Europhysics Letters (EPL)

On the orthogonalization of quantum states

Yong Xiao, Guang Yang

Published Jul 1, 2013• 1 citation

Abstract

Some quantum states can dynamically evolve to an orthogonal state after a time interval, while some states can never evolve to their orthogonal states. In this paper, we investigate three issues on the orthogonalization properties of quantum states. First, we answer the question as to under what conditions an initial state can evolve to an orthogonal state. Second, we discuss the orthogonalization rate of quantum states, i.e. , the total number of orthogonal states that the evolution can go through in unit time. Finally, we show that the Margolus-Levitin bound for orthogonalization time remains tight for superpositions of more than two eigenstates.

DOI: 10.1209/0295-5075/103/10003

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If an initial state $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ and a final state $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ are given, then there exist many Hamiltonians under whose action $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ evolves into $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ . In this case, the problem of the transition of $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ to $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ in the least time is very interesting. It was previously shown that for a Hermitian Hamiltonian, there is an optimum evolution time if $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ and $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ are orthogonal. But for a $$PT$$ -symmetric Hamiltonian, this time can be arbitrarily small, which seems amazing. We discuss the optimum time evolution for pseudo-Hermitian Hamiltonians and obtain a lower bound for the evolution time under the condition that the Hamiltonian is bounded. The optimum evolution time can be attained in the case where two quantum states are orthogonal with respect to some inner product. The results in the Hermitian and pseudo-Hermitian cases coincide if the evolution is unitary with some well-defined inner product. We also analyze two previously studied examples and find that they are consistent with our theory. In addition, we give some explanations of our results with two examples.

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Europhysics Letters (EPL)

On the orthogonalization of quantum states

Yong Xiao, Guang Yang

Published Jul 1, 2013• 1 citation

Publisher

Abstract

DOI: 10.1209/0295-5075/103/10003

Related papers

Generalization of the Margolus-Levitin bound

Bartosz Zieliński, Magdalena Zych

arXiv:quant-ph/0603076Physical Review ASep 11, 200653 citations

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M. Andrecut, Mazhar Ali

Journal of Physics A Mathematical and GeneralMar 29, 200423 citations

Fundamental Limit on the Rate of Quantum Dynamics: The Unified Bound Is Tight

Lev B. Levitin, Tommaso Toffoli

How fast a quantum state can evolve has attracted considerable attention in connection with quantum measurement and information processing. A lower bound on the orthogonalization time, based on the energy spread DeltaE, was found by Mandelstam and Tamm. Another bound, based on the average energy E, was established by Margolus and Levitin. The bounds coincide and can be attained by certain initial states if DeltaE=E. Yet, the problem remained open when DeltaE not equal E. We consider the unified bound that involves both DeltaE and E. We prove that there exist no initial states that saturate the bound if DeltaE not equal E. However, the bound remains tight: for any values of DeltaE and E, there exists a one-parameter family of initial states that can approach the bound arbitrarily close when the parameter approaches its limit. These results establish the fundamental limit of the operation rate of any information processing system.How fast a quantum state can evolve has attracted considerable attention in connection with quantum measurement and information processing. A lower bound on the orthogonalization time, based on the energy spread DeltaE, was found by Mandelstam and Tamm. Another bound, based on the average energy E, was established by Margolus and Levitin. The bounds coincide and can be attained by certain initial states if DeltaE=E. Yet, the problem remained open when DeltaE not equal E. We consider the unified bound that involves both DeltaE and E. We prove that there exist no initial states that saturate the bound if DeltaE not equal E. However, the bound remains tight: for any values of DeltaE and E, there exists a one-parameter family of initial states that can approach the bound arbitrarily close when the parameter approaches its limit. These results establish the fundamental limit of the operation rate of any information processing system.

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Ulvi Yurtsever

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AIP conference proceedingsJan 1, 20090 citations

Optimal time evolution for pseudo-Hermitian Hamiltonians

Wenhua Wang, Z. L. Chen, Yun S. Song +1 more

If an initial state $|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle$ and a final state $|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle$ are given, then there exist many Hamiltonians under whose action $|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle$ evolves into $|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle$ . In this case, the problem of the transition of $|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle$ to $|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle$ in the least time is very interesting. It was previously shown that for a Hermitian Hamiltonian, there is an optimum evolution time if $|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle$ and $|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle$ are orthogonal. But for a $PT$ -symmetric Hamiltonian, this time can be arbitrarily small, which seems amazing. We discuss the optimum time evolution for pseudo-Hermitian Hamiltonians and obtain a lower bound for the evolution time under the condition that the Hamiltonian is bounded. The optimum evolution time can be attained in the case where two quantum states are orthogonal with respect to some inner product. The results in the Hermitian and pseudo-Hermitian cases coincide if the evolution is unitary with some well-defined inner product. We also analyze two previously studied examples and find that they are consistent with our theory. In addition, we give some explanations of our results with two examples.If an initial state $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ and a final state $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ are given, then there exist many Hamiltonians under whose action $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ evolves into $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ . In this case, the problem of the transition of $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ to $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ in the least time is very interesting. It was previously shown that for a Hermitian Hamiltonian, there is an optimum evolution time if $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ and $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ are orthogonal. But for a $$PT$$ -symmetric Hamiltonian, this time can be arbitrarily small, which seems amazing. We discuss the optimum time evolution for pseudo-Hermitian Hamiltonians and obtain a lower bound for the evolution time under the condition that the Hamiltonian is bounded. The optimum evolution time can be attained in the case where two quantum states are orthogonal with respect to some inner product. The results in the Hermitian and pseudo-Hermitian cases coincide if the evolution is unitary with some well-defined inner product. We also analyze two previously studied examples and find that they are consistent with our theory. In addition, we give some explanations of our results with two examples.

Theoretical and Mathematical PhysicsAug 1, 20202 citations

If an initial state

|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle

and a final state

|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle

are given, then there exist many Hamiltonians under whose action

|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle

evolves into

|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle

. In this case, the problem of the transition of

|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle

|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle

in the least time is very interesting. It was previously shown that for a Hermitian Hamiltonian, there is an optimum evolution time if

|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle

and

|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle

are orthogonal. But for a

PT

-symmetric Hamiltonian, this time can be arbitrarily small, which seems amazing. We discuss the optimum time evolution for pseudo-Hermitian Hamiltonians and obtain a lower bound for the evolution time under the condition that the Hamiltonian is bounded. The optimum evolution time can be attained in the case where two quantum states are orthogonal with respect to some inner product. The results in the Hermitian and pseudo-Hermitian cases coincide if the evolution is unitary with some well-defined inner product. We also analyze two previously studied examples and find that they are consistent with our theory. In addition, we give some explanations of our results with two examples.