PASSIVITY OF GROUND STATES OF QUANTUM SYSTEMS

2005 ◽  
Vol 17 (01) ◽  
pp. 1-14 ◽  
Author(s):  
WALTER F. WRESZINSKI
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Ground State ◽  
Quantum Systems ◽  
Point Of View ◽  
Cyclic Vector ◽  
Vector State ◽  
Unitary Evolution ◽  
C Algebra ◽  
Local Perturbations

We consider a quantum system described by a concrete C*-algebra acting on a Hilbert space ℋ with a vector state ω induced by a cyclic vector Ω and a unitary evolution Ut such that UtΩ = Ω, ∀t ∈ ℝ. It is proved that this vector state is a ground state if and only if it is non-faithful and completely passive. This version of a result of Pusz and Woronowicz is reviewed, emphasizing other related aspects: passivity from the point of view of moving observers and stability with respect to local perturbations of the dynamics.

scholarly journals Simulating molecules on a cloud-based 5-qubit IBM-Q universal quantum computer

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
S. Leontica ◽  
F. Tennie ◽  
T. Farrow
Keyword(s):  
Quantum System ◽  
Energy Transport ◽  
Quantum Computer ◽  
Complex Molecule ◽  
Dissipative System ◽  
Quantum Simulation ◽  
Quantum Systems ◽  
Molecular Structures ◽  
Unitary Evolution ◽  
Universal Quantum

AbstractSimulating the behaviour of complex quantum systems is impossible on classical supercomputers due to the exponential scaling of the number of quantum states with the number of particles in the simulated system. Quantum computers aim to break through this limit by using one quantum system to simulate another quantum system. Although in their infancy, they are a promising tool for applied fields seeking to simulate quantum interactions in complex atomic and molecular structures. Here, we show an efficient technique for transpiling the unitary evolution of quantum systems into the language of universal quantum computation using the IBM quantum computer and show that it is a viable tool for compiling near-term quantum simulation algorithms. We develop code that decomposes arbitrary 3-qubit gates and implement it in a quantum simulation first for a linear ordered chain to highlight the generality of the approach, and second, for a complex molecule. We choose the Fenna-Matthews-Olsen (FMO) photosynthetic protein because it has a well characterised Hamiltonian and presents a complex dissipative system coupled to a noisy environment that helps to improve the efficiency of energy transport. The method can be implemented in a broad range of molecular and other simulation settings.

scholarly journals Paths of unitary access to exceptional points

2021 ◽  
Vol 2038 (1) ◽  
pp. 012026
Author(s):  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Phase Transitions ◽  
Quantum Phase Transitions ◽  
Quantum Systems ◽  
Point Of View ◽  
Explicit Description ◽  
Direct Access ◽  
Exceptional Point ◽  
Exceptional Points ◽  
Innovative Idea

Abstract With an innovative idea of acceptability and usefulness of the non-Hermitian representations of Hamiltonians for the description of unitary quantum systems (dating back to the Dyson’s papers), the community of quantum physicists was offered a new and powerful tool for the building of models of quantum phase transitions. In this paper the mechanism of such transitions is discussed from the point of view of mathematics. The emergence of the direct access to the instant of transition (i.e., to the Kato’s exceptional point) is attributed to the underlying split of several roles played by the traditional single Hilbert space of states ℒ into a triplet (viz., in our notation, spaces K and ℋ besides the conventional ℒ ). Although this explains the abrupt, quantum-catastrophic nature of the change of phase (i.e., the loss of observability) caused by an infinitesimal change of parameters, the explicit description of the unitarity-preserving corridors of access to the phenomenologically relevant exceptional points remained unclear. In the paper some of the recent results in this direction are summarized and critically reviewed.

scholarly journals Experimental classical entanglement in a 16 acoustic qubit-analogue

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
M. Arif Hasan ◽  
Keith Runge ◽  
Pierre A. Deymier
Keyword(s):  
Hilbert Space ◽  
Acoustic Field ◽  
Information Science ◽  
Physical Space ◽  
Quantum Systems ◽  
Vector State ◽  
State Representation ◽  
Nonlinear Acoustic ◽  
Spectral Partitioning ◽  
Coupled Waveguides

AbstractThe possibility of achieving and controlling scalable classically entangled, i.e., inseparable, multipartite states, would fundamentally challenge the advantages of quantum systems in harnessing the power of complexity in information science. Here, we investigate experimentally the extent of classical entanglement in a $$16$$ 16 acoustic qubit-analogue platform. The acoustic qubit-analogue, a.k.a., logical phi-bit, results from the spectral partitioning of the nonlinear acoustic field of externally driven coupled waveguides. Each logical phi-bit is a two-level subsystem characterized by two independently measurable phases. The phi-bits are co-located within the same physical space enabling distance independent interactions. We chose a vector state representation of the $$16$$ 16 -phi-bit system which lies in a $${2}^{16}$$ 2 16 -dimensional Hilbert space. The calculation of the entropy of entanglement demonstrates the possibility of achieving inseparability of the vector state and of navigating the corresponding Hilbert space. This work suggests a new direction in harnessing the complexity of classical inseparability in information science.

Reflexive Representations and Banach C*-Modules

1997 ◽  
Vol 40 (4) ◽  
pp. 443-447
Author(s):  
Don Hadwin ◽  
Mehmet Orhon
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Reflexive Banach Space ◽  
Algebra Homomorphism ◽  
Cyclic Vector ◽  
C Algebra

AbstractSuppose A is a unital C*-algebra and m:A → B(X) is unital bounded algebra homomorphism where B(X) is the algebra of all operators on a Banach space X. When X is a Hilbert space, a problem of Kadison [9] asks whether m is similar to a *-homomorphism. Haagerup [5] has shown that the answer is positive when m(A) has a cyclic vector or whenever m is completely bounded. We use this to show m(A) is reflexive (Alg Lat m(A) = m(A)−sot) whenever X is a Hilbert space. Our main result is that whenever A is a separable GCR C*-algebra and X is a reflexive Banach space, then m(A) is reflexive.

scholarly journals Unitary unfoldings of a Bose–Hubbard exceptional point with and without particle number conservation

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200292
Author(s):  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Particle Number ◽  
Quantum Systems ◽  
Exceptional Point ◽  
Dynamical Regime ◽  
Hubbard Hamiltonian ◽  
Number Conservation ◽  
Particle Number Conservation

The conventional non-Hermitian but P T -symmetric three-parametric Bose–Hubbard Hamiltonian H ( γ , v , c ) represents a quantum system of N bosons, unitary only for parameters γ , v and c in a domain D . Its boundary ∂ D contains an exceptional point of order K (EPK; K  =  N  + 1) at c  = 0 and γ  =  v , but even at the smallest non-vanishing parameter c  ≠ ~0 the spectrum of H ( v , v , c ) ceases to be real, i.e. the system ceases to be observable. In this paper, the question is inverted: all of the stable, unitary and observable Bose–Hubbard quantum systems are sought which would lie close to the phenomenologically most interesting EPK-related dynamical regime. Two different families of such systems are found. Both of them are characterized by the perturbed Hamiltonians H ( λ ) = H ( v , v , 0 ) + λ   V for which the unitarity and stability of the system is guaranteed. In the first family the number N of bosons is assumed conserved while in the second family such an assumption is relaxed. Attention is paid mainly to an anisotropy of the physical Hilbert space near the EPK extreme. We show that it is reflected by a specific, operationally realizable structure of perturbations λ   V which can be considered small.

scholarly journals Teleportation in an indivisible quantum system

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
E.O. Kiktenko ◽  
A.K. Fedorov ◽  
V.I. Man’ko
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Quantum State ◽  
Quantum Systems ◽  
High Dimensional ◽  
Dimensional System ◽  
Quantum State Transfer ◽  
State Transfer ◽  
The One ◽  
New Framework

AbstractTeleportation protocol is conventionally treated as a method for quantum state transfer between two spatially separated physical carriers. Recent experimental progress in manipulation with high-dimensional quantum systems opens a new framework for implementation of teleportation protocols. We show that the one-qubit teleportation can be considered as a state transfer between subspaces of the whole Hilbert space of an indivisible eight-dimensional system. We explicitly show all corresponding operations and discuss an alternative way of implementation of similar tasks.

scholarly journals Length filtration of the separable states

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160350 ◽  
Author(s):  
Lin Chen ◽  
Dragomir Ž. Ðoković
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Simple Formula ◽  
Jacobian Matrix ◽  
Critical Length ◽  
Quantum Systems ◽  
Separable States ◽  
Single Qubit ◽  
Finite Dimensional ◽  
Pure Product

We investigate the separable states ρ of an arbitrary multi-partite quantum system with Hilbert space H of dimension d . The length L ( ρ ) of ρ is defined as the smallest number of pure product states having ρ as their mixture. The length filtration of the set of separable states, S , is the increasing chain ∅ ⊊ S 1 ′ ⊆ S 2 ′ ⊆ ⋯ , where S i ′ = { ρ ∈ S : L ( ρ ) ≤ i } . We define the maximum length, L max = max ρ ∈ S L ( ρ ) , critical length, L crit , and yet another special length, L c , which was defined by a simple formula in one of our previous papers. The critical length indicates the first term in the length filtration whose dimension is equal to Dim   S . We show that in general d ≤ L c ≤ L crit ≤ L max ≤ d 2 . We conjecture that the equality L crit = L c holds for all finite-dimensional multi-partite quantum systems. Our main result is that L crit = L c for the bipartite systems having a single qubit as one of the parties. This is accomplished by computing the rank of the Jacobian matrix of a suitable map having S as its range.

scholarly journals A geometric Hamiltonian description of composite quantum systems and quantum entanglement

2015 ◽  
Vol 12 (07) ◽  
pp. 1550069 ◽  
Author(s):  
Davide Pastorello
Keyword(s):  
Hilbert Space ◽  
Phase Space ◽  
Quantum Entanglement ◽  
Hamiltonian Formulation ◽  
Classical Mechanics ◽  
Hamiltonian Formalism ◽  
Quantum Systems ◽  
Point Of View ◽  
Entanglement Measure ◽  
Hamiltonian Description

Finite-dimensional Quantum Mechanics can be geometrically formulated as a proper classical-like Hamiltonian theory in a projective Hilbert space. The description of composite quantum systems within the geometric Hamiltonian framework is discussed in this paper. As summarized in the first part of this work, in the Hamiltonian formulation the phase space of a quantum system is the Kähler manifold given by the complex projective space P(H) of the Hilbert space H of the considered quantum theory. However the phase space of a bipartite system must be P(H1 ⊗ H2) and not simply P(H1) × P(H2) as suggested by the analogy with Classical Mechanics. A part of this paper is devoted to manage this problem. In the second part of the work, a definition of quantum entanglement and a proposal of entanglement measure are given in terms of a geometrical point of view (a rather studied topic in recent literature). Finally two known separability criteria are implemented in the Hamiltonian formalism.

scholarly journals QSES's and the Quantum Jump

2001 ◽  
Vol 56 (1-2) ◽  
pp. 228-229
Author(s):  
David Salgado ◽  
José Luis Sánchez-Gómez
Keyword(s):  
Hilbert Space ◽  
Markov Property ◽  
Quantum Systems ◽  
Point Of View ◽  
Stochastic Methods ◽  
Stochastic Evolution ◽  
Quantum Jumps ◽  
Practical Point ◽  
Quantum Jump ◽  
Evolution Scheme

Abstract The stochastic methods in Hilbert space have been used both from a fundamental and a practical point of view. The result reported here concerns the application of these methods to model the evolution of quantum systems and does not enter into the question of their fundamental or practical status. It can easily be stated as follows: Once a quantum stochastic evolution scheme is assumed, the incompatibility of the Markov property with the notion of quantum jumps is easily established

Ground state molecular parameters of phosphine, PH3, from the simultaneous analysis of the high-resolution infrared, microwave and submillimeterwave spectra

1985 ◽  
Vol 50 (11) ◽  
pp. 2480-2492 ◽  
Author(s):  
Soňa Přádná ◽  
Dušan Papoušek ◽  
Jyrki Kauppinen ◽  
Sergei P. Belov ◽  
Andrei F. Krupnov ◽  
...  
Keyword(s):  
Fourier Transform ◽  
High Resolution ◽  
Ground State ◽  
Unitary Transformation ◽  
Point Of View ◽  
Acoustic Detection ◽  
Molecular Parameters ◽  
Microwave Spectrometer ◽  
Fourier Transform Spectra ◽  
First Time

Fourier transform spectra of the ν2 band of PH3 have been remeasured with 0.0045 cm-1 resolution. Ground state combination differences from these data have been fitted simultaneously with the microwave and submillimeterwave data to determine the ground state spectroscopical parameters of PH3 including the parameters of the Δk = ± 3n interactions. The correlation between the latter parameters has been discussed from the point of view of the existence of two equivalent effective rotational operators which are related by a unitary transformation. The ΔJ = 0, +1, ΔK = 0 (A1 ↔ A2, E ↔ E) rotational transitions in the ν2 and ν4 states have been measured for the first time by using a microwave spectrometer and a radiofrequency spectrometer with acoustic detection.

Sign in / Sign up

Export Citation Format

Share Document