In this article, we study the problem of comparing mixed quantum states: given $n$ unknown mixed quantum states, can one determine whether they are identical or not with an unambiguous quantum measurement? We first study universal comparison of mixed quantum states, and prove that this task is generally impossible to accomplish. Then, we focus on unambiguous comparison of $n$ mixed quantum states arbitrarily chosen from a set of $k$ mixed quantum states. The condition for the existence of an unambiguous measurement operator which can produce a conclusive result when the unknown states are actually the same and the condition for the existence of an unambiguous measurement operator when the unknown states are actually different are studied independently. We derive a necessary and sufficient condition for the existence of the first measurement operator, and a necessary condition and two sufficient conditions for the second. Furthermore, we find that the sufficiency of the necessary condition for the second measurement operator has simple and interesting dependence on $n$ and $k$. At the end, a unified condition is obtained for the simultaneous existence of these two unambiguous measurement operators.
We investigate the correlation properties of separable two qubit states with maximally mixed marginals. These states are divided to two sets with the same geometric quantum correlation. However a closer scrutiny of these states reveals a profound difference between their quantum correlations as measured by more probing measures. Although these two sets of states are prepared by the same type of quantum operations acting on classically correlated states with equal classical correlations, the amount of final quantum correlation is different. We investigate this difference and trace it back to th…
Amplitude damping changes entangled pure states into usually less-entangled mixed states. We show, however, that even local amplitude damping of one or two qubits can result in mixed states more entangled than pure states if one compares the relative entropy of entanglement (REE) for a given degree of the Bell-Clauser-Horne-Shimony-Holt inequality violation (referred to as nonlocality). By applying Monte-Carlo simulations, we find the maximally entangled mixed states and show that they are likely to be optimal by checking the Karush-Kuhn-Tucker conditions, which generalize the method of Lagran…
We discuss the "partial" quantum cloning of the pure two-partite states, when the "part" of initial state related to the one qubit is copied only. The same approach gives the possibility to design the quantum copying machine for the mixed qubit states.
Usually models for quantum computations deal with unitary gates on pure states. In this paper we generalize the usual model. We consider a model of quantum computations in which the state is an operator of density matrix and the gates are quantum operations, not necessarily unitary. A mixed state (operator of density matrix) of n two-level quantum systems is considered as an element of $4^{n}$-dimensional operator Hilbert space. Unitary quantum gates and nonunitary quantum operations for n-qubit system are considered as generalized quantum gates acting on mixed state. In this paper we study un…
We propose generalizations of concurrence for multi-partite quantum systems that can distinguish qualitatively distinct quantum correlations. All introduced quantities can be evaluated efficiently for arbitrary mixed sates.
Quantum coherence is a key element in topical research on quantum resource theories and a primary facilitator for design and implementation of quantum technologies. However, the resourcefulness of quantum coherence is severely restricted by environmental noise, which is indicated by the loss of information in a quantum system, measured in terms of its purity. In this work, we derive the limits imposed by the mixedness of a quantum system on the amount of quantum coherence that it can possess. We obtain an analytical trade-off between the two quantities that upperbound the maximum quantum coher…
We derive a lower bound for the concurrence of mixed bipartite quantum states, valid in arbitrary dimensions. As a corollary, a weaker, purely algebraic estimate is found, which detects mixed entangled states with positive partial transpose.
In the framework of quantum computation with mixed states, a fuzzy representation of CNOT gate is introduced. In this representation, the incidence of non-factorizability is specially investigated.
In this paper we investigate the efficiency of quantum cloning of $N$ identical mixed qubits. We employ a recently introduced measure of distinguishability of quantum states called quantum Chernoff bound. We evaluate the quantum Chernoff bound between the output clones generated by the cloning machine and the initial mixed qubit state. Our analysis is illustrated by performing numerical calculation of the quantum Chernoff bound for different scenarios that involves the number of initial qubits $N$ and the number of output imperfect copies $M$.
We discuss the problem of designing an unambiguous programmable discriminator for mixed quantum states. We prove that there does not exist such a universal unambiguous programmable discriminator for mixed quantum states that has two program registers and one data register. However, we find that we can use the idea of programmable discriminator to unambiguously discriminate mixed quantum states. The research shows that by using such an idea, when the success probability for discrimination reaches the upper bound, the success probability is better than what we do not use the idea to do, except f…
Three-qubit mixed states are used as a channel for controlled quantum teleportation (CQT) of single-qubit states. The connection between different channel parameters to achieve maximum controlled teleportation fidelity is investigated. We show that for a given multipartite entanglement and mixedness, a class of non-maximally entangled mixed $X$ states ($X-$NMEMS) achieves optimum controlled quantum teleportation fidelity, interestingly a class of maximally entangled mixed $X$ states ($X-$MEMS) fails to do so. This demonstrates, for a given spectrum and mixedness, that $X-$MEMS are not sufficie…
We consider the apparatus in a quantum measurement process to be in a mixed state. We propose a simple upper bound on the probability of correctly distinguishing any number of mixed states. We use this to derive fundamental bounds on the efficiency of a measurement in terms of the temperature of the apparatus.
In this paper we introduce a geometric framework for mixed quantum states based on a Kähler structure. The geometric framework includes a symplectic form, an almost complex structure, and a Riemannian metric that characterize the space of mixed quantum states. We argue that the almost complex structure is integrable. We also in detail discuss a visualizing application of this geometric framework by deriving a geometric uncertainty relation for mixed quantum states. The framework is computationally effective and it provides us with a better understanding of general quantum mechanical systems.
We consider the problem of optimal asymptotically faithful compression for ensembles of mixed quantum states. Although the optimal rate is unknown, we prove upper and lower bounds and describe a series of illustrative examples of compression of mixed states. We also discuss a classical analogue of the problem.
We discuss single adaptive measurements for the estimation of mixed quantum states of qubits. The results are compared to the optimal estimation schemes using collective measurements. We also demonstrate that the advantage of collective measurements increases when the degree of mixing of the quantum states increases.
We discuss a model for quantum computing with initially mixed states. Although such a computer is known to be less powerful than a quantum computer operating with pure (entangled) states, it may efficiently solve some problems for which no efficient classical algorithms are known. We suggest a new implementation of quantum computation with initially mixed states in which an algorithm realization is achieved by means of optimal basis independent transformations of qubits.
Entanglement and mixedness of a bipartite mixed state resource are crucial for the success of quantum teleportation. Upper bounds on measures of mixedness, namely, von Neumann entropy and linear entropy beyond which the bipartite state ceases to be useful for quantum teleportation are known in the literature. In this work, we generalize these bounds and obtain rank dependent upper bounds on von Nuemann entropy and linear entropy for an arbitrary bipartite mixed state resource. We observe that the upper bounds on measures of mixedness increase with increase in the rank. For two qubit mixed stat…
Quantum cloning machine for arbitrary mixed states in symmetric subspace is proposed. This quantum cloning machine can be used to copy part of the output state of another quantum cloning machine and is useful in quantum computation and quantum information. The shrinking factor of this quantum cloning achieves the well-known upper bound. When the input is identical pure states, two different fidelities of this cloning machine are optimal.
We prove that the states secretly chosen from a mixed state set can be perfectly discriminated if and only if these states are orthogonal. The sufficient and necessary condition when nonorthogonal quantum mixed states can be unambiguously discriminated is also presented. Furthermore, we derive a series of lower bounds on the inconclusive probability of unambiguous discrimination of states from a mixed state set with \textit{a prior} probabilities.
We provide a solution of finding optimal measurement strategy for distinguishing between symmetric mixed quantum states. It is assumed that the matrix elements of at least one of the symmetric quantum states are all real and nonnegative in the basis of the eigenstates of the symmetry operator.
This paper is intended to be a pedagogical introduction to quantum Bayesian networks (QB nets), as I personally use them to represent mixed states (i.e., density matrices, and open quantum systems). A special effort is made to make contact with notions used in textbooks on quantum Shannon Information Theory (quantum SIT), such as the one by Mark Wilde (arXiv:1106.1445)
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.
We investigate the equivalence of quantum mixed states under local unitary transformations. For a class of rank-two mixed states, a sufficient and necessary condition of local equivalence is obtained by giving a complete set of invariants under local unitary transformations, such that two states in this class are locally equivalent if and only if all these invariants have equal values for them.
We extend a recent method to detect lower bounds to the quantum capacity of quantum communication channels by considering realistic scenarios with general input probe states and arbitrary detection procedures at the output. Realistic certification relies on a new bound for the coherent information of a quantum channel that can be applied with arbitrary bipartite mixed input states and generalized output measurements.
We present the conditions under which probabilistic error-free discrimination of mixed states is possible, and provide upper and lower bounds on the maximum probability of success for the case of two mixed states. We solve certain special cases exactly, and demonstrate how the problems of state filtering and state comparison can be recast as problems of mixed state unambiguous discrimination.
We describe a purely algebraic method for finding the best separable approximation to a mixed state of a composite 2x2 quantum system, consisting of a decomposition of the state into a linear combination of a mixed separable part and a pure entangled one. We prove that, in a generic case, the weight of the pure part in the decomposition equals the concurrence of the state.
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.
Geometric phase has been proposed as one of the promising methodologies to perform fault tolerant quantum computations. However, since decoherence plays a crucial role in such studies, understanding of mixed state geometric phase has become important. While mixed state geometric phase was first introduced mathematically by Uhlmann, recently Sjoqvist et al. [Phys. Rev. Lett. 85(14), 2845 (2000)] have described the mixed state geometric phase in the context of quantum interference and shown theoretically that the visibility as well as the shift of the interference pattern are functions of geomet…
We analyze the effects of quantum correlations, such as entanglement and discord, on the efficiency of phase estimation by studying four quantum circuits that can be readily implemented using NMR techniques. These circuits define a standard strategy of repeated single-qubit measurements, a classical strategy where only classical correlations are allowed, and two quantum strategies where nonclassical correlations are allowed. In addition to counting space (number of qubits) and time (number of gates) requirements, we introduce mixedness as a key constraint of the experiment. We compare the effi…
In this paper, we consider Wigner-Yanase-Dyson information as a measure of quantum uncertainty of a mixed state. We study some of the interesting properties of this generalized measure. The construction is reminiscent of the generalized entropies that have shown to be useful in many applications.
We consider the problem of designing a measurement to minimize the probability of a detection error when distinguishing between a collection of possibly non-orthogonal mixed quantum states. We show that if the quantum state ensemble consists of linearly independent density operators then the optimal measurement is an orthogonal Von Neumann measurement consisting of mutually orthogonal projection operators and not a more general positive operator-valued measure.
Information-theoretic aspects of quantum inseparability of mixed states are investigated in terms of the $α$-entropy inequalities and teleportation fidelity. Inseparability of mixed states is defined and a complete characterization of the inseparable $2\times2$ systems with maximally disordered subsystems is presented within the Hilbert-Schmidt space formalism. A connection between teleportation and negative conditional $α$-entropy is also emphasized.
In the theory of quantum information, the mixed-unitary quantum channels, for any positive integer dimension $n$, are those linear maps that can be expressed as a convex combination of conjugations by $n\times n$ complex unitary matrices. We consider the mixed-unitary rank of any such channel, which is the minimum number of distinct unitary conjugations required for an expression of this form. We identify several new relationships between the mixed-unitary rank~$N$ and the Choi rank~$r$ of mixed-unitary channels, the Choi rank being equal to the minimum number of nonzero terms required for a K…
Entangled pure-states, Werner-states and generalized mixed-states of any structure, spanning a 2x2 Hilbert space are created by a novel high-brilliance universal source of polarization-entangled photon pairs. The violation of a Bell inequality has been tested for the first time with a pure-state, indeed a conceptually relevant ideal condition, and with Werner-states. The generalized ''maximally entangled mixed states'' (MEMS) were also synthetized for the first time and their exotic properties investigated by means of a quantum tomographic technique.
Maximally entangled mixed states are those states that, for a given mixedness, achieve the greatest possible entanglement. For two-qubit systems and for various combinations of entanglement and mixedness measures, the form of the corresponding maximally entangled mixed states is determined primarily analytically. As measures of entanglement, we consider entanglement of formation, relative entropy of entanglement, and negativity; as measures of mixedness, we consider linear and von Neumann entropies. We show that the forms of the maximally entangled mixed states can vary with the combination of…
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…
It is often assumed that the ancilla qubits required for encoding a qubit in quantum error correction (QEC) have to be in pure states, $|00...0>$ for example. In this letter, we seek an encoding scheme, in which the ancillae may be in a uniformly mixed state. We demonstrate our scheme experimentally by making use of a three-qubit NMR quantum computer. Moreover, the encoded state has an interesting nature in terms of Quantum Discord, or purely quantum correlations between the data-qubit and ancillae.
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…
We derive a general lower bound on the minimum-error probability for {\it ambiguous discrimination} between arbitrary $m$ mixed quantum states with given prior probabilities. When $m=2$, this bound is precisely the well-known Helstrom limit. Also, we give a general lower bound on the minimum-error probability for discriminating quantum operations. Then we further analyze how this lower bound is attainable for ambiguous discrimination of mixed quantum states by presenting necessary and sufficient conditions related to it. Furthermore, with a restricted condition, we work out a upper bound on th…
We report on an experiment to detect non-classical correlations in a highly mixed state. The correlations are characterized by the quantum discord and are observed using four qubits in a liquid state nuclear magnetic resonance quantum information processor. The state analyzed is the output of a DQC1 computation, whose input is a single quantum bit accompanied by n maximally mixed qubits. This model of computation outperforms the best known classical algorithms, and although it contains vanishing entanglement it is known to have quantum correlations characterized by the quantum discord. This ex…
Using an algebraic framework we solve a problem posed in [5] and [7] about the axiomatizability of a quantum computational type logic related to fuzzy logic. A Hilbert-style calculus is developed obtaining an algebraic strong completeness theorem.
Teleportation for pure states, mixed states with standard and optimal protocols are introduced and investigated systematically. An explicit equation governing the teleportation of finite dimensional quantum pure states by a generally given non-local entangled state is presented. For the teleportation of a mixed state with an arbitrary mixed state resource, an explicit expression is obtained for the quantum channel associated with the standard teleportation protocol. The corresponding transmission fidelity is calculated. It is shown that the standard teleportation protocol is not optimal. The o…
We investigate the unambiguous comparison of quantum states in a scenario that is more general than the one that was originally suggested by Barnett et al. First, we find the optimal solution for the comparison of two states taken from a set of two pure states with arbitrary a priori probabilities. We show that the optimal coherent measurement is always superior to the optimal incoherent measurement. Second, we develop a strategy for the comparison of two states from a set of N pure states, and find an optimal solution for some parameter range when N=3. In both cases we use the reduction metho…
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 …
In this paper, we introduce a mixed form of ambiguous and unambiguous quantum state discriminations, and show that the mixed form has higher success probability than the unambiguous quantum state discriminations.
Spin chains have been proposed as quantum wires in many quantum information processing architectures. Coherent transmission of quantum information over short distances is enabled by their internal dynamics, which drives the transport of single-spin excitations in perfectly polarized chains. Given the practical challenge of preparing the chain in a pure state, we propose to use a chain that is initially in the maximally mixed state. We compare the transport properties of pure and mixed-state chains, finding similarities that enable the experimental study of pure-state transfer by its simulation…
The symplectic wavelet transformation [Opt. Lett. 31 (2006) 3432], which is related to quantum optical Fresnel transform, is developed to the symplectic-dilation mixed wavelet transform (SDWT). The SDWT involves both a real-variable dilation-transform and a complex-variable symplectic transform, and possesses well-behaved properties such as Parseval theorem and inversion formula. The entangled-coherent state representation (ESCR) not only underlies the SDWT, but also helps to derive the corresponding quantum transform operator whose counterpart in classical optics is the lens-Fresnel mixed tra…
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) …
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 $…
Bounds analogous to entropic uncertainty relations allow one to design practical tests to detect quantum entanglement by a collective measurement performed on several copies of the state analyzed. This approach, initially worked out for pure states only [Phys. Rev. Lett. 107, 150502 (2011)], is extended here for mixed quantum states. We define collectibility for any mixed states of a multipartite system. Deriving bounds for collectibility for positive partially transposed states of given purity provides a new insight into the structure of entangled quantum states. In case of two qubits the app…
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 physical resources required to implement general quantum operations, and provide new bounds on the minimum possible size which an environment must be in order to perform certain quantum operations. We prove that contrary to a previous conjecture, not all quantum operations on a single-qubit can be implemented with a single-qubit environment, even if that environment is initially prepared in a mixed state. We show that a mixed single-qutrit environment is sufficient to implement a special class of operations, the generalized depolarizing channels.
