A physical requirement on the Hamiltonian operator in quantum mechanics is that it must generate real energy spectrum and unitary time evolution. While the Hamiltonians are Dirac Hermitian in conventional quantum mechanics, they observe P T -symmetry in P T -symmetric quantum theory.The embedding property was first studied by Günther and Samsonov [Phys. Rev. Lett. 101, 230404 (2008)] to visualise the evolution of unbroken P T -symmetric Hamiltonians on C 2 by Hermitian Hamiltonians on C 4 . This paper investigates the properties of P Tsymmetric quantum systems including the embedding property. We provide a full characterization of the embedding property in the general case and show that only unbroken P T -symmetric quantum systems admit this property in a finite dimensional space. Furthermore, utilizing this property, we establish a physically realizable simulation process of the unbroken P T -symmetric Hamiltonians. An observation that the unbroken P T -symmetric quantum systems can be viewed as open systems in the conventional quantum mechanics accounts for the consistency of P T -symmetric quantum theory.
A. Parity (P), Time reversal (T ) and P T -symmetric Hamiltonian (H)A linear operator P is said to be a parity operator if P 2 = I. And, an anti-linear operator T is said to
A fundamental task in any physical theory is to quantify certain physical quantity in a meaningful way. In this paper we show that both fidelity distance and affinity distance satisfy the strong contractibility, and the corresponding resource quantifiers can be used to characterize a large class of resource theories. Under two assumptions, namely, convexity of "free states" and closure of free states under "selective free operations", our general framework of resource theory includes quantum resource theories of entanglement, coherence, partial coherence and superposition. In partial coherence theory, we show that fidelity partial coherence of a bipartite state is equal to the minimal error probability of a mixed quantum state discrimination (QSD) task and vice versa, which complements the main result in [Xiong and Wu, J. Phys. A: Math. Theor. 51, 414005 (2018)]. We also compute the analytic expression of fidelity partial coherence for (2, n) bipartite X-states. At last, we study the correlated coherence in the framework of partial coherence theory. We show that partial coherence of a bipartite state, with respect to the eigenbasis of a subsystem, is actually a measure of quantum correlation. a family of resource quantifiers such as robustness monotones and norm-based quantifiers, etc.
By embedding a P T -symmetric (pseudo-Hermitian) system into a large Hermitian one, we disclose the relations between P T -symmetric quantum theory and weak measurement theory. We show that the weak measurement can give rise to the inner product structure of P T -symmetric systems, with the pre-selected state and its post-selected 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 P T -symmetric Hamiltonian systems can be effectively simulated by this weak measurement paradigm.
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