Non-Hermitian Hamiltonians and no-go theorems in quantum information

Ju, Chia-Yi; Miranowicz, Adam; Chen, Guang-Yin; Nori, Franco

doi:10.1103/physreva.100.062118

Cited by 100 publications

(83 citation statements)

References 81 publications

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“…For systems with Hermitian Hamiltonians, these probabilities satisfies the conditions of orthogonalities of two-dimensional eigenvectors of the 2 × 2 matrix H. For P T -symmetric Hamiltonians, which are non-Hermitian ones, the probabilities are such that the trace of pure-state density matrices ρ 1 = |ψ E 1 ψ E 1 | and ρ 2 = |ψ E 2 ψ E 2 |, i.e., k 12 = Tr (ρ 1 ρ 2 ), where E 1 and E 2 are real eigenvalues of the matrix H, is not equal to zero, and the value k 12 characterizes properties of the P T -symmetric system. The nonorthogonality of the non-Hermitian Hamiltonian eigenvectors associated with P T -symmetry properties of quantum systems was mentioned, e.g., in [19,27].…”

Section: Discussionmentioning

confidence: 99%

“…Employing Equations ( 19) and (20), we obtain a new form of the Schrödinger equation for probabilities p 1 , p 2 , p 3 , (1 − p 1 ), (1 − p 2 ), and (1 − p 3 ) determining stationary states of the spin-1/2 particle. First, we consider the Hermitian Hamiltonian for which in (19)…”

Section: The Schrödinger Equation For States With Eigenvalues Of Energy As Equations For Eigenvectors With Components-probabilities Determentioning

confidence: 99%

“…was considered in [4,5,19]. Here, we consider a particular example of the Hamiltonian with z = 1 + i; i.e., we have the energy levels…”

Section: An Example Of a Non-hermitian Hamiltonian With P T -Symmetrymentioning

confidence: 99%

“…The problems of non-Hermitian quantum mechanics were studied in [14][15][16][17][18]. The differences between conventional quantum mechanics and non-Hermitian quantum mechanics in the Hilbert-space representation of pure and mixed quantum states were analyzed in [19]. The applications of the P T -symmetry approach in quantum optics and physics of oscillators were discussed in [20].…”

Section: Introductionmentioning

confidence: 99%

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PT -Symmetric Qubit-System States in the Probability Representation of Quantum Mechanics

Chernega

Man’ko

2020

Symmetry

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PT-symmetric qubit-system states are considered in the probability representation of quantum mechanics. The new energy eigenvalue equation for probability distributions identified with qubit and qutrit states is presented in an explicit form. A possibility to test PT-symmetry and its violation by measuring the probabilities of spin projections for qubits in three perpendicular directions is discussed.

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Section: Discussionmentioning

confidence: 99%

Section: The Schrödinger Equation For States With Eigenvalues Of Energy As Equations For Eigenvectors With Components-probabilities Determentioning

confidence: 99%

“…was considered in [4,5,19]. Here, we consider a particular example of the Hamiltonian with z = 1 + i; i.e., we have the energy levels…”

Section: An Example Of a Non-hermitian Hamiltonian With P T -Symmetrymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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PT -Symmetric Qubit-System States in the Probability Representation of Quantum Mechanics

Chernega

Man’ko

2020

Symmetry

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“…To achieve this, we combine two techniques: a dilation method that is universal (applicable to any Hamiltonian) 12 and an optimal method for generating any two-qubit gate with combinations of single-qubit gates and at most three CNOT (controlled-NOT) gates 18,19 . This combination enables us to observe and fully characterize the broken PT -symmetry transition and to settle decisively the relationship between non-Hermitian quantum mechanics and no-go theorems on state distinguishability and monotony of entanglement [20][21][22][23] . We achieve this by making use of the emergent technology of superconducting processors, on which significant technical progress has been shown in recent times by IBM 24 .…”

mentioning

confidence: 99%

Quantum simulation of parity–time symmetry breaking with a superconducting quantum processor

2021

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The observation of genuine quantum effects in systems governed by non-Hermitian Hamiltonians has been an outstanding challenge in the field. Here we simulate the evolution under such Hamiltonians in the quantum regime on a superconducting quantum processor by using a dilation procedure involving an ancillary qubit. We observe the parity–time ($${\mathcal{PT}}$$ PT )-symmetry breaking phase transition at the exceptional points, obtain the critical exponent, and show that this transition is associated with a loss of state distinguishability. In a two-qubit setting, we show that the entanglement can be modified by local operations.

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Generalized Gauge Transformation with PT$PT$‐Symmetric Non‐Unitary Operator and Classical Correspondence of Non‐Hermitian Hamiltonian for a Periodically Driven System

Hao

Liang

2022

Annalen der Physik

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This paper demonstrates that the parity‐time (PT$PT$)‐symmetric non‐Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time‐dependent transformation operator has to be PT$PT$ symmetric and non‐unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of the non‐Hermitian Hamiltonian. The wave functions and associated non‐adiabatic Berry phase γn$\gamma _{n}$ for the nth eigenstate are obtained analytically. The classical version of the non‐Hermitian Hamiltonian becomes a complex function of canonical variables and time. The corresponding kernel Hamiltonian is derived with PT$PT$ symmetric canonical‐variable transfer in the classical gauge transformation. Moreover, with the change of position‐momentum to angle‐action variables it is revealed that the non‐adiabatic Hannay's angle normalΔθH$\Delta \theta _{\text{H}}$ and Berry phase satisfy precisely the quantum‐classical correspondence, γn=(n+1/2)normalΔθH$\gamma _{n}= (n+1/2)\Delta \theta _{\text{H}}$.

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Non-Hermitian Hamiltonians and no-go theorems in quantum information

Cited by 100 publications

References 81 publications

PT -Symmetric Qubit-System States in the Probability Representation of Quantum Mechanics

PT -Symmetric Qubit-System States in the Probability Representation of Quantum Mechanics

Quantum simulation of parity–time symmetry breaking with a superconducting quantum processor

Generalized Gauge Transformation with PT$PT$‐Symmetric Non‐Unitary Operator and Classical Correspondence of Non‐Hermitian Hamiltonian for a Periodically Driven System

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