We show that the von Neumann's algorithm of reduction (i.e. the algorithm of calculating the density matrix of the observable subsystem from the density matrix of the closed quantum system) corresponds to the special approximation at which the unobservable subsystem is supposed to be in the steady state of minimum information (infinite temperature). We formulate the generalized algorithm of reduction that includes as limiting cases the von Neumann's reduction and the self-congruent correlated reduction most corresponding to the quantum nondemolition measurement. We demonstrate the correlation in dynamics of subsystems with exactly soluble models of quantum optics: 1) about the dynamics of a pair of interacting two-level atoms, and 2) about the dynamics of a two-level atom interacting with a single-mode resonant field.
In [PRL, 113, 050401 (2014)] the authors have shown that instead of L$\ddot{u}$der rule, if degeneracy breaking von Neumann projection rule is adopted for state reduction, the quantum value of three-time Leggett-Garg inequality can exceed it's L$\ddot{u}$ders bound. Such violation of L$\ddot{u}$ders bound may even approach algebraic maximum of the inequality in the asymptotic limit of system size. They also claim that for Clauser-Horne-Shimony-Holt (CHSH) inequality such violation of L$\ddot{u}$ders bound (known as Cirelson's bound) cannot be obtained even when the measurement is performed seq…
The Stochastic Liouville-von Neumann equation provides an exact numerical simulation strategy for quantum systems interacting with Gaussian reservoirs [J.T. Stockburger & H. Grabert, PRL 88, 170407 (2002)]. Its scaling with the extension of the time interval covered has recently improved dramatically through time-domain projection techniques [J.T. Stockburger, EPL 115, 40010 (2016)]. Here we present a sampling strategy which results in a significantly improved scaling with the strength of the dissipative interaction, based on reducing the non-unitary terms in sample propagation through convex …
We derive the projection hypothesis in projective quantum measurement by restricting the set of observables. This projection hypothesis accompanies a bipartite system with the von Neumann-type interaction, which consists of a quantum mechanical system, with a meter variable to be measured, and a quantum field theoretically macroscopic extended object, that is, a spatiotemporally inhomogeneous Bose-Einstein condensate in quantum field theory with the quantum coordinate, that is, the zero-energy Goldstone mode(s) of the spontaneously broken global spatial translational symmetry. The key steps in…
The von Neumann lattice refers to a discrete basis of Gaussians located on a lattice in phase space. It provides an attractive approach for solving quantum mechanical problems, allowing the pruning of tensor-product basis sets using phase space considerations. In a series of recent articles Shimshovitz et al. [Phys. Rev. Lett. 109 7 (2012)], Takemoto et al. [Journal of Chemical Physics 137 1 (2012)] Machnes et al. [Journal of Chemical Physics, accepted (2016)]), we have introduced two key new elements into the method: a formalism for converging the basis and for efficient pruning by use of the…
Entropy is a fundamental property of both classical and quantum systems, spanning myriad theoretical and practical applications in physics and computer science. We study the problem of obtaining estimates to within a multiplicative factor $γ>1$ of the Shannon entropy of probability distributions and the von Neumann entropy of mixed quantum states. Our main results are: $\quad\bullet$ an $\widetilde{\mathcal{O}}\left( n^{\frac{1+η}{2γ^2}}\right)$-query quantum algorithm that outputs a $γ$-multiplicative approximation of the Shannon entropy $H(\mathbf{p})$ of a classical probability distribution…
We revisit the pointer-based measurement concept of von Neumann which allows us to model a quantum counterpart of the classical time-of-flight (ToF) momentum. Our approach is based on the Hamiltonian for a particle interacting with two quantum pointers serving as basic measurement devices. The corresponding dynamics leads to a pointer-based ToF observable for the operational momentum of the particle. We can consider single measurements of our quantum pointers and show that this process will result in a state reduction for a single particle being downstream of the time-of-flight setup.
Von Neumann projections are the main operations by which information can be extracted from the quantum to the classical realm. They are however static processes that do not adapt to the states they measure. Advances in the field of adaptive measurement have shown that this limitation can be overcome by "wrapping" the von Neumann projectors in a higher-dimensional circuit which exploits the interplay between measurement outcomes and measurement settings. Unfortunately, the design of adaptive measurement has often been ad hoc and setup-specific. We shall here develop a unified framework for desi…
In this paper we study von Neumann un-biasing normalisation for ideal and real quantum random number generators, operating on finite strings or infinite bit sequences. In the ideal cases one can obtain the desired un-biasing. This relies critically on the independence of the source, a notion we rigorously define for our model. In real cases, affected by imperfections in measurement and hardware, one cannot achieve a true un-biasing, but, if the bias "drifts sufficiently slowly", the result can be arbitrarily close to un-biasing. For infinite sequences, normalisation can both increase or decrea…
Quantum computers can be used to simulate nonlinear non-Hamiltonian classical dynamics on phase space by using the generalized Koopman-von Neumann formulation of classical mechanics. The Koopman-von Neumann formulation implies that the conservation of the probability distribution function on phase space, as expressed by the Liouville equation, can be recast as an equivalent Schrödinger equation on Hilbert space with a Hermitian Hamiltonian operator and a unitary propagator. This Schrödinger equation is linear in the momenta because it derives from a constrained Hamiltonian system with twice th…
A modern computer system, based on the von Neumann architecture, is a complicated system with several interactive modular parts. Quantum computing, as the most generic usage of quantum information, follows a hybrid architecture so far, namely, quantum algorithms are stored and controlled classically, and mainly the executions of them are quantum, leading to the so-called quantum processing units. Such a quantum-classical hybrid is constrained by its classical ingredients, and cannot reveal the computational power of a fully quantum computer system as conceived from the beginning of the field. …
The entropy of a quantum system is a measure of its randomness, and has applications in measuring quantum entanglement. We study the problem of measuring the von Neumann entropy, $S(ρ)$, and Rényi entropy, $S_α(ρ)$ of an unknown mixed quantum state $ρ$ in $d$ dimensions, given access to independent copies of $ρ$. We provide an algorithm with copy complexity $O(d^{2/α})$ for estimating $S_α(ρ)$ for $α<1$, and copy complexity $O(d^{2})$ for estimating $S(ρ)$, and $S_α(ρ)$ for non-integral $α>1$. These bounds are at least quadratic in $d$, which is the order dependence on the number of copies req…
