the criteria for weak value measurements of pure states to be optimal in the sense considered by Wootters and Fields. We had proved, for arbitrary spin cases, that the measurements are optimal when the post-selected state is mutually unbiased wrt the eigenstates of the observable being measured.Here we extend the discussion to mixed states. For these, weak value measurements have several problems which we illustrate with the protocol proposed by Shengjun Wu. We discuss tomography of mixed states based on weak measurements and sho…
The separability detecting problem of mixed states is one of the fundamental problems in quantum information theory. In the last 20 years, almost all methods are based on the sufficient or necessary conditions for entanglement. However, in this paper, we only need one algorithm to solve the problem. We propose a tensor optimization method to check whether an $m$-partite quantum mixed state is separable or not and give a decomposition for it if it is. We first convert the separability discrimination problem of mixed states to the positive Hermitian decomposition problem of Hermitian tensors. Th…
We explore sufficient conditions for inseparability in mixed states with a globally conserved charge, such as a particle number. We argue that even separable states may contain entanglement in fixed charge sectors, as long as the state can not be separated into charge conserving components. As a witness of symmetric inseparability we study the number entanglement (NE), $ΔS_m$, defined as the entropy change due to a subsystem's charge measurement. Whenever $ΔS_m > 0$, there exist inseparable charge sectors, having finite (logarithmic) negativity, even when the full state is either separable or …
In this short note, we propose a generalization of Atiyah type TQFTs from pure states to mixed states in the sense that the Hilbert space of pure states associated to a space manifold is replaced by a quantum coherent space related to density matrices. Atiyah type TQFT is a symmetric monoidal functor from the Bord category of manifolds to the category Vec of finite dimensional vector spaces. In this paper, we define mixed-state TQFTs by replacing the target category Vec by QCS--the category of quantum coherent spaces, then a mixed-state TQFT is simply a symmetric monoidal functor from Bord to …
Within a geometrical context, we derive an explicit formula for the computation of the symmetric logarithmic derivative for arbitrarily mixed quantum systems, provided that the structure constants of the associated unitary Lie algebra are known. To give examples of this procedure, we first recover the known formulae for two-level mixed and three-level pure state systems and then apply it to the novel case of U(3), that is for arbitrarily mixed three-level systems (q-trits). Exploiting the latter result, we finally calculate an expression for the Fisher tensor for a q-trit considering also all …
Separability criteria are typically of the necessary, but not sufficient, variety, in that satisfying some separability criterion, such as positivity of eigenvalues under partial transpose, does not strictly imply separability. Certifying separability amounts to proving the existence of a decomposition of a target mixed state into some convex combination of separable states; determining the existence of such a decomposition is "hard." We show that it is effective to ask, instead, if the target mixed state "fits" some preconstructed separable form, in that one can generate a sufficient separabi…
Given an ensemble of mixed qubit states, it is possible to increase the purity of the constituent states using a procedure known as state purification. The reverse operation, which we refer to as dilution, reduces the level of purity present in the constituent states. In this paper we find asymptotically optimal procedures for purification and dilution of an ensemble of i.i.d. mixed qubit states, for some given input and output purities and an asymptotic output rate. Our solution involves using the statistical tool of local asymptotic normality, which recasts the qubit problem in terms of atte…
