This paper presents a constructive proof of complete kinematic state controllability of finite-dimensional open quantum systems whose dynamics are represented by Kraus maps. For any pair of states (pure or mixed) on the Hilbert space of the system, we explicitly show how to construct a Kraus map that transforms one state into another. Moreover, we prove by construction the existence of a Kraus map that transforms all initial states into a predefined target state (such a process may be used, for example, in quantum information dilution). Thus, in sharp contrast to unitary control, Kraus-map dynamics allows for the design of controls which are robust to variations in the initial state of the system. The general formalism is illustrated with examples of state-to-state Kraus transformations in a two-level system. In particular, we construct a family of non-unitary Kraus maps, which transform one pure state into another. The problem of dynamic state controllability of open quantum systems (i.e., controllability of state-to-state transformations, given a set of available dynamical resources such as coherent controls, incoherent interactions with the environment, and measurements) is also discussed.
A quantum control landscape is defined as the expectation value of a target observable $Θ$ as a function of the control variables. In this work control landscapes for open quantum systems governed by Kraus map evolution are analyzed. Kraus maps are used as the controls transforming an initial density matrix $ρ_{\rm i}$ into a final density matrix to maximize the expectation value of the observable $Θ$. The absence of suboptimal local maxima for the relevant control landscapes is numerically illustrated. The dependence of the optimization search effort is analyzed in terms of the dimension of t…
We give a simple direct proof of the Jamiolkowski criterion to check whether a linear map between matrix algebras is completely positive or not. This proof is more accesible for physicists than others found in the literature and provides a systematic method to give any set of Kraus matrices of its Kraus decomposition.
Quantum operations are usually defined as completely positive (CP), trace preserving (TP) maps on quantum states, and can be represented by operator-sum or Kraus representations. In this paper, we calculate operator-sum representation and master equation of an exactly solvable dynamic of one-qubit open system in layered environment . On the other hand, we obtain exact Nakajima-Zwanzig (NZ) and time-convolutionless (TCL) master equation from the maps. Finally, we study a simple example to consider the relation between CP maps and initial quantum correlation and show that vanishing initial quant…
Quantum dissipation is studied for a discrete system that linearly interacts with a reservoir of harmonic oscillators at thermal equilibrium. Initial correlations between system and reservoir are assumed to be absent. The dissipative dynamics as determined by the unitary evolution of system and reservoir is described by a Kraus map consisting of an infinite number of matrices. For all Laplace-transformed Kraus matrices exact solutions are constructed in terms of continued fractions that depend on the pair correlation functions of the reservoir. By performing factorizations in the Kraus map a p…
A unital completely positive map governing the time evolution of a quantum system is usually called a quantum channel, and it can be represented by a tuple of operators which are then referred to as the Kraus operators of the channel. We look at states of the system whose correlations with respect to the channel have a certain symmetry. Then we show that detailed balance holds if the Kraus operators satisfy a very interesting algebraic relation which plays an important role in the representation theory of any compact quantum group.
We show that the Truncated Wigner Approximation developed in the flat phase-space is mapped into a Lindblad-type evolution with an indefinite metric in the space of linear operators. As a result, the classically evolved Wigner function corresponds to a non-positive operator $\hat{R}(t)$, which does not describe a physical state. The rate of appearance of negative eigenvalues of $\hat{R}(t)$ can be efficiently estimated. The short-time dynamics of the Kerr and second harmonic generation Hamiltonains are discussed.
The polarization of light is utilized in many technologies throughout science and engineering. The ability to transform one state of polarization to another is a key enabling technology. Common polarization transformers are simple polarizers and polarization rotators. Simple polarizers change the intensity depending on the input state and can only output a fixed polarized state, while polarization rotators rotates the input Stokes vector in the 3D Stokes space. We experimentally demonstrate an all-optical input-agnostic polarization transformer (AI-APT), which transforms all input states of po…
We propose a way to understand the evolution of an open quantum system using a description that dispenses a continuous evolution in time, by discrete operators entangled states, in its most direct and fundamental way. We show that the successive application of these operators in very small time intervals reproduce continuous evolution. It describes and compares the temporal evolution of an open quantum system of three levels, for which the Lindblad equation is solved to obtain the density matrix function of time, a method is developed to find Kraus operators time dependent and also finds const…
Kraus map closed-form solution for general master equation dynamics
Institute for Theoretical Physics, University of Cologne（科隆大学理论物理研究所）；Forschungszentrum Jülich GmbH, Peter Grünberg Institute, Quantum Control (PGI-8)（于利希研究中心有限公司彼得·格林伯格研究所量子控制部门（PGI-8））
The study of dissipation and decoherence in generic open quantum systems recently led to the investigation of spectral and steady-state properties of random Lindbladian dynamics. A natural question is then how realistic and universal those properties are. Here, we address these issues by considering a different description of dissipative quantum systems, namely, the discrete-time Kraus map representation of completely positive quantum dynamics. Through random matrix theory (RMT) techniques and numerical exact diagonalization, we study random Kraus maps, allowing for a varying dissipation stren…
The characterization of transformations among entangled pure states via local operations assisted by classical communication (LOCC) is a crucial problem in quantum information theory for both theoretical and practical reasons. As LOCC has a highly intricate structure, sometimes the larger set of separable (SEP) maps is considered, which has a mathematically much simpler description. In the literature, mainly SEP maps consisting of invertible Kraus operators have been taken into account. In this paper we show that the consideration of those maps is not sufficient when deciding whether a state c…
For any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, a general procedure is reviewed for constructing the corresponding linear map from the initial state to the state at time t, including its Kraus-type representations. Formally, this is equivalent to solving the master equation. For an N-dimensional Hilbert space it requires (i) solving a first order N^2 x N^2 matrix time evolution (to obtain the completely positive map), and (ii) diagonalising a related N^2 x N^2 matrix (to obtain a Kraus-type representation). Conversely, for a given time…
Microscopic Hamiltonian models of the composite system "open system + environment" typically do not provide the operator-sum Kraus form of the open system's dynamical map. With the use of a recently de- veloped method [16], we derive the Kraus operators starting from the mi- croscopic Hamiltonian model, i.e. from the proper master equation, of the one-qubit depolarizing channel. Those Kraus operators generalize the stan- dard counterparts, which are widely used in the literature. Comparison of the standard and the here obtained Kraus operators is performed via inves- tigating dynamical change …
Operator-sum or Kraus representations for single-mode Bosonic Gaussian channels are developed, and several of their consequences explored. Kraus operators are employed to bring out the manner in which the unphysical matrix transposition map when accompanied by injection of a threshold classical noise becomes a physical channel. The action of the quantum-limited attenuator and amplifier channels as simply scaling maps on suitable quasi-probabilities in phase space is examined in the Kraus picture. Consideration of cumulants is used to examine the issue of fixed points. In the cases of entanglem…
We investigate elementary topological properties of sets of completely positive (CP) maps that arise in quantum Perron-Frobenius theory. We prove that the set of primitive CP maps of fixed Kraus rank is path-connected and we provide a complete classification of the connected components of irreducible CP maps at given Kraus rank and fixed peripheral spectrum in terms of a multiplicity index. These findings are then applied to analyse 1D quantum phases by studying equivalence classes of translational invariant Matrix Product States that correspond to the connected components of the respective CP…
On account of the Abel-Galois no-go theorem for the algebraic solution to quintic and higher order polynomials, the eigenvalue problem and the associated characteristic equation for a general noise dynamics in dimension $d$ via the Choi-Jamiolkowski approach cannot be solved in general via radicals. We provide a way around this impasse by decomposing the Choi matrix into simpler, not necessarily positive, Hermitian operators that are diagonalizable via radicals, which yield a set of `positive' and `negative' Kraus operators. The price to pay is that the sufficient number of Kraus operators is …
We consider the case of a $\sqrt{\mathrm{SWAP}}$ quantum gate and its optimized entangling action, via continuous dynamical decoupling, in the presence of dephasing noise. We illustrate the procedure in the specific case where only the two-qubit operation is controlled and no single-qubit operations are included in the description. To compare the optimized dynamics in the presence of noise with the ideal case, we use the standard fidelity measure. Then we discuss the importance of using optimized gates in the quantum operational-probabilistic theory. Because of their importance for the explici…
We present a general formalism for charecterizing 2-time quantum states, describing pre- and post-selected quantum systems. The most general 2-time state is characterized by a `density vector' that is independent of measurements performed between the preparation and post-selection. We provide a method for performing tomography of an unknown 2-time density vector. This procedure, which cannot be implemented by weak or projective measurements, brings new insight to the fundamental role played by Kraus operators in quantum measurements. Finally, after showing that general states and measurements …
From the key composite quantum system made of a two-level system (qubit) and a harmonic oscillator (photon) with resonant or dispersive interactions, one derives the corresponding quantum Stochastic Master Equations (SME) when either the qubits or the photons are measured. Starting with an elementary discrete-time formulation based on explicit formulae for the interaction propagators, one shows how to include measurement imperfections and decoherence. This qubit/photon quantum system illustrates the Kraus-map structure of general discrete-time SME governing the dynamics of an open quantum syst…
Fidelity is known to increase through any Kraus map: the fidelity between two density matrices is less than the fidelity between their images via a Kraus map. We prove here that, in average, fidelity is also increasing for any discrete-time quantum filter: fidelity between the density matrix of the underlying Markov chain and the density matrix of its associated quantum filter is a sub-martingale. This result is not restricted to pure states. It also holds true for mixed states.
Quantum process tomography is a procedure by which an unknown quantum operation can be fully experimentally characterized. We reinterpret Choi's proof of the fact that any completely positive linear map has a Kraus representation [Lin. Alg. and App., 10, 1975] as a method for quantum process tomography. Furthermore, the analysis for obtaining the Kraus operators are particularly simple in this method.
A quantum channel from a system $A$ of dimension $d_A$ to a system $B$ of dimension $d_B$ is a completely positive trace-preserving map from complex $d_A\times d_A$ to $d_B\times d_B$ matrices, and the set of all such maps with Kraus rank $r$ has the structure of a smooth manifold. We describe this set in two ways. First, as a quotient space of (a subset of) the $rd_B\times d_A$ dimensional Stiefel manifold. Secondly, as the set of all Choi-states of a fixed rank $r$. These two descriptions are topologically equivalent. This allows us to show that the set of all Choi-states corresponding to ex…
Using pure entangled Schmidt states, we show that m-positivity of a map is bounded by the ranks of its negative Kraus matrices. We also give an algebraic condition for a map to be m-positive. We interpret these results in the context of positive maps as entanglement witnesses, and find that only 1-positive maps are needed for testing entanglement.
A new bound on quantum version of Wielandt inequality for positive (not necessarily completely positive) maps has been established. Also bounds for entanglement breaking and PPT channels are put forward which are better bound than the previous bounds known. We prove that a primitive positive map $\mathcal{E}$ acting on $\mathcal{M}_d$ that satisfies the Schwarz inequality becomes strictly positive after at most $2(d-1)^2$ iterations. This is to say, that after $2(d-1)^2$ iterations, such a map sends every positive semidefinite matrix to a positive definite one. This finding does not depend on …
Quantum dissipation is studied for a discrete system that linearly interacts with a reservoir of harmonic oscillators at thermal equilibrium. Initial correlations between system and reservoir are assumed to be absent. The dissipative dynamics as determined by the unitary evolution of system and reservoir is described by a Kraus map consisting of an infinite number of matrices. For all Laplace-transformed Kraus matrices exact solutions are constructed in terms of continued fractions that depend on the pair correlation functions of the reservoir.
A quantum control landscape is defined as the physical objective as a function of the control variables. In this paper the control landscapes for two-level open quantum systems, whose evolution is described by general completely positive trace preserving maps (i.e., Kraus maps), are investigated in details. The objective function, which is the expectation value of a target system operator, is defined on the Stiefel manifold representing the space of Kraus maps. Three practically important properties of the objective function are found: (a) the absence of local maxima or minima (i.e., false tra…
A system of diagrams is introduced that allows the representation of various elements of a quantum circuit, including measurements, in a form which makes no reference to time (hence ``atemporal''). It can be used to relate quantum dynamical properties to those of entangled states (map-state duality), and suggests useful analogies, such as the inverse of an entangled ket. Diagrams clarify the role of channel kets, transition operators, dynamical operators (matrices), and Kraus rank for noisy quantum channels. Positive (semidefinite) operators are represented by diagrams with a symmetry that aid…
We construct relativistic quantum Markov semigroups from covariant completely positive maps. We proceed by generalizing a step in Stinespring's dilation to a general system of imprimitivity and basing it on Poincaré group. The resulting noise channels are relativistically consistent and the method is applicable to any fundamental particle, though we demonstrate it for the case of light-like particles. The Krauss decomposition of the relativistically consistent completely positive identity preserving maps (our set up is in Heisenberg picture) enables us to construct the covariant quantum Markov…
Quantum channels, complex Stiefel manifolds, and optimization
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；University of Science and Technology MISIS（国立研究型技术大学米西斯）；Ivannikov Institute for System Programming of the Russian Academy of Sciences（俄罗斯科学院伊万尼科夫系统编程研究所）
We associate with k hermitian N\times N matrices a probability measure on R^k. It is supported on the joint numerical range of the k-tuple of matrices. We call this measure the joint numerical shadow of these matrices. Let k=2. A pair of hermitian N\times N matrices defines a complex N\times N matrix. The joint numerical range and the joint numerical shadow of the pair of hermitian matrices coincide with the numerical range and the numerical shadow, respectively, of this complex matrix. We study relationships between the dynamics of quantum maps on the set of quantum states, on one hand, and t…
Continuity properties of the output entropy of positive linear maps between Banach spaces of trace class operators are investigated with the special attention to the classes of quantum channels and operations. It is shown that finiteness of the output entropy of a positive map on the whole input state space implies its continuity. Sufficient conditions for this property expressed in terms of the Kraus representation of quantum channels and operations are presented. The characterization of a positive map "preserving continuity of the entropy" (in the sense that continuity of the entropy on a se…
We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular…
Einstein introduced the locality principle which states that all physical effect in some finite space-time region does not influence its space-like separated finite region. Recently, in algebraic quantum field theory, Rédei captured the idea of the locality principle by the notion of operational separability. The operation in operational separability is performed in some finite space-time region, and leaves unchanged the state in its space-like separated finite space-time region. This operation is defined with a completely positive map. In the present paper, we justify using a completely posit…
Institute of Theoretical Physics, Jagiellonian University（雅盖隆大学理论物理研究所）；Doctoral School of Exact and Natural Sciences, Jagiellonian University（雅盖隆大学精确与自然科学博士学院）
In a topological space, a family of continuous mappings is called universal if its action, in at least one element of the space, is dense. If the mappings are unitary or trace-preserving completely positive, the notion of universality is closely related to the notion of controllability in either closed or open quantum systems. Quantum controllability in infinite dimensions is discussed in this setting and minimal generators are found for full control universal families. Some of the requirements of the operators needed for control in infinite dimensions follow from the properties of the infinit…
Digital controllability of transverse field Ising chains
SISSA（的里雅斯特国际高等研究院）；CNR-IOM - Istituto Officina dei Materiali, Consiglio Nazionale delle Ricerche（意大利国家研究委员会材料研究所）；International Centre for Theoretical Physics (ICTP)（国际理论物理中心）
We introduce and study two new classes of unital quantum channels. The first class describes a 2-parameter family of channels given by completely positive (CP) maps $M_3({\bf C}) \mapsto M_3({\bf C})$ which are both unital and trace-preserving. Almost every member of this family is factorizable and extreme in the set of CP maps which are both unital and trace-preserving, but is not extreme in either the set of unital CP maps or the set of trace-preserving CP maps. We also study a large class of maps which generalize the Werner-Holevo channel for $d = 3$ in the sense that they are defined in te…
We obtain the analytical expression for the Kraus decomposition of the quantum map of an environment modeled by an arbitrary quadratic fermionic Hamiltonian acting on one or two qubits, and derive simple functions to check the non-positivity of the intermediate map. These functions correspond to two different sufficient criteria for non-Markovianity. In the particular case of an environment represented by the Ising Hamiltonian, we discuss the two sources of non-Markovianity in the model, one due to the finite size of the lattice, and another due to the kind of interactions.
The non-Markovian dynamics of quantum entanglement is studied by the Shabani-Lidar master equation when one of entangled quantum systems is coupled to a local reservoir with memory effects. The completely positive reduced dynamical map can be constructed in the Kraus representation. Quantum entanglement decays more slowly in the non-Markovian environment. The decoherence time for quantum entanglement can be markedly increased by the change of the memory kernel. It is found out that the entanglement sudden death between quantum systems and entanglement sudden birth between the system and reserv…
Measurement-Induced Boolean Dynamics and Controllability for Quantum Networks
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；The Australian National University（澳大利亚国立大学）；The University of Sydney（悉尼大学）
Efficient implementation of arbitrary Hermitian-preserving and trace-preserving maps
University of Science and Technology of China（中国科学技术大学）；QICI Quantum Information and Computation Initiative, School of Computing and Data Science, The University of Hong Kong（香港大学计算与数据科学学院量子信息与计算倡议中心）；The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Hefei National Laboratory（合肥国家实验室）；Tsinghua University（清华大学）
We consider complete positivity of dynamics regarding subsystems of an open composite quantum system, which is subject of a completely positive dynamics. By "completely positive dynamics", we assume the dynamical maps called the completely positive and trace preserving maps, with the constraint that domain of the map is the whole Banach space of the system's density matrices. We provide a technically simple and conceptually clear proof for the subsystems' completely positive dynamics. Actually, we prove that every subsystem of a composite open system can be subject of a completely positive dyn…
We address the question of the existence of quantum channels that are divisible in two quantum channels but not in three or, more generally, channels divisible in $n$ but not in $n+1$ parts. We show that for the qubit those channels \textit{do not} exist, whereas for general finite-dimensional quantum channels the same holds at least for full Kraus rank channels. To prove these results, we introduce a novel decomposition of quantum channels which separates them into a boundary and Markovian part, and it holds for any finite dimension. Additionally, the introduced decomposition amounts to the w…
Every completely positive map G that commutes which the Hamiltonian time evolution is an integral or sum over (densely defined) CP-maps G_σwhere σis the energy that is transferred to or taken from the environment. If the spectrum is non-degenerated each G_σis a dephasing channel followed by an energy shift. The dephasing is given by the Hadamard product of the density operator with a (formally defined) positive operator. The Kraus operator of the energy shift is a partial isometry which defines a translation on R with respect to a non-translation-invariant measure. As an example, I calculate t…
