scholarly journals On Bell’s Inequality in PT-Symmetric Quantum Systems

2021 ◽  
Vol 3 (3) ◽  
pp. 417-424
Author(s):  
Sarang S. Bhosale ◽  
Biswanath Rath ◽  
Prasanta K. Panigrahi
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Systems ◽  
Inner Product ◽  
Bell’S Inequality ◽  
Standard Quantum ◽  
Bell's Inequality ◽  
Qubit System ◽  
Symmetric Quantum ◽  
No Signaling

Bell’s inequality is investigated in parity-time (PT) symmetric quantum mechanics, using a recently developed form of the inequality by Maccone [Am. J. Phys. 81, 854 (2013) ] , with two PT-qubits in the unbroken phase with real energy spectrum. It is shown that the inequality produces a bound that is consistent with the standard quantum mechanics even after using Hilbert space equipped with CPT inner product and therefore, the entanglement has identical structure with standard quantum mechanics. Consequently, the no-signaling principle for a two-qubit system in PT-symmetric quantum theory is preserved.

scholarly journals Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 471 ◽  
Author(s):  
Ali Mostafazadeh
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Hermitian Operator ◽  
Quantum Systems ◽  
Time Dependent ◽  
Inner Product ◽  
Geometric Framework ◽  
Heisenberg Picture ◽  
Metric Operator ◽  
Recent Developments

A non-Hermitian operator H defined in a Hilbert space with inner product ⟨ · | · ⟩ may serve as the Hamiltonian for a unitary quantum system if it is η -pseudo-Hermitian for a metric operator (positive-definite automorphism) η . The latter defines the inner product ⟨ · | η · ⟩ of the physical Hilbert space H η of the system. For situations where some of the eigenstates of H depend on time, η becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM.

NONLOCAL MEASUREMENTS IN THE TIME-SYMMETRIC QUANTUM MECHANICS

2006 ◽  
Vol 20 (11n13) ◽  
pp. 1528-1535 ◽  
Author(s):  
LEV VAIDMAN ◽  
IZHAR NEVO
Keyword(s):  
Quantum Mechanics ◽  
Quantum Systems ◽  
Quantum Measurements ◽  
Quantum States ◽  
Standard Quantum ◽  
Time Direction ◽  
Quantum Formalism ◽  
Symmetric Quantum ◽  
Time Symmetric Quantum Mechanics

Although for some nonlocal variables the standard quantum measurements which are reliable, instantaneous, and nondemolition, are impossible, demolition reliable instantaneous measurements of all variables are possible. It is shown that this is correct also in the framework of the time-symmetric quantum formalism, i.e. nonlocal variables of composite quantum systems with quantum states evolving both forward and backward in time are measurable in a demolition way. The result follows from the possibility to reverse with certainty the time direction of backward evolving quantum states.

scholarly journals A study of quantum correlations in open quantum systems

2011 ◽  
Vol 11 (7&8) ◽  
pp. 541-562
Author(s):  
Indranil Chakrabarty ◽  
Subhashish Banerjee ◽  
Nana Siddharth
Keyword(s):  
Open Quantum Systems ◽  
Quantum Correlations ◽  
Quantum Systems ◽  
Bell’S Inequality ◽  
Mixed States ◽  
Classical Correlation ◽  
Bell's Inequality ◽  
Qubit System ◽  
Dynamical Parameters ◽  
Potential Resource

In this work, we study quantum correlations in mixed states. The states studied are modeled by a two-qubit system interacting with its environment via a quantum non demolition (purely dephasing) as well as dissipative type of interaction. The entanglement dynamics of this two qubit system is analyzed. We make a comparative study of various measures of quantum correlations, like Concurrence, Bell's inequality, Discord and Teleportation fidelity, on these states, generated by the above evolutions. We classify these evoluted states on basis of various dynamical parameters like bath squeezing parameter $r$, inter-qubit spacing $r_{12}$, temperature $T$ and time of system-bath evolution $t$. In this study, in addition we report the existence of entangled states which do not violate Bell's inequality, but can still be useful as a potential resource for teleportation. Moreover we study the dynamics of quantum as well as classical correlation in presence of dissipative coherence.

scholarly journals Buonomano against Bell: Nonergodicity or nonlocality?

2017 ◽  
Vol 15 (08) ◽  
pp. 1740010 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Quantum Mechanics ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Bell Inequality ◽  
Quantum Measurements ◽  
Neutron Interferometry ◽  
Bell’S Inequality ◽  
Weak Mixing ◽  
Bell's Inequality

The aim of this note is to attract attention of the quantum foundational community to the fact that in Bell’s arguments, one cannot distinguish two hypotheses: (a) quantum mechanics is nonlocal, (b) quantum mechanics is nonergodic. Therefore, experimental violations of Bell’s inequality can be as well interpreted as supporting the hypothesis that stochastic processes induced by quantum measurements are nonergodic. The latter hypothesis was discussed actively by Buonomano since 1980. However, in contrast to Bell’s hypothesis on nonlocality, it did not attract so much attention. The only experiment testing the hypothesis on nonergodicity was performed in neutron interferometry (by Summhammer, in 1989). This experiment can be considered as rejecting this hypothesis. However, it cannot be considered as a decisive experiment. New experiments are badly needed. We point out that a nonergodic model can be realistic, i.e. the distribution of hidden (local!) variables is well-defined. We also discuss coupling of violation of the Bell inequality with violation of the condition of weak mixing for ergodic dynamical systems.

scholarly journals Tsirelson's bound and supersymmetric entangled states

2014 ◽  
Vol 470 (2170) ◽  
pp. 20140253 ◽  
Author(s):  
L. Borsten ◽  
K. Brádler ◽  
M. J. Duff
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Transition Probabilities ◽  
Entangled States ◽  
Supersymmetric Extension ◽  
Standard Quantum ◽  
Local Resource ◽  
Non Local ◽  
Minimal Supersymmetric Extension ◽  
Tsirelson Bound

A superqubit, belonging to a (2|1)-dimensional super-Hilbert space, constitutes the minimal supersymmetric extension of the conventional qubit. In order to see whether superqubits are more non-local than ordinary qubits, we construct a class of two-superqubit entangled states as a non-local resource in the CHSH game. Since super Hilbert space amplitudes are Grassmann numbers, the result depends on how we extract real probabilities and we examine three choices of map: (1) DeWitt (2) Trigonometric and (3) Modified Rogers. In cases (1) and (2), the winning probability reaches the Tsirelson bound p win = cos 2 π / 8 ≃ 0.8536 of standard quantum mechanics. Case (3) crosses Tsirelson's bound with p win ≃0.9265. Although all states used in the game involve probabilities lying between 0 and 1, case (3) permits other changes of basis inducing negative transition probabilities.

scholarly journals The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics

2018 ◽  
Vol 474 (2217) ◽  
pp. 20180264 ◽  
Author(s):  
David Krejčiřík ◽  
Vladimir Lotoreichik ◽  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Adjoint Operator ◽  
Inner Product ◽  
Sufficient Condition ◽  
Lagrange Equations ◽  
Entire Class ◽  
Inner Products ◽  
Metric Operator ◽  
A Charge

We propose a unique way to choose a new inner product in a Hilbert space with respect to which an originally non-self-adjoint operator similar to a self-adjoint operator becomes self-adjoint. Our construction is based on minimizing a ‘Hilbert–Schmidt distance’ to the original inner product among the entire class of admissible inner products. We prove that either the minimizer exists and is unique or it does not exist at all. In the former case, we derive a system of Euler–Lagrange equations by which the optimal inner product is determined. A sufficient condition for the existence of the unique minimally anisotropic metric is obtained. The abstract results are supported by examples in which the optimal inner product does not coincide with the most popular choice fixed through a charge-like symmetry.

scholarly journals Remarks on the preservation of no-signaling principle in parity-time-symmetric quantum mechanics

2020 ◽  
Vol 35 (12) ◽  
pp. 2050090
Author(s):  
Bijan Bagchi ◽  
Suvendu Barik
Keyword(s):  
Quantum Mechanics ◽  
Density Matrix ◽  
The Other ◽  
Reduced Density Matrix ◽  
Entanglement Generation ◽  
Symmetric Quantum ◽  
No Signaling ◽  
Time Symmetric Quantum Mechanics ◽  
Reduced Density

Working within the framework of parity-time-symmetric quantum mechanics, we look into the possibility of entanglement generation and demonstrate that the feature of non-violation of no-signaling principle may hold for the simplest nontrivial case of bipartite systems. Basically, our arguments are based on the computation of the reduced density matrix of one party to justify that the entropy of the other does not change.

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