scholarly journals Hilbert Space Structure Induced by Quantum Probes

Proceedings ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 4
Author(s):  
Kato ◽  
Owari ◽  
Maruyama
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Degrees Of Freedom ◽  
Quantum Control ◽  
Space Structure ◽  
Direct Access ◽  
High Dimensional ◽  
Direct Control ◽  
Physical Systems ◽  
Controllability And Observability

It is unrealistic to control all of the degrees of freedom of a high-dimensional quantum system. Here, we consider a scenario where our direct access is restricted to a small subsystem S that is constantly interacting with the rest of the system E. What we investigate is the fundamental structures of the Hilbert space and the algebra of hamiltonians that are caused solely by the restrictedness of the direct control. One key finding is that hamiltonians form a Jordan algebra, and this leads to a significant observation that there is a sharp distinction between the cases of dimHS ≥ 3 and dimHS = 2 in terms of the nature of possible operations in E. Since our analysis is totally free from specific properties of any physical systems, it would form a solid basis for obtaining deeper insights into quantum control related issues, such as controllability and observability.

scholarly journals QUANTUM BACKREACTION (CASIMIR) EFFECT WITHOUT INFINITIES: ALGEBRAIC ANALYSIS

2012 ◽  
Vol 14 ◽  
pp. 376-382
Author(s):  
ANDRZEJ HERDEGEN
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
Quantum System ◽  
Casimir Effect ◽  
Quantum Field ◽  
Expectation Value ◽  
Physical Systems ◽  
Algebra Of Observables ◽  
General Terms ◽  
External Conditions

Casimir effect, in most general terms, is the backreaction of a quantum system responding to an adiabatic change of external conditions. This backreaction is expected to be quantitatively measured by a change in the expectation value of a certain energy observable of the system. However, for this concept to be applicable, the system has to retain its identity in the process. Most prevailing tendencies in the analysis of the effect seem to ignore this question. In general, a quantum theory is defined by an algebra of observables, whose representations by operators in a Hilbert space define concrete physical systems described by the theory. A quantum system retains its identity if both the algebra as well as its representation do not change. We discuss the resulting restrictions for admissible models of changing external conditions. These ideas are applied to quantum field models. No infinities arise, if the algebraic demands are respected.

scholarly journals Spacetime Quantum Reference Frames and superpositions of proper times

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 508
Author(s):  
Flaminia Giacomini
Keyword(s):  
Hilbert Space ◽  
Reference Frame ◽  
Degrees Of Freedom ◽  
Reference Frames ◽  
Proper Time ◽  
Relativistic Quantum ◽  
Dynamical Evolution ◽  
Quantum Superposition ◽  
Local Frame ◽  
Physical Systems

In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.

scholarly journals Continuous and Pulsed Quantum Control

Proceedings ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 15
Author(s):  
Gramegna ◽  
Burgarth ◽  
Facchi ◽  
Pascazio
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Quantum Control ◽  
The Other ◽  
Control Potential ◽  
Finite Dimensional ◽  
Alternative Procedures

We consider two alternative procedures which can be used to control the evolution of a generic finite-dimensional quantum system, one hinging upon a strong continuous coupling with a control potential and the other based on the application of frequently repeated pulses onto the system. Despite the practical and conceptual difference between them, they lead to the same dynamics, characterised by a partitioning of the Hilbert space into sectors among which transitions are inhibited by dynamical superselection rules.

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 Quantum system compression: A Hamiltonian guided walk through Hilbert space

2021 ◽  
Vol 103 (1) ◽  
Author(s):  
Robert L. Kosut ◽  
Tak-San Ho ◽  
Herschel Rabitz
Keyword(s):  
Hilbert Space ◽  
Quantum System

scholarly journals Exact mapping between a laser network loss rate and the classical XY Hamiltonian by laser loss control

Nanophotonics ◽  
2020 ◽  
Vol 9 (13) ◽  
pp. 4117-4126 ◽  
Author(s):  
Igor Gershenzon ◽  
Geva Arwas ◽  
Sagie Gadasi ◽  
Chene Tradonsky ◽  
Asher Friesem ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Loss Rate ◽  
Oscillation Amplitude ◽  
Xy Model ◽  
Spin Hamiltonian ◽  
Laser Oscillator ◽  
Physical Systems ◽  
Network State ◽  
Spin Hamiltonians ◽  
Loss Control

AbstractRecently, there has been growing interest in the utilization of physical systems as heuristic optimizers for classical spin Hamiltonians. A prominent approach employs gain-dissipative optical oscillator networks for this purpose. Unfortunately, these systems inherently suffer from an inexact mapping between the oscillator network loss rate and the spin Hamiltonian due to additional degrees of freedom present in the system such as oscillation amplitude. In this work, we theoretically analyze and experimentally demonstrate a scheme for the alleviation of this difficulty. The scheme involves control over the laser oscillator amplitude through modification of individual laser oscillator loss. We demonstrate this approach in a laser network classical XY model simulator based on a digital degenerate cavity laser. We prove that for each XY model energy minimum there corresponds a unique set of laser loss values that leads to a network state with identical oscillation amplitudes and to phase values that coincide with the XY model minimum. We experimentally demonstrate an eight fold improvement in the deviation from the minimal XY energy by employing our proposed solution scheme.

scholarly journals Real algebras with a Hilbert space structure

1966 ◽  
Vol 6 (4-5) ◽  
pp. 459-465
Author(s):  
Lars Ingelstam
Keyword(s):  
Hilbert Space ◽  
Space Structure ◽  
Real Algebras

Computation and cognition: issues in the foundations of cognitive science

1980 ◽  
Vol 3 (1) ◽  
pp. 111-132 ◽  
Author(s):  
Zenon W. Pylyshyn
Keyword(s):  
Tacit Knowledge ◽  
Degrees Of Freedom ◽  
Ad Hoc ◽  
Physical Systems ◽  
Symbolic Representations ◽  
Natural Domain ◽  
Human Functioning ◽  
Biological Laws ◽  
Post Hoc ◽  
Level Of Analysis

AbstractThe computational view of mind rests on certain intuitions regarding the fundamental similarity between computation and cognition. We examine some of these intuitions and suggest that they derive from the fact that computers and human organisms are both physical systems whose behavior is correctly described as being governed by rules acting on symbolic representations. Some of the implications of this view are discussed. It is suggested that a fundamental hypothesis of this approach (the “proprietary vocabulary hypothesis”) is that there is a natural domain of human functioning (roughly what we intuitively associate with perceiving, reasoning, and acting) that can be addressed exclusively in terms of a formal symbolic or algorithmic vocabulary or level of analysis.Much of the paper elaborates various conditions that need to be met if a literal view of mental activity as computation is to serve as the basis for explanatory theories. The coherence of such a view depends on there being a principled distinction between functions whose explanation requires that we posit internal representations and those that we can appropriately describe as merely instantiating causal physical or biological laws. In this paper the distinction is empirically grounded in a methodological criterion called the “cognitive impenetrability condition.” Functions are said to be cognitively impenetrable if they cannot be influenced by such purely cognitive factors as goals, beliefs, inferences, tacit knowledge, and so on. Such a criterion makes it possible to empirically separate the fixed capacities of mind (called its “functional architecture”) from the particular representations and algorithms used on specific occasions. In order for computational theories to avoid being ad hoc, they must deal effectively with the “degrees of freedom” problem by constraining the extent to which they can be arbitrarily adjusted post hoc to fit some particular set of observations. This in turn requires that the fixed architectural function and the algorithms be independently validated. It is argued that the architectural assumptions implicit in many contemporary models run afoul of the cognitive impenetrability condition, since the required fixed functions are demonstrably sensitive to tacit knowledge and goals. The paper concludes with some tactical suggestions for the development of computational cognitive theories.

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