Computational procedure to determine quantum state evolution in Fock space

A new algorithm to solve the quantum state evolution of a system described by a general quadratic Hamiltonian form in creation and the annihilation operators of Fock space is presented. The nonlinear equation for the dynamic operators are obtained in the matrix representation, and by a recursive relation the time evolution operator in the Fock basis is constructed. The method permits to obtain the evolution of entangled quantum states of interacting subsystems when the Hamiltonian of the whole system is in the above mentioned form. Numerical solution with the method is sufficiently accurate to safely analyze the important question of quantum state transfer between the interacting subsystems. A qubits transfer is discussed as an illustrative example when the method is applied to a system described by a particular quadratic Hamiltonian form.     

Score operators of a qubit with applications

The score operators of a quantum system are the symmetric logarithmic derivatives of the system’s parametrically defined quantum state. Score operators are central to the calculation of the quantum Fisher information (QFI) associated with the state of the system, and the QFI determines the maximum precision with which the state parameters can be estimated. We give a simple, explicit expression for score operators of a qubit and apply this expression in a series of settings. We treat in detail the task of identifying a quantum Pauli channel from the state of its qubit output, and we show that a “balanced” probe state is highly robust for this purpose. The QFI for this task is a matrix, and we study its determinant, for which we establish a Cramér-Rao inequality.     

Protected subspaces in quantum information

In certain situations the state of a quantum system, after transmission through a quantum channel, can be perfectly restored. This can be done by “coding” the state space of the system before transmission into a “protected” part of a larger state space, and by applying a proper “decoding” map afterwards. By a version of the Heisenberg Principle, which we prove, such a protected space must be “dark” in the sense that no information leaks out during the transmission. We explain the role of the Knill–Laflamme condition in relation to protection and darkness, and we analyze several degrees of protection, whether related to error correction, or to state restauration after a measurement. Recent results on higher rank numerical ranges of operators are used to construct examples. In particular, dark spaces are constructed for any map of rank 2, for a biased permutations channel and for certain separable maps acting on multipartite systems. Furthermore, error correction subspaces are provided for a class of tri-unitary noise models.     

d维量子同态加密算法的设计与仿真

现有量子同态加密算法局限于二维或三维的希尔伯特空间,突破这种低维度的限制,基于任意两个[d]维酉算子的可交换性提出了两个[d]维量子同态加密算法。一种是单粒子上的[d]维量子同态加密算法,另一种是多粒子上的[d]维量子同态加密算法。每个算法包括密钥生成子算法、加密子算法、评估子算法和解密子算法,证明了两个量子同态加密算法的正确性并举例予验证算法的可行性。由于评估算法不依赖于客户端的密钥,并且输出态具有完全混合态性质,保证两个算法的安全性。仿真结果显示解密子算法的输出与评估子算法对明文的直接计算结果完全一致,验证了两个算法的同态特性。     

Symmetries, Group Actions, and Entanglement

We address several problems concerning the geometry of the space of Hermitian operators on a finite-dimensional Hilbert space, in particular the geometry of the space of density states and canonical group actions on it. For quantum composite systems we discuss and give examples of entanglement measures.     

Symmetries of fundamental interactions in quantum phase space

Quantum operators of coordinates and momentum components of a particle in Minkowski space-time belong to a noncommutative algebra and give rise to a quantum phase space. Under some constraints, in particular, the Lorentz invariance condition, the algebra of observables, including the Lorentz group generators, depends on additional fundamental physical constants with the dimensions of mass, length and action. Generalized symmetries in a quantum phase space and some consequences for fundamental interactions of particles are considered.     

Cluster态的量子签名方案

提出一种利用Cluster state纠缠态实现的量子签名方案。该方案中用Cluster态作为量子信道,每一组量子比特串分别分发给消息拥有和签名者Alice、公证人TA、验签名者Bob。加载消息的方法是Alice在TA规定量子比特串序列下,分别对拥有的量子比特对的第一个量子比特进行幺正变换操作而进行。对拥有的量子比特对进行的Bell测量结果是消息的签名。Bob对拥有的对应的两个量子比特对进行Bell测量来验证签名,但要得到公证人TA对其约束才能完成。Cluster state纠缠态在纠缠特性、局域操作保真性和安全性有较好的性能。     

量子计算的表象无关性探讨

大因数分解和数据检索量子算法的提出带来了量子计算与量子信息的研究高潮。由于量子计算具有并行性、不可克隆性及量子态的不可测性,使得量子信息及量子计算在某些方面具有传统计算所无法比拟的优势。量子的态空间作为一个完备的Hilbert空间,在定义了内积和范数并赋予相应的物理意义后,就构成了理论意义上的量子计算系统。该文抽象了量子系统的本质,描述了量子计算及遵循的计算规则以及如何实现量子信息表示和进行信息的处理与测量,从理论上阐述了量子态系统迁移的线性同构和等距同构,说明了量子计算与量子信息的研究与具体的量子表象空间无关。     

Coexistence of qubit effects

Two quantum events, represented by positive operators (effects), are coexistent if they can occur as possible outcomes in a single measurement scheme. Equivalently, the corresponding effects are coexistent if and only if they are contained in the ranges of a single (joint) observable. Here we give several equivalent characterizations of coexistent pairs of qubit effects. We also establish the equivalence between our results and those obtained independently by other authors. Our approach makes explicit use of the Minkowski space geometry inherent in the four-dimensional real vector space of selfadjoint operators in a two-dimensional complex Hilbert space.     

Coprime factorization and robust stabilization for discrete-time infinite-dimensional systems

We solve the problem of robust stabilization with respect to right-coprime factor perturbations for irrational discrete-time transfer functions. The key condition is that the associated dynamical system and its dual should satisfy a finite-cost condition so that two optimal cost operators exist. We obtain explicit state space formulas for a robustly stabilizing controller in terms of these optimal cost operators and the generating operators of the realization. Along the way we also obtain state space formulas for Bezout factors.     

Information Gain versus State Disturbance for a Single Qubit

The trade-off between the information gain and the state disturbance is derived for quantum operations on a single qubit prepared in a uniformly distributed pure state. The derivation is valid for a class of measures quantifying the state disturbance and the information gain which satisfy certain invariance conditions. This class includes in particular the Shannon entropy versus the operation fidelity. The central role in the derivation is played by efficient quantum operations, which leave the system in a pure output state for any measurement outcome. It is pointed out that the optimality of efficient quantum operations among those inducing a given operator-valued measure is related to Davies' characterization of convex invariant functions on hermitian operators.     

Geometry of quantum state space and quantum correlations

Quantum state space is endowed with a metric structure, and Riemannian monotone metric is an important geometric entity defined on such a metric space. Riemannian monotone metrics are very useful for information-theoretic and statistical considerations on the quantum state space. In this article, considering the quantum state space being spanned by \(2\times 2\) density matrices, we determine a particular Riemannian metric for a state \(\rho \) and show that if \(\rho \) gets entangled with another quantum state, the negativity of the generated entangled state is, upto a constant factor, equal to square root of that particular Riemannian metric . Our result clearly relates a geometric quantity to a measure of entanglement. Moreover, the result establishes the possibility of understanding quantum correlations through geometric approach.     

Correlations in a General Theory of Quantum Measurement

The paper investigates correlations in a general theory of quantum measurement based on the notion of instrument. The analysis is performed in the algebraic formalism of quantum theory in which the observables of a physical system are described by a von Neumann algebra, and the states—by normal positive normalized functionals on this algebra. The results extend and generalise those obtained for the classical case where one deals with the full algebra of operators on a Hilbert space.     

Generic Quantum Markov Semigroups: the Gaussian Gauge Invariant Case

We study a class of generic quantum Markov semigroups on the algebra of all bounded operators on a Hilbert space arising from the stochastic limit of a discrete system with generic Hamiltonian H _S , acting on , interacting with a Gaussian, gauge invariant, reservoir. The selfadjoint operator H _S determines a privileged orthonormal basis of . These semigroups leave invariant diagonal and off-diagonal bounded operators with respect to this basis. The action on diagonal operators describes a classical Markov jump process. We construct generic semigroups from their formal generators by the minimal semigroup method and discuss their conservativity (uniqueness). When the semigroup is irreducible we prove uniqueness of the equilibrium state and show that, starting from an arbitrary initial state, the semigroup converges towards this state. We also prove that the exponential speed of convergence of the quantum Markov semigroup coincides with the exponential speed of convergence of the classical (diagonal) semigroup towards its unique invariant measure. The exponential speed is computed or estimated in some examples.     

Termination of nondeterministic quantum programs

We define a language-independent model of nondeterministic quantum programs in which a quantum program consists of a finite set of quantum processes. These processes are represented by quantum Markov chains over the common state space, which formalize the quantum mechanical behaviors of the machine. An execution of a nondeterministic quantum program is modeled by a sequence of actions of individual processes, and at each step of an execution a process is chosen nondeterministically to perform the next action. This execution model formalize the users’ behavior of calling the processes in the classical world. Applying the model to a quantum walk as an instance of physically realizable systems, we describe an execution step by step. A characterization of reachable space and a characterization of diverging states of a nondeterministic quantum program are presented. We establish a zero-one law for termination probability of the states in the reachable space. A combination of these results leads to a necessary and sufficient condition for termination of nondeterministic quantum programs. Based on this condition, an algorithm is found for checking termination of nondeterministic quantum programs within a fixed finite-dimensional state space.     

Quantum Subspace Measurements

Fundamental studies of quantum measurements and their capacity to acquire information are typically based on scenarios in which the full Hilbert space of the measured quantum system is open to measurement interactions. In this work, we consider a class of incomplete quantum measurements – quantum subspace measurements (QSM’s) – for which all measurement interactions are restricted to an arbitrary but specified subspace of the measured system Hilbert space. We define QSM’s formally through a condition on the measurement Hamiltonian, obtain forms for the post-measurement states and positive operators (POVM elements) associated with QSM’s acting in a specified subspace, and upper bound the accessible information for such measurements. Characteristic features of QSM’s are identified and discussed.     

About Weyl and Wigner Tomography in Finite-Dimensional Hilbert Spaces

We study different techniques that allow us to gain complete knowledge about an unknown quantum state, e.g. to perform full tomography of this state. In a first time, we focus on two simple cases, full tomography of one- and two-qubit systems. We analyze and compare those techniques according to two criteria. Our first criterion is the minimisation of the redundancy of the data acquired during the tomographic process. In the case of two-qubits tomography, we also analyze this process from the point of view of factorisability, so to say we analyze the possibility to realise the tomographic process through local operations and classical communications between local observers. Finally, we present new results that concern the extension of the one- and two-qubit cases to higher dimensions.     

A phase-space formulation and Gaussian approximation of the filtering equations for nonlinear quantum stochastic systems

This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic systems with multichannel nondemolition measurements. The system-observation dynamics are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation driven by quantum Wiener processes of bosonic fields in vacuum state. The Hamiltonian and system-field coupling operators, as functions of the system variables, are assumed to be represented in a Weyl quantization form. Using the Wigner-Moyal phase-space framework, we obtain a stochastic integro-differential equation for the posterior quasi-characteristic function (QCF) of the system conditioned on the measurements. This equation is a spatial Fourier domain representation of the Belavkin-Kushner-Stratonovich stochastic master equation driven by the innovation process associated with the measurements. We discuss a specific form of the posterior QCF dynamics in the case of linear system-field coupling and outline a Gaussian approximation of the posterior quantum state.