On Duality between Quantum Maps and Quantum States

We investigate the space of quantum operations, as well as the larger space of maps which are positive, but not completely positive. A constructive criterion for decomposability is presented. A certain class of unistochastic operations, determined by unitary matrices of extended dimensionality, is defined and analyzed. Using the concept of the dynamical matrix and the Jamiokowski isomorphism we explore the relation between the set of quantum operations (dynamics) and the set of density matrices acting on an extended Hilbert space (kinematics). An analogous relation is established between the classical maps and an extended space of the discrete probability distributions.     

Informational correlation between two parties of a quantum system: spin-1/2 chains

We introduce the informational correlation \(E^{AB}\) between two interacting quantum subsystems \(A\) and \(B\) of a quantum system as the number of arbitrary parameters \(\varphi _i\) of a unitary transformation \(U^A\) (locally performed on the subsystem \(A\) ) which may be detected in the subsystem \(B\) by the local measurements. This quantity indicates whether the state of the subsystem \(B\) may be effected by means of the unitary transformation applied to the subsystem \(A\) . Emphasize that \(E^{AB}\ne E^{BA}\) in general. The informational correlations in systems with tensor product initial states are studied in more details. In particular, it is shown that the informational correlation may be changed by the local unitary transformations of the subsystem \(B\) . However, there is some non-reducible part of \(E^{AB}(t)\) which may not be decreased by any unitary transformation of the subsystem \(B\) at a fixed time instant \(t\) . Two examples of the informational correlations between two parties of the four-node spin-1/2 chain with mixed initial states are studied. The long chains with a single initially excited spin (the pure initial state) are considered as well.     

Efficient joint remote preparation of an arbitrary two-qubit state via cluster and cluster-type states

In this paper, we present a possible improvement of the successful probability of joint remote state preparation via cluster states following some ideals from probabilistic joint remote state preparation (Wang et al. in Opt Commun, 284:5835, 2011). The success probability can be improved from $1/4$ to 1 via the same quantum entangled channel by adding some classical information and performing some unitary operations. Moreover, we also discussed the scheme for joint remote preparation via cluster-type states. Compared with other schemes, our schemes have the advantage of having high successful probability for joint preparation of an arbitrary two-qubit state via cluster states and cluster-type states.     

A model of discrete quantum computation

We present a model of discrete quantum computing focused on a set of discrete quantum states. For this, we choose the set that is the most outstanding in terms of simplicity of the states: the set of Gaussian coordinate states, which includes all the quantum states whose coordinates in the computation base, except for a normalization factor \(\sqrt{2^{-k}}\), belong to the ring of Gaussian integers \(\mathbb {Z}[i]=\{a+bi\ |\ a,b\in \mathbb {Z}\}\). We also introduce a finite set of quantum gates that transforms discrete states into discrete states and generates all discrete quantum states, and the set of discrete quantum gates, as the quantum gates that leave the set of discrete states invariant. We prove that the quantum gates of the model generate the expected discrete states and the discrete quantum gates of 2-qubits and conjecture that they also generate the discrete quantum gates of n-qubits.     

Quantum Fisher information matrix for unitary processes: closed relation for <Emphasis Type="Italic">SU</Emphasis>(2)

Quantum Fisher information plays a central role in the field of quantum metrology. In this paper, we study the problem of quantum Fisher information of unitary processes. Associated with each parameter \(\theta _i\) of unitary process \(U(\varvec{\theta })\), there exists a unique Hermitian matrix \(M_{\theta _i}=i(U^\dagger \partial _{\theta _i} U)\). Except for some simple cases, such as when the parameter under estimation is an overall multiplicative factor in the Hamiltonian, calculation of these matrices is not an easy task to treat even for estimating a single parameter of qubit systems. Using the Bloch vector \(\varvec{m}_{\theta _i}\), corresponding to each matrix \(M_{\theta _i}\), we find a closed relation for the quantum Fisher information matrix of the SU(2) processes for an arbitrary number of estimation parameters and an arbitrary initial state. We extend our results and present an explicit relation for each vector \(\varvec{m}_{\theta _i}\) for a general Hamiltonian with arbitrary parametrization. We illustrate our results by obtaining the quantum Fisher information matrix of the so-called angle-axis parameters of a general SU(2) process. Using a linear transformation between two different parameter spaces of a unitary process, we provide a way to move from quantum Fisher information of a unitary process in a given parametrization to the one of the other parametrizations. Knowing this linear transformation enables one to calculate the quantum Fisher information of a composite unitary process, i.e., a unitary process resulted from successive action of some simple unitary processes. We apply this method for a spin-half system and obtain the quantum Fisher matrix of the coset parameters in terms of the one of the angle-axis parameters.     

Coherence of one-dimensional quantum walk on cycles

Quantum coherence plays a central role in quantum mechanics and provides essential power for quantum information processing. In this paper, we study the dynamics of the \(l_1\) norm coherence in one-dimensional quantum walk on cycles for two initial states. For the first initial state, the walker starts from a single position. The coherence increases with the number of steps at the beginning and then fluctuates over time after approaching to saturation. The coherence with odd number of sites is much larger than that with even number of sites. Another initial state, i.e., the equally superposition state, is also considered. The coherence of the whole system is proved to be \(N-1\) (\(2N-1\)) for any odd (even) time step where N is the number of sites. We also investigate the influence of two unitary noises, i.e., noisy Hadamard operator and broken link, on the coherence evolution.     

Cyclic joint remote state preparation in noisy environment

We propose a scheme of cyclic joint remote state preparation for three sides, which takes advantage of three GHZ states to compose product state as quantum channel. Suppose there are six legitimate participants, says Alice, Bob, Charlie, David, Emma and Fred in the scheme. It can be shown that Alice and David can remotely prepare a single-qubit state on Bob’s side; meanwhile, Bob and Emma can remotely prepare a desired quantum state on Charlie’s side, and Charlie and Fred can also remotely prepare a single-qubit state on Alice’s side at the same time. Further, it can be achieved in the opposite direction of the cycle by changing the quantum channel. Based on it, we generalize this protocol to \(N (N\ge 3)\) sides utilizing three multi-qubit GHZ-type states as quantum channel. Therefore, the scheme can achieve cyclic joint remote state preparation, which remotely prepares N states in quantum network with N-party, simultaneously. In addition, we consider that the effect of amplitude-damping noise of the initial states is prepared in four different laboratory. Clearly, we use fidelity to describe how much information has been lost in the cyclic process. Our investigation about the effect of noise shows that the preparing of the initial state in different laboratories will affect the loss of information.     

Disentanglement and quantum states transitions dynamics in spin-qutrit systems: dephasing random telegraph noise and the relevance of the initial state

Using negativity and realignment criterion as quantifiers of free and bound entanglements respectively, we present in details the analytical study of the entanglements and quantum states transitions dynamics in a two-qutrit system driven by dephasing random telegraph noise channel(s). Both collective and independent system–environment couplings as well as the Markovian and the non-Markovian regimes of the noise channel(s) are considered. Two non-equivalent initial states and their locally equivalent through a local unitary operation (LUO) are also considered. We demonstrate a stronger entanglement under independent Markovian environments than with a collective one; meanwhile, for the non-Markovian regime, entanglement is stronger under a collective environment than with independent ones. States transitions as well as the (re)activation of bound entanglement (for initially free entangled states) can be found for a specific class of initial states, but can, however, be avoided by means of a LUO on the initial state. While unavoidable disentanglement occurs for independents coupling, we demonstrate the possibility of indefinite free entanglement survival in the qutrit system under a common environment by converting the initial entangled state using the local unitary operation.     

Optimal unambiguous discrimination of pure qudits

Linearly independent pure quantum states can be discriminated unambiguously, while linearly dependent states cannot. We use a physical accessible unitary transformation to map the nonorthogonal quantum states onto a set of orthogonal ones so that measuring the output states can discriminate the initial states with the deterministic and inconclusive results. The failure states that give an inconclusive result are linearly dependent ones. In finding the optimal unambiguous discrimination (UD), we show that a main constraint condition that the determinant constructed by the complex inner products of the failure states must be zero, along with two additional conditions, can provide solutions to the problem of the optimal UD for pure qudits. For any d, we give one analytical solution as all the Berry phases being zero. We also derive the lowest bound of the total failure probability of the optimal UD.     

Steady states of continuous-time open quantum walks

Continuous-time open quantum walks (CTOQW) are introduced as the formulation of quantum dynamical semigroups of trace-preserving and completely positive linear maps (or quantum Markov semigroups) on graphs. We show that a CTOQW always converges to a steady state regardless of the initial state when a graph is connected. When the graph is both connected and regular, it is shown that the steady state is the maximally mixed state. As shown by the examples in this article, the steady states of CTOQW can be very unusual and complicated even though the underlying graphs are simple. The examples demonstrate that the structure of a graph can affect quantum coherence in CTOQW through a long-time run. Precisely, the quantum coherence persists throughout the evolution of the CTOQW when the underlying topology is certain irregular graphs (such as a path or a star as shown in the examples). In contrast, the quantum coherence will eventually vanish from the open quantum system when the underlying topology is a regular graph (such as a cycle).     

The hierarchical control of ST-finite-state machines

This paper follows (Caines and Wei, 1995) where a new notion of state aggregation for finite machines was introduced via the concept of the dynamical consistency (DC) relation between the blocks of states in any given state-space partition Π. This formulation results in a definition of high-level dynamics on the finite (partition) machine whose states correspond to the given partition elements. This paper treats the more general case of ST-systems where there is a preferred sense of flow from a set of source states (S) to a set of target states (T) which is to be achieved by hierarchical control. A generalisation of the theory of Caines and Wei to ST-systems is given which includes the generalisation of the notions of dynamical consistency in block controllability and hierarchical feedback control on the associated hierarchical lattices.     

Quantum image with high retrieval performance

Quantum image retrieval is an exhaustive work due to exponential measurements. Casting aside the background of image processing, quantum image is a pure many-body state, and the retrieval task is a physical process named as quantum state tomography. Tomography of a special class of states, permutationally symmetric states, just needs quadratic measurement scales with the number of qubits. In order to take advantage of this result, we propose a method to map the main energy of the image to these states. First, we deduce that \(n+1\) permutationally symmetric states can be constructed as bases of \(2^n\) Hilbert space (n qubits) at least. Second, we execute Schmidt decomposition by continually bipartite splitting of the quantum image (state). At last, we select \(n+1\) maximum coefficients, do base transformation to map these coefficients to new bases (permutationally symmetric states). By these means, the quantum image with high retrieval performance can be gotten.     

Quantum correlations in a family of bipartite separable qubit states

Quantum correlations (QCs) in some separable states have been proposed as a key resource for certain quantum communication tasks and quantum computational models without entanglement. In this paper, a family of nine-parameter separable states, obtained from arbitrary mixture of two sets of bi-qubit product pure states, is considered. QCs in these separable states are studied analytically or numerically using four QC quantifiers, i.e., measurement-induced disturbance (Luo in Phys Rev A77:022301, 2008), ameliorated MID (Girolami et al. in J Phys A Math Theor 44:352002, 2011),quantum dissonance (DN) (Modi et al. in Phys Rev Lett 104:080501, 2010), and new quantum dissonance (Rulli in Phys Rev A 84:042109, 2011), respectively. First, an inherent symmetry in the concerned separable states is revealed, that is, any nine-parameter separable states concerned in this paper can be transformed to a three-parameter kernel state via some certain local unitary operation. Then, four different QC expressions are concretely derived with the four QC quantifiers. Furthermore, some comparative studies of the QCs are presented, discussed and analyzed, and some distinct features about them are exposed. We find that, in the framework of all the four QC quantifiers, the more mixed the original two pure product states, the bigger QCs the separable states own. Our results reveal some intrinsic features of QCs in separable systems in quantum information.     

Threshold quantum state sharing based on entanglement swapping

A threshold quantum state sharing scheme is proposed. The dealer uses the quantum-controlled-not operations to expand the d-dimensional quantum state and then uses the entanglement swapping to distribute the state to a random subset of participants. The participants use the single-particle measurements and unitary operations to recover the initial quantum state. In our scheme, the dealer can share different quantum states among different subsets of participants simultaneously. So the scheme will be very flexible in practice.     

Analytic expressions of discord and geometric discord in Werner derivatives

Werner derivatives are a special kind of mixing states transformed from Werner states by unitary operations (Hiroshima and Ishizaka in Phys Rev A 62:044302, 2000). In this paper, the inherent quantum correlations in Werner derivatives are quantified by two different quantifiers, i.e., quantum discord and geometric discord. Different analytic expressions of the two discords in Werner derivatives are derived out. Some distinct features of the discords and their underlying physics are exposed via discussions and analyses. Moreover, it is found that the amount of quantum correlations quantified by either quantifier in each derivative cannot exceed that in the original Werner state. In other words, no unitary operation can increase quantum correlation in a Werner state as far as the two quantifiers are concerned.     

Relationship between the field local quadrature and the quantum discord of a photon-added correlated channel under the influence of scattering and phase fluctuation noise

We study quantum correlations and discord in a bipartite continuous variable hybrid system formed by linear combinations of coherent states \(\mathinner {|{\alpha }\rangle }\) and single photon-added coherent states of the form \(\mathinner {|{\psi }\rangle }_{\text {dp(pa)}}= \mathcal {N}/\sqrt{2} (\hat{a}^\dagger \mathinner {|{\alpha }\rangle }_a\mathinner {|{\alpha }\rangle }_b \pm \hat{b}^\dagger \mathinner {|{\alpha }\rangle }_a\mathinner {|{\alpha }\rangle }_b)\). We stablish a relationship between the quantum discord with a local observable (the quadrature variance for one subsystem) under the influence of scattering and phase fluctuation noise. For the pure states the quantum correlations are characterized by means of measurement induced disturbance (MID) with simultaneous quadrature measurements. In a scenario where homodyne conditional measurements are available we show that the MID provides an easy way to select optimal phases to obtain information of the maximal correlations in the channels. The quantum correlations of these entangled states with channel losses are quantitatively characterized with the quantum discord (QD) with a displaced qubit projector. We observe that as scattering increases, QD decreases monotonically. At the same time for the state \(\mathinner {|{\psi }\rangle }_{\text {dp}}\), QD is more resistant to high phase fluctuations when the average photon number \(n_0\) is bigger than zero, but if phase fluctuations are low, QD is more resistant if \(n_0=0\). For the dp model with scattering, we obtain an analytical expression of the QD as a function of the observable quadrature variance in a local subsystem. This relation allows us to have a way to obtain the degree of QD in the channel by just measuring a local property observable such as the quadrature variance. For the other model this relation still exists but is explored numerically. This relation is an important result that allows to identify quantum processing capabilities in terms of just local observables.     

OptQC: An optimized parallel quantum compiler

The software package Qcompiler (Chen and Wang 2013) provides a general quantum compilation framework, which maps any given unitary operation into a quantum circuit consisting of a sequential set of elementary quantum gates. In this paper, we present an extended software OptQC  , which finds permutation matrices P$P$ and Q$Q$ for a given unitary matrix U$U$ such that the number of gates in the quantum circuit of U=Q^TP^TU^′PQ$U=Q^T{P}^T{U}^{\prime }PQ$ is significantly reduced, where U^′$U^{\prime }$ is equivalent to U$U$ up to a permutation and the quantum circuit implementation of each matrix component is considered separately. We extend further this software package to make use of high-performance computers with a multiprocessor architecture using MPI. We demonstrate its effectiveness in reducing the total number of quantum gates required for various unitary operators.     

Robustness of Greenberger$$\textendash $$Horne$$\textendash $$Zeilinger and W states against Dzyaloshinskii-Moriya interaction

In this article, the robustness of tripartite Greenberger–Horne–Zeilinger (GHZ) and W states is investigated against Dzyaloshinskii-Moriya (i.e. DM) interaction. We consider a closed system of three qubits and an environmental qubit. The environmental qubit interacts with any one of the three qubits through DM interaction. The tripartite system is initially prepared in GHZ and W states, respectively. The composite four qubits system evolve with unitary dynamics. We detach the environmental qubit by tracing out from four qubits, and profound impact of DM interaction is studied on the initial entanglement of the system. As a result, we find that the bipartite partitions of W states suffer from entanglement sudden death (i.e. ESD), while tripartite entanglement does not. On the other hand, bipartite partitions and tripartite entanglement in GHZ states do not feel any influence of DM interaction. So, we find that GHZ states have robust character than W states. In this work, we consider generalised GHZ and W states, and three \(\pi \) is used as an entanglement measure. This study can be useful in quantum information processing where unwanted DM interaction takes place.     

On <Emphasis Type="Italic">S</Emphasis>-mixing entropy of quantum channels

In this paper, an S-mixing entropy of quantum channels is introduced as a generalization of Ohya’s S-mixing entropy. We investigate several properties of the introduced entropy. Moreover, certain relations between the S-mixing entropy and the existing map and output entropies of quantum channels are investigated as well. These relations allowed us to find certain connections between separable states and the introduced entropy. Hence, there is a sufficient condition to detect entangled states. Moreover, several properties of the introduced entropy are investigated. Besides, entropies of qubit and phase-damping channels are calculated.     

Sequential generation of polynomial invariants and N-body non-local correlations

We report an inductive process that allows for sequential construction of local unitary invariant polynomials of state coefficients for multipartite quantum states. The starting point can be a physically meaningful invariant of a smaller part of the system. The process is applied to construct a chain of invariants that quantify non-local N-way correlations in an N-qubit pure state. It also yields the invariants to quantify the sum of N-way and (N-1)-way correlations. Analytic expressions for four-way and three-way correlation quantifiers for four-qubit states, as well as, five-way and four-way correlation quantifiers for five-qubit pure states are given.