GENERIC QUANTUM MARKOV SEMIGROUPS, CYCLE DECOMPOSITION AND DEVIATION FROM EQUILIBRIUM

We show that the commutant of the range of the infinitesimal generator of a norm-continuous quantum Markov semigroup on [Formula: see text], not consisting of identity maps, with a faithful normal invariant state is trivial whenever the fixed point algebra is atomic. As a consequence, two formulations of the irreversible [Formula: see text]-KMS condition proposed in Ref. 2 are equivalent for this class of quantum Markov semigroups.

Imaginary-time evolution (ITE) on a quantum computer is a promising formalism for obtaining the ground state of a quantum system. The probabilistic ITE (PITE) exploits measurements to implement nonunitary operations, and it can avoid the restriction of dynamics to a low-dimensional subspace imposed by variational parameters unlike other types of ITE. In this paper, we propose a PITE approach that uses only one ancillary qubit. Unlike the existing PITE approaches, the one proposed here constructs, under a practical approximation, the circuit from forward and backward real-time evolution (RTE) gates as black boxes for the original Hamiltonian. Thus all efficient unitary algorithms for RTE can be transferred to the ITE without any modifications. Our approach can be used to obtain the Gibbs state at a finite temperature and partition function. We validate the approach via several illustrative systems where the trial states are found to converge rapidly to the ground states. In addition, we discuss its applicability to quantum chemistry by focusing on the scaling of computational cost; this leads to the development of a framework referred to as a first-quantized eigensolver. The nonvariational generic approach will expand the scope of practical quantum computation for versatile objectives.

The main contribution of this paper is to present the systematic and low-complexity translation techniques between a class of hybrid systems referred to as automaton-based DHA and piecewise affine (PWA) systems. As an starting point the general modeling framework of the automaton-based DHA is represented which models the controlled and uncontrolled switching phenomena between linear continuous dynamics including discrete and continuous states, inputs and outputs. The basic theoretical definitions on the state trajectories of the proposed DHA with forward and backward evolutions which yield forward and backward piecewise affine (FPWA and BPWA) systems are given. Next, the well-posedness and equivalency properties are proposed and the sufficient conditions under which the well-posedness property is achieved with the automaton-based DHA and PWA systems are given. It is shown that the graphical structure of the proposed automaton-based DHA makes it possible to obtain analytically the equivalent PWA system with a polynomial complexity in contrast to the existing numerical translation techniques via decomposed structure of the DHA with an exponential complexity. Examples are presented to confirm the effectiveness of the proposed translation techniques.

Within a differential expression of the Heisenberg operator, the forward and backward evolution can be joined together along a closed time contour. This manipulation leads to a dramatic cancellation of oscillations due to the two individual propagators in the Heisenberg operator and the resulting forward-backward propagator is more tractable to semiclassical approximations. This article gives a detailed description of the forward-backward semiclassical dynamics (FBSD) formalism. The semiclassical propagators, especially those of the initial value representations (IVRs), are briefly discussed. The derivation of the FBSD based on the Herman–Kluk propagator is reviewed. Different FBSD formulations with other semiclassical IVRs are worked out and numerical calculations show that they are also capable of describing quantum dynamics semiquantitatively and all display accuracy similar to the classical Wigner method.

Information scrambling refers to the rapid spreading of initially localized information over an entire system, via the generation of global entanglement. This effect is usually detected by measuring a temporal decay of the out-of-time order correlators. However, in experiments, decays of these correlators suffer from fake positive signals from various sources, e.g., decoherence due to inevitable couplings to the environment, or errors that cause mismatches between the purported forward and backward evolutions. In this Letter, we provide a simple and robust approach to single out the effect of genuine scrambling. This allows us to benchmark the scrambling process by quantifying the degree of the scrambling from the noisy backgrounds. We also demonstrate our protocol with simulations on IBM cloud-based quantum computers.

This chapter is devoted to presenting some results on controllability for forward and backward stochastic linear evolution equations with possibly unbounded control operators. By the duality argument, these controllability problems can be reduced to suitable observability for the dual equations. Explicit forms of controls for the controllability problems are also provided. Finally, the controllability of some forward stochastic evolution equations are shown to be equivalent to that of suitably chosen backward stochastic evolution equations.

Solutions of semilinear parabolic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations. Parabolic equations are intended in a mild sense that reveals to be suitable also towards applications to optimal control.

We introduce a class of abstract nonlinear fractional pseudo-differential equations in Banach spaces that includes both the Mc-Kean-Vlasov-type equations describing nonlinear Markov processes and the Hamilton-Jacobi-Bellman(HJB)-Isaacs equation of stochastic control and games thus allowing for a unified analysis of these equations. This leads to an effective theory of coupled forward-backward systems (forward McKean-Vlasov evolution and backward HJB-Isaacs evolution) that are central to the modern theory of mean-field games.

Let M be a von Neumann algebra, let $\varphi $ be a normal faithful state on M and let $L^p(M,\varphi )$ be the associated Haagerup noncommutative $L^p$ -spaces, for $1\leq p\leq \infty $ . Let $D\in L^1(M,\varphi )$ be the density of $\varphi $ . Given a positive map $T\colon M\to M$ such that $\varphi \circ T\leq C_1\varphi $ for some $C_1\geq 0$ , we study the boundedness of the $L^p$ -extension $T_{p,\theta }\colon D^{\frac {1-\theta }{p}} MD^{\frac {\theta }{p}}\to L^p(M,\varphi )$ which maps $D^{\frac {1-\theta }{p}} x D^{\frac {\theta }{p}}$ to $D^{\frac {1-\theta }{p}} T(x) D^{\frac {\theta }{p}}$ for all $x\in M$ . Haagerup–Junge–Xu showed that $T_{p,{\frac {1}{2}}}$ is always bounded and left open the question whether $T_{p,\theta }$ is bounded for $\theta \not ={\frac {1}{2}}$ . We show that for any $1\leq p<2$ and any $\theta \in [0, 2^{-1}(1-\sqrt {p-1})]\cup [2^{-1}(1+\sqrt {p-1}), 1]$ , there exists a completely positive T such that $T_{p,\theta }$ is unbounded. We also show that if T is $2$ -positive, then $T_{p,\theta }$ is bounded provided that $p\geq 2$ or $1\leq p<2$ and $\theta \in [1-p/2,p/2]$ .

We provide a comprehensive study on uniformly left [Formula: see text]-bounded (respectively, [Formula: see text]-bounded) orthonormal bases in infinite-dimensional cyclic bimodules associated with c.p. maps between two von Neumann algebras [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are faithful normal states on [Formula: see text] and [Formula: see text], respectively. Separate investigations on cyclic bimodules associated with Markov maps and arbitrary c.p. maps are also provided since the results differ. This generalizes the results of the authors’ previous work on Kadison’s unitary basis problem.

Let (M, ϕ) = (M1, ϕ1) * (M2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q ⊂ M is a von Neumann subalgebra with separable predual such that both Q and Q ∩ M1 are the ranges of faithful normal conditional expectations and such that both the intersection Q ∩ M1 and the central sequence algebra Q′ ∩ Mω are diffuse (e.g. Q is amenable), then Q must sit inside M1. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion M1 ⊂ M in arbitrary free product von Neumann algebras.

Almost uniform convergence on the predual of a von Neumann algebra and an ergodic theorem

We prove a generalization of N. Ozawa’s Kurosh-type theorem to the setting of free products of semiexact II 1 \text {II}_1 factors with respect to arbitrary (non-tracial) faithful normal states. We are thus able to distinguish certain resulting type III factors. For example, if M = L F n ⊗ L F m M = LF_n \otimes LF_m and { φ i } \{\varphi _i\} is any sequence of faithful normal states on M M , then the l l -various ( M , φ 1 ) ∗ . . . ∗ ( M , φ l ) (M,\varphi _1) * ... * (M,\varphi _l) are all mutually non-isomorphic.

Let A A be a von Neumann algebra with a faithful normal state ϕ \phi . It is shown that if a sequence ( x n ) ({x_n}) in A A is orthogonal relative to ϕ \phi and satisfies the condition \[ ∑ k ϕ ( | x k | 2 ) ( log ⁡ k k ) 2 > ∞ , \sum \limits _{k} {\phi (|{x_k}{|^2}){{\left ( {\frac {{\log k}}{k}} \right )}^2} > \infty ,} \] then n 1 ∑ k = 1 n x k → 0 {}_n^1\sum \nolimits _{k = 1}^n {{x_k} \to 0} almost uniformly in A A . Some other results related to this theorem are also discussed.

We investigate martingale inequalities\nin noncommutative $L^p$-spaces associated with a von Neumann\nalgebra equipped with a faithful normal state. We prove the\nnoncommutative analogue of the classical Burkholder inequality\non the conditioned (or little) square function and extend the\nnoncommutative Burkholder--Gundy inequalities from Comm. Math. Phys.\n189 (1997) 667--698 to this\nnontracial setting. We include several related results.