We introduce a classification of mixed three-qubit states, in which we define the classes of separable, biseparable, W- and GHZ-states. These classes are successively embedded into each other. We show that contrary to pure W-type states, the mixed W-class is not of measure zero. We construct witness operators that detect the class of a mixed state. We discuss the conjecture that all entangled states with positive partial transpose (PPTES) belong to the W-class. Finally, we present a new family of PPTES "edge" states with maximal ranks.
We address the problem of optimal estimation of the relative phase for two-dimensional quantum systems in mixed states. In particular, we derive the optimal measurement procedures for an arbitrary number of qubits prepared in the same mixed state.
Quantum cloning of two identical mixed qubits $ρ\otimes ρ$ is studied. We propose the quantum cloning transformations not only for the triplet (symmetric) states but also for the singlet (antisymmetric) state. We can copy these two identical mixed qubits to M (M>2) copies. This quantum cloning machine is optimal in the sense that the shrinking factor between the input and the output single qubit achieves the upper bound. The result shows that we can copy two identical mixed qubits with the same quality as that of two identical pure states.
In this paper, we discuss the partial separability and its criteria problems of multipartite qubit mixed-states. First we strictly define what is the partial separability of a multipartite qubit system. Next we give a reduction way from N-partite qubit density matrixes to bipartite qubit density matrixes, and prove a necessary condition that a N-partite qubit mixed-state to be partially separable is its reduction to satisfy the PPT condition. PACC numbers: 03.67.Mn; 03.65.Ud; 03.67.Hk
We analyze the class of single qubit channels with the environment modeled by a one-qubit mixed state. The set of affine transformations for this class of channels is computed analytically, employing the canonical form for the two-qubit unitary operator. We demonstrate that, 3/8 of the generalized depolarizing channels can be simulated by the one-qubit mixed state environment by explicitly obtaining the shape of the volume occupied by this class of channels within the tetrahedron representing the generalized depolarizing channels. Further, as a special case, we show that the two-Pauli Channel …
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.
It is commonly assumed that Shor's quantum algorithm for the efficient factorization of a large number $N$ requires a pure initial state. Here we demonstrate that a single pure qubit together with a collection of $log_2 N$ qubits in an arbitrary mixed state is sufficient to implement Shor's factorization algorithm efficiently.
We show that there does not exist any universal quantum cloning machine that can broadcast an arbitrary mixed qubit with a constant fidelity. Based on this result, we investigate the dependent quantum cloner in the sense that some parameter of the input qubit $ρ_s(θ,ω,λ)$ is regarded as constant in the fidelity. For the case of constant $ω$, we establish the $1\to2$ optimal symmetric dependent cloner with a fidelity 1/2. It is also shown that the $1\to M$ optimal quantum cloning machine for pure qubits is also optimal for mixed qubits, when $λ$ is the unique parameter in the fidelity. For gene…
We find the necessary and sufficient condition under which two two-qubit mixed states can be purified into a pure maximally entangled state by local operations and classical communication. The optimal protocol for such transformation is obtained. This result leads to a necessary and sufficient condition for the exact purification of $n$ copies of a two-qubit state.
In this work, we provide a complete analysis to minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities. For the complete anaysis, the most important work to do is to find the necessary and sufficient conditions for the existence of null measurement operator. From the geometric structure of qubit states, we obtain the analytic condition for deciding the existence of a null operator in minimum-error measurement for mixed four qubit states, which also gives the necessary and sufficient conditions for every optimal POVM to have non-zero elements. Using the condi…
We study a generalized amplitude damping channel when environment is initially in the single-qubit mixed state. Representing the affine transformation of the generalized amplitude damping by a three-dimensional volume, we plot explicitly the volume occupied by the channels simulatable by a single-qubit mixed-state environment. As expected, this volume is embedded in the total volume by the channels which is simulated by two-qubit enviroment. The volume ratio is approximately 0.08 which is much smaller than 3/8, the volume ratio for generalized depolarizing channels.
For a class of mixed two -qubit states we show that it is not possible to discriminate between states violating or non - violating Bell - CHSH inequalities, knowing only their entanglement and mixedness. For a large set of possible values of these quantities, we construct pairs of states with the same entanglement and mixedness such that one state is violating but the other is non - violating Bell - CHSH inequality.
Estimating multiparamter simultaneously as precise as possible is an important goal of quantum metrolo- gy. As a first step to this end, here we give a condition determining whether two arbitrary parameters can be estimated simultaneously for a qubit in the mixed state. An application of this condition is shown.
We give a complete, hierarchic classification for arbitrary multi-qubit mixed states based on the separability properties of certain partitions. We introduce a family of N-qubit states to which any arbitrary state can be depolarized. This family can be viewed as the generalization of Werner states to multi-qubit systems. We fully classify those states with respect to their separability and distillability properties. This provides sufficient conditions for nonseparability and distillability for arbitrary states.
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 prove that universal quantum computation is possible using only (i) the physically natural measurement on two qubits which distinguishes the singlet from the triplet subspace, and (ii) qubits prepared in almost any three different (potentially highly mixed) states. In some sense this measurement is a `more universal' dynamical element than a universal 2-qubit unitary gate, since the latter must be supplemented by measurement. Because of the rotational invariance of the measurement used, our scheme is robust to collective decoherence in a manner very different to previous proposals - in effe…
We investigate the quantum correlation in a qutrit-qubit mixed spin chain based on measurement-induced disturbance (MID) [S. Luo, Phys. Rev. A, 77, (2008) 022301]. We also compare MID and thermal entanglement measured by negativity and illustrate their different characteristics.
Two-qubit states occupy a large and relatively unexplored Hilbert space. Such states can be succinctly characterized by their degree of entanglement and purity. In this letter we investigate entangled mixed states and present a class of states that have the maximum amount of entanglement for a given linear entropy.
We analyze a controllable generation of maximally entangled mixed states of a circuit containing two-coupled superconducting charge qubits. Each qubit is based on a Cooper pair box connected to a reservoir electrode through a Josephson junction. Illustrative variational calculations were performed to demonstrate the effect on the two-qubits entanglement. At sufficiently deviation between the Josephson energies of the qubits and/or strong coupling regime, maximally entangled mixed states at certain instances of time is synthesized. We show that entanglement has an interesting subsequent time ev…
We experimentally demonstrate optimal entanglement distillation from two forms of two-qubit mixed states under local filtering operations according to the constructive method intruduced by F. Verstraete et al. [Phys. Rev. A 64, 010101(R) (2001)]. In principle, our set-up can be easily applied to distilling entanglement from arbitrary two-qubit partially mixed states. We also test the violation of the Clauser-Horne-Shinmony-Holt (CHSH) inequality for the distilled state from the first form of mixed state to show its "hidden non-locality".
We consider the mixed states of the bipartite quantum system with the first party a qubit and the second a qutrit. The group of local unitary transformations of the system, ignoring the overall phase factor, is the direct product G of SU(2) and SU(3). We compute the simply graded Poincare series of the algebra of G-invariant polynomial functions on the set of mixed states of the system, and construct several low degree invariants.
We propose novel mixed states in two qubits, ``maximally entangled mixed states'', which have a property that the amount of entanglement of these states cannot be increased further by applying any unitary operations. The property is proven when the rank of the states is less than 4, and confirmed numerically in the other general cases. The corresponding entanglement of formation specified by its eigenvalues gives an upper bound of that for density matrices with same eigenvalues.
The paper [Phys. Lett. A 373 (2009) 3610] by D.-C. Li analyzes the transformation between two-qubit mixed states by local operations and classical communication. We show that the proof of the main theorem, Theorem 2.6 in [Phys. Lett. A 373 (2009) 3610] is not complete. Therefore the generalization of Nielsen's theorem to mixed states still remains an open problem.
We investigate the entanglement, CHSH nonlocality, fully entangled fraction and symmetric extendibility of two-qubit states that have a single maximally mixed marginal. Within this set of states, the steering ellipsoid formalism has recently highlighted an interesting family of so-called 'maximally obese' states. These are found to have extremal quantum correlation properties that are significant in the steering ellipsoid picture and for the study of two-qubit states in general.
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…
A simple scheme to prepare an entanglement between two separated qubits from a given mixed state is proposed. A single qubit (entanglement mediator) is repeatedly made to interact locally and consecutively with the two qubits through rotating-wave couplings and is then measured. It is shown that we need to repeat this kind of process only three times to establish a maximally entangled state directly from an arbitrary initial mixed state, with no need to prepare the state of the qubits in advance or to rearrange the setup step by step. Furthermore, the maximum yield realizable with this scheme …
Most quantum error correcting codes are predicated on the assumption that there exists a reservoir of qubits in the state $\ket{0}$, which can be used as ancilla qubits to prepare multi-qubit logical states. In this report, we examine the consequences of relaxing this assumption, and propose a method to increase the fidelity produced by a given code when the ancilla qubits are initialized in mixed states, using the same number of qubits, at most doubling the number of gates. The procedure implemented consists of altering the encoding operator to include the inverse of the unitary operation use…
We describe a general framework to study covariant symmetric broadcasting maps for mixed qubit states. We explicitly derive the optimal N to M superbroadcasting maps, achieving optimal purification of the single-site output copy, in both the universal and the phase covariant cases. We also study the bipartite entanglement properties of the superbroadcast states.
Interaction with environment may lead to the transition of quantum system from pure state to the mixed one. In this case, the problem of definition of entanglement may arise. In particular, quantitative measure of entanglement concurrence is introduced differently for pure and mixed states. To solve this problem, alternative definition of concurrence of mixed states is introduced in present work. In particular, concurrence of mixed states is determined as an average of pure states concurrences. Averaging is carried out over the quantum ensemble, corresponding to all possible realization of ran…
We consider a single copy of a mixed state of two qubits and show how its fidelity or maximal singlet fraction is related to the entanglement measures concurrence and negativity. We characterize the extreme points of the convex set of states with constant fidelity, and use this to prove tight lower and upper bounds on the fidelity for a given amount of entanglement.
The problem of discriminating with minimum error between two mixed quantum states is reviewed, with emphasize on the detection operators necessary for performing the measurement. An analytical result is derived for the minimum probability of errors in deciding whether the state of a quantum system is either a given pure state or a uniform statistical mixture of any number of mutually orthogonal states. The result is applied to two-qubit states, and the minimum error probabilities achievable by collective and local measurements on the qubits are compared.
In this note we consider the problem of preparing a {\em single} copy of an arbitrary two-qubit mixed state $ρ$ starting from an entangled pure state $ψ$ and using only local operations assisted with classical communication. We present an analytical expression for the minimal amount of pure state entanglement required, and describe the corresponding local strategy. We also examine optimal probabilistic generalizations of the previous process.
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…
The bipartite quantum teleportation with a three-qubit mixture of GHZ and W states as a quantum channel is discussed. When the quantum channel is a mixed state consisting of the GHZ state plus small perturbed W state, the entanglement of the channel becomes zero when the average fidelity $\bar{F}_{GHZ}$ is less than 0.775. This means that the mixed state cannot play an any role as a quantum channel at $\bar{F}_{GHZ} \leq 0.775$. For the case of the mixed state consisting of the W state plus small perturbed GHZ state it turn out that this mixed state cannot play a role as a quantum channel at $…
We consider quantum computing in the k-qubit model where the starting state of a quantum computer consists of k qubits in a pure state and n-k qubits in a maximally mixed state. We ask the following question: is there a general method for simulating an arbitrary m-qubit pure state quantum computation by a quantum computation in the k-qubit model? We show that, under certain constraints, this is impossible, unless m=O(k+ log n).
Given a multipartite quantum system that consists of two-level particles (qubits), one may or may not have access to all the subsystems. What can we know about the entanglement of the multiqubit system and residual correlations beyond two-tangle if we have access only to two-qubits at a time? Algebraic analysis of two-qubit states yields monogamy constraints on distribution of entanglement between sub-systems of an N-qubit state and criterion to determine if the state has multipartite entanglement. Monogamy constraints, reported in this letter, are relations between well known entanglement mea…
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 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 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…
In this article, by treating minimum error state discrimination as a complementarity problem, we obtain the geometric optimality conditions. These can be used as the necessary and sufficient conditions to determine whether every optimal measurement operator can be nonzero. Using these conditions and an inductive approach, we demonstrate a geometric method and the intrinsic polytope for N-qubit mixed-state discrimination. When the intrinsic polytope becomes a point, a line segment, or a triangle, the guessing probability, the necessary and sufficient condition for the exact solution, and the op…
Non-classical joint measurements can hugely improve the efficiency with which certain figures of merit of quantum systems are measured. We use such a measurement to determine a particular figure of merit, the purity, for a polarization qubit. In the process we highlight some of subtleties involved in common methods for generating decoherence in quantum optics.
We consider the problems of maximizing the entanglement negativity of X-form qubit-qutrit density matrices with (i) a fixed spectrum and (ii) a fixed purity. In the first case, the problem is solved in full generality whereas, in the latter, partial solutions are obtained by imposing extra spectral constraints such as rank-deficiency and degeneracy, which enable a semidefinite programming treatment for the optimization problem at hand. Despite the technically-motivated assumptions, we provide strong numerical evidence that three-fold degenerate X states of purity $P$ reach the highest entangle…
Quantum secret sharing is one of the most important and interesting quantum information processing task. In quantum secret sharing, information is split among several parties such that only one of them is able to recover the qubit exactly provided all the other parties agree to cooperate. To achieve this task, all the parties need to share entangled state. As far as my knowledge, all the previous quantum secret sharing protocol used either pure tripartite or pure bipartite entangled state. In this work we use for the first time bipartite two qubit mixed state (formed due to noisy environment) …
We present computable criterion for completely classifying multi-qubit quantum states under local unitary operations. The criterion can be used to detect whether two quantum states in multi-qubit systems are local unitary equivalent or not. Once obtaining the positive answer, we are further able to compute the corresponding unitary operators precisely. Since the scheme is based on the mean values of some quantum mechanical observables, it supplies an experimental way to judge the local equivalence of quantum states.
We present a new tripartite entanglement measure for three-qubit mixed states. The new measure $t_{\mathrm{r}}(ρ)$, which we refer to as the r-tangle, is given as a kind of the tangle, but has a feature which the tangle does not have; if we can derive an analytical form of $ t_{\mathrm{r}}(ρ)$ for a three-qubit mixed state $ρ$, we can also derive $t_{\mathrm{r}}(ρ')$ analytically for any states $ρ'$ which are SLOCC-equivalent to the state $ρ$. The concurrence of two-qubit states also satisfies the feature, but the tangle does not. These facts imply that the r-tangle $t_{\mathrm{r}}$ is the app…
We show that a highly-mixed state in terms of a large min-entropy is useless as a resource state for measurement-based quantum computation in the sense that if a classically efficiently verifiable problem is efficiently solved with such a highly-mixed measurement-based quantum computation then such a problem can also be classically efficiently solved. We derive a similar result also for the DQC1$_k$ model, which is a generalized version of the DQC1 model where $k$ output qubits are measured. We also show that the measurement-based quantum computing on a highly-mixed resource state in terms of …
The optimal entanglement manipulation for a single copy of mixed states of two qubits is to transform it to a Bell diagonal state. In this paper we derive an explicit form of the local operation that can realize such a transformation. The result obtained is universal for arbitrary entangled two-qubit states and it discloses that the corresponding local filter is not unique for density matrices with rank $n=2$ and can be exclusively determined for that with $n=3$ and 4. As illustrations, a four-parameters family of mixed states are explored, the local filter as well as the transformation probab…
The N to M (M>N) universal quantum broadcasting of mixed states are proposed for qubits system. The broadcasting of mixed states is universal and optimal in the sense that the shrinking factor is independent of input state and achieves the upper bound. The quantum broadcasting of mixed qubits is a generalization of the universal quantum cloning machine for identical pure input states. A new pure state decompositions of the identical mixed qubits are obtained.
We study the entanglement between a certain qubit and the remaining system in rank- 2 mixed states prepared on the quantum computer. The protocol, which we propose for this purpose, is based on the relation of geometric measure of entanglement with correlations between qubits. As a special case, we consider a two-qubit rank-2 mixed state and find the relation of concurrence with the geometric measure of entanglement. On the ibmq-melbourne quantum computer we measure the geometric measure of entanglement in the cases of 2- and 4-qubit mixed quantum states which consist of Schrödinger cat states…
The quantum probability distribution arising from single-copy von Neumann measurements on an arbitrary two-qubit state is decomposed into the local and nonlocal parts, in the approach of Elitzur, Popescu and Rohrlich [A. Elitzur, S. Popescu, and D. Rohrlich, Phys. Lett. A 162, 25 (1992)]. A lower bound of the local weight is proved being connected with the concurrence of the state $p_L^{\max}= 1-\mathcal{C}(ρ)$. The local probability distributions for two families of mixed states are constructed independently, which accord with the lower bound.
The Uhlmann-Jozsa fidelity (or, equivalently, the Bures distance) is a basic concept of quantum communication and quantum information, which however is very difficult to measure efficiently without recourse to quantum tomography. Here we propose a direct experimental method to estimate the fidelity between two unknown two-qubit mixed states via the measurement of the upper and lower bounds of the fidelity, which are referred to as the superfidelity and subfidelity, respectively. Our method enables a direct measurement of the first- and second-order overlaps between two arbitrary two-qubit stat…
To explore the properties of a two-qubit mixed state, we consider quantum teleportation. The fidelity of a teleported state depends on the resource state purity and entanglement, as characterized by concurrence. Concurrence and purity are functions of state parameters. However, it turns out that a state with larger purity and concurrence, may have comparatively smaller fidelity. By computing teleportation fidelity, concurrence and purity for two-qubit X-states, we show it explicitly. We further show that fidelity changes monotonically with respect to functions of parameters - other than concur…
Recently, an explicit protocol ${\cal E}_0$ for faithfully teleporting arbitrary two-qubit states using genuine four-qubit entangled states was presented by us [Phys. Rev. Lett. {\bf 96}, 060502 (2006)]. Here, we show that ${\cal E}_0$ with an arbitrary four-qubit mixed state resource $Ξ$ is equivalent to a generalized depolarizing bichannel with probabilities given by the maximally entangled components of the resource. These are defined in terms of our four-qubit entangled states. We define the generalized singlet fraction ${\cal G}[Ξ]$, and illustrate its physical significance with several e…
In this paper we extend our recent results for quantum teleportation of single qubit pure information state using non-maximally entangled pure state resource [Quantum Inf Process, 11 (2012) 1251] to the case of a general mixed information state and mixed Werner states resource. Fidelity is found dependent on information state in general. We minimize it over all possible information states and maximize it over possible unitary transformations which can be done by the receiver to obtain a value dependent on purity of information state and mixing parameter of Werner states. For pure information s…
Leakage errors take qubits out of the computational subspace and will accumulate if not addressed. A leaked qubit will reduce the effectiveness of quantum error correction protocols due to the cost of implementing leakage reduction circuits and the harm caused by interacting leaked states with qubit states. Ion trap qubits driven by Raman gates have a natural choice between qubits encoded in magnetically insensitive hyperfine states that can leak and qubits encoded in magnetically sensitive Zeeman states of the electron spin that cannot leak. In our previous work, we compared these two qubits …
Characterization of the multipartite mixed state entanglement is still a challenging problem. Since due to the fact that the entanglement for the mixed states, in general, is defined by a convex-roof extension. That is the entanglement measure of a mixed state \r{ho} of a quantum system can be defined as the minimum average entanglement of an ensemble of pure states. In this Letter, we give an explicit formula for the polynomial entanglement measures of degree 2 of even-N qubits mixed states that is similar to Wooters formula in [1]. Then we discuss our findings in the framework of X density m…
We analyze the multipartite entanglement evolution of three-qubit mixed states composed of a GHZ state and a W state. For a composite system consisting of three cavities interacting with independent reservoirs, it is shown that the entanglement evolution is restricted by a set of monogamy relations. Furthermore, as quantified by the negativity, the entanglement dynamical property of the mixed entangled state of cavity photons is investigated. It is found that the three cavity photons can exhibit the phenomenon of entanglement sudden death (ESD). However, compared with the evolution of a genera…
