Simple Hartree-type equations lead to dynamics of a subsystem that is not completely positive in the sense accepted in mathematical literature. In the linear case this would imply that negative probabilities have to appear for some system that contains the subsystem in question. In the nonlinear case this does not happen because the mathematical definition is physically unfitting as shown on a concrete example.
We specify the structure of completely positive operators and quantum Markov semigroup generators that are symmetric with respect to a family of inner products, also providing new information on the order strucure an extreme points in some previously studied cases.
We show that any Hermiticity and trace preserving continuous semigroup gamma_t in d dimensions is completely positive if and only if the semigroup gamma_t otimes gamma_t is positivity preserving.
In relation with experiments on correlated kaons at phi-factories, it is shown that the request of complete positivity is necessary in any physically consistent description of neutral kaons as open quantum systems.
We consider the propagation of polarized photons in optical fibers under the action of randomly generated noise. In such situation, the change in time of the photon polarization can be described by a quantum dynamical semigroup. We show that the hierarchy among the decay constants of the polarization density matrix elements as prescribed by complete positivity can be experimentally probed using standard laboratory set-ups.
We consider a two-dimensional quantum control system evolving under an entropy-increasing irreversible dynamics in the semigroup form. Considering a phenomenological approach to the dynamics, we show that the accessibility property of the system depends on whether its evolution is assumed to be positive or completely positive. In particular, we characterize the family of maps having different accessibility and show the impact of that property on observable quantities by means of a simple physical model.
We study how some recently proposed noncontextuality tests based on quantum interferometry are affected if the test particles propagate as open systems in presence of a gaussian stochastic background. We show that physical consistency requires the resulting markovian dissipative time-evolution to be completely positive.
We study the modified dynamical evolution of the neutral kaon system under the condition of complete positivity. The accuracy of the data from planned future experiments is expected to be sufficiently precise to test such a hypothesis.
We derive a purely algebraic framework for the identification of hierarchy equations of motion that induce completely positive dynamics and demonstrate the applicability of our approach with several examples. We find bounds on the violation of complete positivity for microscopically derived hierarchy equations of motion and construct well-behaved phenomenological models with strongly non-Markovian revivals of quantum coherence.
We analyze the dynamics of neutron beams in interferometry experiments using quantum dynamical semigroups. We show that these experiments could provide stringent limits on the non-standard, dissipative terms appearing in the extended evolution equations.
We present an analysis of complete positivity (CP) constraints on qutrit quantum channels that have a form of affine transformations of generalized Bloch vector. For diagonal (damping) channels we derive conditions analogous to the ones that in qubit case produce tetrahedron structure in the channel parameter space.
We consider two non-interacting systems embedded in a heat bath. If they remain dynamically independent, physical inconsistencies are avoided only if the single-system reduced dynamics is completely positive also beyond the weak-coupling limit.
After reviewing the main properties of time-evolutions of open quantum systems, some considerations about the positivity of factorized Markovian dynamics for bipartite systems are made. In particular, it is shown that the positivity of the whole time-evolution in general does not ask for the complete positivity of the single system time-evolutions, if they are allowed to differ. However, they must be completely positive if one is a small perturbation of the other, which is the typical situation for open systems in a heat bath.
We study quantum processes, as one parameter families of differentiable completely positive and trace preserving (CPTP) maps. Using different representations of the generator, and the Sylvester criterion for positive semi-definite matrices, we obtain conditions for the divisibility of the process into completely positive (CP-divisibility) and positive (P-divisibility) infinitesimal maps. Both concepts are directly related to the definition of quantum non-Markovianity. For the single qubit case we show that CP- and P-divisibility only depend on the dissipation matrix in the master equation form…
Positivity or the stronger notion of complete positivity, and contextuality are central properties of quantum dynamics. In this work, we demonstrate that a physical unitary-universe dilation model could be employed to characterize the completely positive map, regardless of the initial correlation condition. Particularly, the problem of initial correlation can be resolved by a swap operation. Furthermore, we discuss the physical essence of completely positive map and highlights its limitations. Then we develop the quantum measurement-chain formula beyond the framework of completely positive map…
When a composite Lindblad system consists of weakly coupled sub-systems with fast and slow timescales, the description of slow dynamics can be simplified by discarding fast degrees of freedom. This model reduction technique is called adiabatic elimination. While second-order perturbative expansion with respect to the timescale separation has revealed that the evolution of a reduced state is completely positive, this paper presents an example exhibiting complete positivity violation in the fourth-order expansion. Despite the non-uniqueness of slow dynamics parametrization, we prove that complet…
Two fundamental contributions to categorical quantum mechanics are presented. First, we generalize the CP-construction, that turns any dagger compact category into one with completely positive maps, to arbitrary dimension. Second, we axiomatize when a given category is the result of this construction.
While it is well known that complete positivity guarantees the fulfilment of the second law of thermodynamics, its possible violations have never been proposed as a check of the complete positivity of a given open quantum dynamics. We hereby consider an open quantum micro-circuit, effectively describable as a two-level open quantum system, whose asymptotic current might be experimentally accessible. This latter could indeed be used to discriminate between its possible non-completely positive Redfield dynamics and a completely positive one obtained by standard weak-coupling limit techniques, at…
It is shown that if the decoherence matrix corresponding to a qubit master equation has a block-diagonal real part, then the evolution is determined by a one-dimensional oscillator equation. Further, when the full decoherence matrix is block-diagonal, then the necessary and sufficient conditions for completely positive evolution may be formulated in terms of the oscillator Hamiltonian or Lagrangian. When the solution of the oscillator equation is not known, an explicit sufficient condition for complete positivity can still be obtained, based on a Hamiltonian/Lagrangian inequality. A rotational…
The relevance that the property of complete positivity has had in the determination of quantum structures is briefly reviewed, together with recent applications to neutron optics and quantum Brownian motion. A possible useful application and generalization of this property to the description of macroscopic systems inside quantum mechanics is discussed on the basis of recent work on the derivation of subdynamics in Heisenberg picture of slowly varying degrees of freedom inside nonrelativistic quantum field theory.
Open quantum dynamics without Complete Positivity: a criticism
University of Trieste（的里雅斯特大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Institute of Physics, Nicolaos Copernicus University（尼古拉·哥白尼大学物理研究所）；Department of Physics, University of Bari（巴里大学物理系）；Istituto Nazionale di Fisica Nucleare, Sezione di Bari（意大利国家核物理研究所巴里分部）
Complete positivity of quantum dynamics is often viewed as a litmus test for physicality, yet it is well known that correlated initial states need not give rise to completely positive evolutions. This observation spurred numerous investigations over the past two decades attempting to identify necessary and sufficient conditions for complete positivity. Here we describe a complete and consistent mathematical framework for the discussion and analysis of complete positivity for correlated initial states of open quantum systems. This formalism is built upon a few simple axioms and is sufficiently …
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.
The reduced dynamics of an open quantum system $S$, interacting with its environment $E$, is not completely positive, in general. In this paper, we demonstrate that if the two following conditions are satisfied, simultaneously, then the reduced dynamics is completely positive: (1) the reduced dynamics of the system is linear, for arbitrary system-environment unitary evolution $U$; and (2) the reduced dynamics of the system is linear, for arbitrary initial state of the system $ρ_S$.
Consider the set $\mathcal{S}=\lbraceρ_{SE}\rbrace$ of possible initial states of the system-environment. The map which assigns to each $ρ_{S}\in \mathrm{Tr}_{E}\mathcal{S}$ a $ρ_{SE}\in \mathcal{S}$ is called the assignment map. The assignment map is Hermitian, in general. In this paper, we restrict ourselves to the case that the assignment map is, in addition, positive and show that this implies that the so-called reference state is a Markov state. Markovianity of the reference state leads to existence of another assignment map which is completely positive. So, the reduced dynamics of the sy…
We present a general model of qubit dynamics which entails pure dephasing and dissipative time-local master equations. This allows us to describe the combined effect of thermalisation and dephasing beyond the usual Markovian approximation. We investigate the complete positivity conditions and introduce a heuristic model that is always physical and provides the correct Markovian limit. We study the effects of temperature on the non-Markovian behaviour of the system and show that the noise additivity property discussed by Yu and Eberly in Ref. [1] holds beyond the Markovian limit.
We review the standard treatment of open quantum systems in relation to quantum entanglement, analyzing, in particular, the behaviour of bipartite systems immersed in a same environment. We first focus upon the notion of complete positivity, a physically motivated algebraic constraint on the quantum dynamics, in relation to quantum entanglement, i.e. the existence of statistical correlations which can not be accounted for by classical probability. We then study the entanglement power of heat baths versus their decohering properties, a topic of increasing importance in the framework of the fast…
We show that complete positivity is not only sufficient but also necessary for the validity of the quantum data-processing inequality. As a consequence, the reduced dynamics of a quantum system are completely positive, even in the presence of initial correlations with its surrounding environment, if and only if such correlations do not allow any anomalous backward flow of information from the environment to the system. Our approach provides an intuitive information-theoretic framework to unify and extend a number of previous results.
Entanglement of states and Complete Positivity of maps are concepts that have achieved physical importance with the recent growth of quantum information science. They are however mathematically relevant whenever tensor products of complex linear (Hilbert) spaces are involved. We present such situations in classical scalar paraxial wave optics where these concepts play a role: propagation characteristics of coherent and partially coherent Gaussian beams; and the definition and separability of the family of Twisted Gaussian Schell Model (TGSM) beams. In the former, the evolution of the width of …
A quantum channel will have a Choi representation from which the complete positivity (CP) can be determined in a number of different ways. Every method relies on Choi's proof which relates CP to the positive semi-definiteness of a specially constructed matrix (i.e. the "Choi representation of the channel"). It can be shown that different CP tests can be used in different situations to avoid the computationally expensive brute force method of calculating the full spectrum of the Choi representation, which is the traditional method for testing whether or not a channel is CP. It is important to h…
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…
To avoid the computational burden of many-body quantum simulation, the interaction of an electron with a photon (phonon) is typically accounted for by disregarding the explicit simulation of the photon (phonon) degree of freedom and just modelling its effect on the electron dynamics. For quantum models developed from the (reduced) density matrix or its Wigner-Weyl transformation, the modelling of collisions may violate complete positivity (precluding the typical probabilistic interpretation). In this paper, we show that such quantum transport models can also strongly violate the energy conserv…
The dual of a matrix ordered space has a natural matrix ordering that makes the dual space matrix ordered as well. The purpose of these notes is to give a condition that describes when the linear map taking a basis of the n by n matrices to its dual basis is a complete order isomorphism and complete co-order isomorphism. In the case of the standard matrix units this map is a complete order isomorphism and this is a restatement of the correspondence between completely positive maps and the Choi matrix. However, we exhibit natural orthonormal bases for the matrices such that this map is an order…
Using the positivity of relative entropy arising from the Ryu-Takayanagi formula for spherical entangling surfaces, we obtain constraints at the nonlinear level for the gravitational dual. We calculate the Green's function necessary to compute the first order correction to the entangling surface and use this to find the relative entropy for non-constant stress tensors in a derivative expansion. We show that the Einstein value satisfies the positivity condition while the multi-dimensional parameter space away from it gets constrained.
In this work we are interested the problem of testing quantum entanglement. More specifically, we study the separability problem in quantum property testing, where one is given $n$ copies of an unknown mixed quantum state $\varrho$ on $\mathbb{C}^d \otimes \mathbb{C}^d$, and one wants to test whether $\varrho$ is separable or $ε$-far from all separable states in trace distance. We prove that $n = Ω(d^2/ε^2)$ copies are necessary to test separability, assuming $ε$ is not too small, viz.\ $ε= Ω(1/\sqrt{d})$. We also study completely positive distributions on the grid $[d] \times [d]$, as a class…
We provide a general and consistent formulation for linear subsystem quantum dynamical maps, developed from a minimal set of postulates, primary among which is a relaxation of the usual, restrictive assumption of uncorrelated initial system-bath states. We describe the space of possibilities admitted by this formulation, namely that, far from being limited to only completely positive (CP) maps, essentially any $\mathbb{C}$-linear, Hermiticity-preserving, trace-preserving subsystem map can arise as a legitimate subsystem dynamical map from a joint unitary evolution of a system coupled to a bath…
The traditional scheme for realizing open-system quantum dynamics takes the initial state of the system-bath composite as a simple product. Currently, however, the issue of system-bath initial correlations possibly affecting the reduced dynamics of the system has been attracting considerable interest. The influential work of Shabani and Lidar [PRL {\bf 102}, 100402 (2009)] famously related this issue to quantum discord, a concept which has in recent years occupied the centre-stage of quantum information theory and has led to several fundamental results. They suggested that reduced dynamics is …
We consider the time evolution of the density matrix $ρ$ in a 2-dimensional complex Hilbert space. We allow for dissipation by adding to the von Neumann equation a term $D[ρ]$, which is of Lindblad type in order to assure complete positivity of the time evolution. We present five equivalent forms of $D[ρ]$. In particular, we connect the familiar dissipation matrix $L$ with a geometric version of $D[ρ]$, where $L$ consists of a positive sum of projectors onto planes in $\mathbf{R}^3$. We also study the minimal number of Lindblad terms needed to describe the most general case of $D[ρ]$. All proo…
We introduce a classification scheme for the generators of bosonic open Gaussian dynamics, providing instructive diagrams description for each type of dynamics. Using this classification, we discuss the consequences of imposing complete positivity on Gaussian dynamics. In particular, we show that non-symplectic operations must be active to allow for complete positivity. In addition, non-symplectic operations can, in fact, conserve the volume of phase space only if the restriction of complete positivity is lifted. We then discuss the implications for the relationship between information and ene…
Consider the set $\mathcal{S}=\lbraceρ_{SE}\rbrace$ of possible initial states of the system-environment, steered from a tripartite reference state $ω_{RSE}$. Buscemi [F. Buscemi, Phys. Rev. Lett. 113, 140502 (2014)] showed that the reduced dynamics of the system, for each $ρ_{S}\in \mathrm{Tr}_{E}\mathcal{S}$, is always completely positive if and only if $ω_{RSE}$ is a Markov state. There, during the proof, it has been assumed that the dimensions of the system and the environment can vary through the evolution. Here, we show that this assumption is necessary: we give an example for which, tho…
Positive maps which are not completely positive are used in quantum information theory as witnesses for convex sets of states, in particular as entanglement witnesses and more generally as witnesses for states having Schmidt number not greater than k. It is known that such witnesses are related to k-positive maps. In this article we propose a new proof for the correspondence between vectors having Schmidt number k and k-positive maps using Jamiolkowski's criterion for positivity of linear maps; to this aim, we also investigate the precise notion of the term "Jamiolkowski isomorphism". As conse…
We show how to recover complete positivity (and hence positivity) of the Redfield equation via a coarse grain average technique. We derive general bounds for the coarse graining time scale above which the positivity of the Redfield equation is guaranteed. It turns out that a coarse grain time scale has strong impact on the characteristics of the Lamb shift term and implies in general non-commutation between the dissipating and the Hamiltonian components of the generator of the dynamical semi-group. Finally we specify the analysis to a two-level system or a quantum harmonic oscillator coupled t…
There are several inequalities in physics which limit how well we can process physical systems to achieve some intended goal, including the second law of thermodynamics, entropy bounds in quantum information theory, and the uncertainty principle of quantum mechanics. Recent results provide physically meaningful enhancements of these limiting statements, determining how well one can attempt to reverse an irreversible process. In this paper, we apply and extend these results to give strong enhancements to several entropy inequalities, having to do with entropy gain, information gain, entropic di…
We study the reduced time-evolution of open quantum systems by combining quantum-information and statistical field theory. Inspired by prior work [EPL 102, 60001 (2013) and Phys. Rev. Lett. 111, 050402 (2013)] we establish the explicit structure guaranteeing the complete positivity (CP) and trace-preservation (TP) of the real-time evolution expansion in terms of the microscopic system-environment coupling. This reveals a fundamental two-stage structure of the coupling expansion: Whereas the first stage defines the dissipative timescales of the system --before having integrated out the environm…
We obtain new types of exponential decay laws for solutions of density-matrix master equations in the weak-coupling limit: after comparing with results already present in the literature and developing the necessary techniques, we study the crucial aspect of complete positivity under fairly general conditions. We propose a new type of time average that guarantees complete positivity and approximates, in markovian fashion, the exact dynamics for a plethora of physical applications, no matter which are the spectral properties of the subsystem, or its dimensions. We shall comment on some interesti…
The common wisdom in the field of quantum information theory is that when a system is initially correlated with its environment, the map describing its evolution may fail to be completely positive. If true, this would have practical and foundational significance. We here demonstrate, however, that the common wisdom is mistaken. We trace the error to the standard argument for how the evolution map ought to be defined. We show that it sometimes fails to define a linear map or any map at all and that these pathologies persist even in completely classical examples. Drawing inspiration from the fra…
