scholarly journals Is the Quantum State Real in the Hilbert Space Formulation?

Quanta ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 37-46
Author(s):  
Mani L. Bhaumik
Keyword(s):  
Hilbert Space ◽  
Quantum State ◽  
Energy Levels ◽  
Quantum States ◽  
Direct Perception ◽  
Quantum Fields ◽  
Von Neumann ◽  
Space Formulation ◽  
Atomic Energy Levels ◽  
Almost All

The persistent debate about the reality of a quantum state has recently come under limelight because of its importance to quantum information and the quantum computing community. Almost all of the deliberations are taking place using the elegant and powerful but abstract Hilbert space formalism of quantum mechanics developed with seminal contributions from John von Neumann. Since it is rather difficult to get a direct perception of the events in an abstract vector space, it is hard to trace the progress of a phenomenon. Among the multitude of recent attempts to show the reality of the quantum state in Hilbert space, the Pusey–Barrett–Rudolph theory gets most recognition for their proof. But some of its assumptions have been criticized, which are still not considered to be entirely loophole free. A straightforward proof of the reality of the wave packet function of a single particle has been presented earlier based on the currently recognized fundamental reality of the universal quantum fields. Quantum states like the atomic energy levels comprising the wave packets have been shown to be just as real. Here we show that an unambiguous proof of reality of the quantum states gleaned from the reality of quantum fields can also provide an explicit substantiation of the reality of quantum states in Hilbert space.Quanta 2020; 9: 37–46.

scholarly journals Revisiting Online Quantum State Learning

2020 ◽  
Vol 34 (04) ◽  
pp. 6607-6614
Author(s):  
Feidiao Yang ◽  
Jiaqing Jiang ◽  
Jialin Zhang ◽  
Xiaoming Sun
Keyword(s):  
Theoretical Analysis ◽  
Quantum State ◽  
Learning Algorithm ◽  
Closed Form Solution ◽  
Form Solution ◽  
Quantum States ◽  
Frobenius Norm ◽  
Deterministic Learning ◽  
State Learning ◽  
Almost All

In this paper, we study the online quantum state learning problem which is recently proposed by Aaronson et al. (2018). In this problem, the learning algorithm sequentially predicts quantum states based on observed measurements and losses and the goal is to minimize the regret. In the previous work, the existing algorithms may output mixed quantum states. However, in many scenarios, the prediction of a pure quantum state is required. In this paper, we first propose a Follow-the-Perturbed-Leader (FTPL) algorithm that can guarantee to predict pure quantum states. Theoretical analysis shows that our algorithm can achieve an O(√T) expected regret under some reasonable settings. In the case that the pure state prediction is not mandatory, we propose another deterministic learning algorithm which is simpler and more efficient. The algorithm is based on the online gradient descent (OGD) method and can also achieve an O(√T) regret bound. The main technical contribution of this result is an algorithm of projecting an arbitrary Hermitian matrix onto the set of density matrices with respect to the Frobenius norm. We think this subroutine is of independent interest and can be widely used in many other problems in the quantum computing area. In addition to the theoretical analysis, we evaluate the algorithms with a series of simulation experiments. The experimental results show that our FTPL method and OGD method outperform the existing RFTL approach proposed by Aaronson et al. (2018) in almost all settings. In the implementation of the RFTL approach, we give a closed-form solution to the algorithm. This provides an efficient, accurate, and completely executable solution to the RFTL method.

Unifying with Mathematics

2018 ◽  
Author(s):  
Otávio Bueno ◽  
Steven French
Keyword(s):  
Hilbert Space ◽  
Degrees Of Freedom ◽  
Von Neumann Algebras ◽  
Quantum States ◽  
Logical Inference ◽  
Top Down ◽  
Von Neumann ◽  
Mathematical Structures ◽  
Space Formalism

In this chapter, we examine a different case study, where the aim was to unify apparently unrelated domains, such as quantum states, probability assignments, and logical inference. This is John von Neumann’s development of an alternative framework to the Hilbert space formalism he pioneered: one articulated in terms of his theory of operators and what we now call von Neumann algebras. This allowed him to accommodate probabilities in the context of systems with infinite degrees of freedom. Here we find, in addition to ‘top-down’ moves from the mathematics to the physics, ‘bottom-up’ developments from empirical features, to a particular logic and thence to mathematical structures. Through a combination of such moves, crucially involving exploration of the structural relations that hold between the mathematical and the physical domains, von Neumann articulated the kind of unification across such domains that represents a further important aspect of the application of mathematics.

scholarly journals Quantum Mechanics and Its Evolving Formulations

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 124
Author(s):  
Jean-Pierre Antoine
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Time Evolution ◽  
Von Neumann ◽  
Rigged Hilbert Space ◽  
Space Formulation ◽  
Space Approach

In this paper, we discuss the time evolution of the quantum mechanics formalism. Starting from the heroic beginnings of Heisenberg and Schrödinger, we cover successively the rigorous Hilbert space formulation of von Neumann, the practical bra-ket formalism of Dirac, and the more recent rigged Hilbert space approach.

scholarly journals Lagrangian description of Heisenberg and Landau–von Neumann equations of motion

2020 ◽  
Vol 35 (19) ◽  
pp. 2050161
Author(s):  
F. M. Ciaglia ◽  
F. Di Cosmo ◽  
A. Ibort ◽  
G. Marmo ◽  
L. Schiavone ◽  
...  
Keyword(s):  
Quantum System ◽  
Quantum State ◽  
Equations Of Motion ◽  
Quantum States ◽  
Heisenberg Equation ◽  
Lagrangian Description ◽  
Von Neumann ◽  
Fixed Reference ◽  
Von Neumann Equation ◽  
Algebra Of Operators

An explicit Lagrangian description is given for the Heisenberg equation on the algebra of operators of a quantum system, and for the Landau–von Neumann equation on the manifold of quantum states which are isospectral with respect to a fixed reference quantum state.

AN EFFICIENT SEPARABILITY CRITERION FOR n-PARTITE ARBITRARILY DIMENSIONAL QUANTUM STATES

2011 ◽  
Vol 09 (04) ◽  
pp. 1101-1112
Author(s):  
YINXIANG LONG ◽  
DAOWEN QIU ◽  
DONGYANG LONG
Keyword(s):  
Hilbert Space ◽  
Quantum State ◽  
Pure State ◽  
Coefficient Matrix ◽  
Quantum States ◽  
Mixed States ◽  
Density Matrices ◽  
Separability Criterion ◽  
Dimensional Hilbert Space ◽  
Pure States

In this paper, we obtain an efficient separability criterion for bipartite quantum pure state systems, which is based on the two-order minors of the coefficient matrix corresponding to quantum state. Then, we generalize this criterion to multipartite arbitrarily dimensional pure states. Our criterion is directly built upon coefficient matrices, but not density matrices or observables, so it has the advantage of being computed easily. Indeed, to judge separability for an arbitrary n-partite pure state in a d-dimensional Hilbert space, it only needs at most O(d) times operations of multiplication and comparison. Our criterion can be extended to mixed states. Compared with Yu's criteria, our methods are faster, and can be applied to any quantum state.

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

scholarly journals Gaussian quantum states can be disentangled using symplectic rotations

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Maurice A. de Gosson
Keyword(s):  
Quantum State ◽  
Covariance Matrices ◽  
Quantum States ◽  
Weyl Quantization ◽  
Symplectic Transformation ◽  
Metaplectic Operator

AbstractWe show that every Gaussian mixed quantum state can be disentangled by conjugation with a passive symplectic transformation, that is a metaplectic operator associated with a symplectic rotation. The main tools we use are the Werner–Wolf condition on covariance matrices and the symplectic covariance of Weyl quantization. Our result therefore complements a recent study by Lami, Serafini, and Adesso.

scholarly journals Zero uncertainty states in the presence of quantum memory

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Huangjun Zhu
Keyword(s):  
Uncertainty Principle ◽  
Quantum States ◽  
Quantum Memory ◽  
System State ◽  
Von Neumann ◽  
Practical Applications ◽  
Classical Information ◽  
Fundamental Limit ◽  
Minimum Uncertainty ◽  
Incompatible Observables

AbstractThe uncertainty principle imposes a fundamental limit on predicting the measurement outcomes of incompatible observables even if complete classical information of the system state is known. The situation is different if one can build a quantum memory entangled with the system. Zero uncertainty states (in contrast with minimum uncertainty states) are peculiar quantum states that can eliminate uncertainties of incompatible von Neumann observables once assisted by suitable measurements on the memory. Here we determine all zero uncertainty states of any given set of nondegenerate observables and determine the minimum entanglement required. It turns out all zero uncertainty states are maximally entangled in a generic case, and vice versa, even if these observables are only weakly incompatible. Our work establishes a simple and precise connection between zero uncertainty and maximum entanglement, which is of interest to foundational studies and practical applications, including quantum certification and verification.

scholarly journals Purification and redistribution of entanglement via single local filtering

2014 ◽  
Vol 12 (01) ◽  
pp. 1450004 ◽  
Author(s):  
K. O. Yashodamma ◽  
P. J. Geetha ◽  
Sudha
Keyword(s):  
Quantum State ◽  
The State ◽  
Quantum States ◽  
Local Action ◽  
Entanglement Transfer ◽  
Local Filtering

The effect of filtering operation with respect to purification and concentration of entanglement in quantum states are discussed in this paper. It is shown, through examples, that the local action of the filtering operator on a part of the composite quantum state allows for purification of the remaining part of the state. The redistribution of entanglement in the subsystems of a noise affected state is shown to be due to the action of local filtering on the non-decohering part of the system. The varying effects of the filtering parameter, on the entanglement transfer between the subsystems, depending on the choice of the initial quantum state is illustrated.

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