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arXiv:1711.06817

The broken $\cal PT$ -symmetric hamiltonians and the weak measurement

Minyi Huang, Junde Wu

Published Nov 18, 2017• 0 citations
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Abstract

In this paper, we discuss the relation of broken $\cal PT$-symmetric hamiltonians and the weak values. It was showed that a broken $\cal PT$-symmetric quantum system can be simulated by weak measurement in a large conventional quantum system. Moreover, such a paradigm reduces to the embedding paradigm when the system is unbroken $\cal PT$-symmetric.

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The broken PT\cal PTPT -symmetric hamiltonians and the weak measurement
In this paper, we discuss the relation of broken PT\cal PTPT-symmetric hamiltonians and the weak values. It was showed that a broken PT\cal PTPT-symmetric quantum system can be simulated by weak measurement in a large conventional quantum system. Moreover, such a paradigm reduces to the embedding paradigm when the system is unbroken PT\cal PTPT-symmetric.
By embedding a PT-symmetric (pseudo-Hermitian) system into a large Hermitian one, we disclose the relations between PT-symmetric quantum theory and weak measurement theory. We show that the weak measurement can give rise to the inner product structure of PT-symmetric systems, with the preselected state and its postselected state resident in the dilated conventional system. Typically in quantum information theory, by projecting out the irrelevant degrees and projecting onto the subspace, even local broken PT-symmetric Hamiltonian systems can be effectively simulated by this weak measurement paradigm.
The Embedding Property and the Consistency Problem of PT\cal PTPT -symmetric Theory
In this paper, we show that a PT\cal PTPT-symmetric hamiltonian can be embedded into a hermitian hamiltonian if and only if it is unbroken. Moreover, by utilizing the embedding property, we can settle the consistency problem of the PT\cal PTPT-symmetry theory completely within the scope of conventional quantum mechanics. In particular, we show how a violation of the no-signaling principle arises in the PT\cal PTPT-symmetric quantum theory and fits into the existing theories.
If a quantum-mechanical Hamiltonian is PT symmetric, there are two possibilities: either all of the eigenvalues are real, in which case the Hamiltonian is said to be in an unbroken-PT-symmetric phase, or else some of the eigenvalues are real and some are complex, in which case the Hamiltonian is said to be in a broken-PT-symmetric phase. As one varies the parameters of the Hamiltonian, one can pass through the phase transition that separates the unbroken and broken phases. This transition has recently been observed in a variety of laboratory experiments. This paper explains the phase transition in a simple and intuitive fashion and then describes an elementary experiment in which the phase transition is easily observed.
In this report we examine the concept of PT -symmetric quantum mechanics and analyze two such systems. PT -symmetric quantum mechanics is a new branch of quantum mechanics, first proposed by Bender and Boettcher in 1998 and has been the subject of much research since then. Various important aspects of PT -symmetric quantum mechanics is explained, such as asymptotic behaviour of solutions of the Schrödinger equation, as well as the concept of the C operator that is unique to PT -symmetric quantum mechanics. We briefly discuss both the concept of broken and unbroken PT -symmetry, and the relation between PT -symmetric and Hermitian quantum mechanics through similarity transforms. We specifically direct our attention to two theoretical quantum mechanical systems described by Hamiltonians with the property of being PT -symmetric. For these systems we determine the energy levels, one by analytical means and the other by use of numerical methods. To determine the energy levels numerically we have used the builtin ODE-solver ode45 in MATLAB in conjunction with the equation solver fzero. Our results are compared to those of previous studies and our methods and results are discussed.
Abstract Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.