Geometric characteristics of quantum evolution: curvature and torsion

Laba, H. P.; Tkachuk, V. M.

doi:10.5488/cmp.20.13003

Cited by 21 publications

(25 citation statements)

References 16 publications

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“…Such evolution, Eq. (45), proceeds precisely along the geodesic determined by the Fubini-Study metric [19,38], saturating the Mandelstam-Tamm bound [39]…”

Section: Discrete History Statesmentioning

confidence: 99%

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History states of systems and operators

Boette

Rossignoli

2018

Phys. Rev. A

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We discuss some fundamental properties of discrete system-time history states. Such states arise for a quantum reference clock of finite dimension and lead to a unitary evolution of system states when satisfying a static discrete Wheeler-DeWitt-type equation. We consider the general case where system-clock pairs can interact, analyzing first their different representations and showing there is always a special clock basis for which the evolution for a given initial state can be described by a constant Hamiltonian H. It is also shown, however, that when the evolution operators form a complete orthogonal set, the history state is maximally entangled for any initial state, as opposed to the case of a constant H, and can be generated through a simple double-clock setting. We then examine the quadratic system-time entanglement entropy, providing an analytic evaluation and showing it satisfies strict upper and lower bounds determined by the energy spread and the geodesic evolution connecting the initial and final states. We finally show that the unitary operator that generates the history state can itself be considered as an operator history state, whose quadratic entanglement entropy determines its entangling power. Simple measurements on the clock enable to efficiently determine overlaps between system states and also evolution operators at any two times.

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“…Such evolution, Eq. (45), proceeds precisely along the geodesic determined by the Fubini-Study metric [19,38], saturating the Mandelstam-Tamm bound [39]…”

Section: Discrete History Statesmentioning

confidence: 99%

“…The latter vanishes just for the geodesic evolution, where ∆H = ∆H min = φ t f σ y and hence (∆H min ) 4 = (∆H min ) 2 2 = φ 4 /t 4 f , implying E 2 (S, T ) = E min 2 (S, T ). Such fluctuation represents a curvature coefficient which measures the deviation from the geodesic [38,40].…”

Section: Discrete History Statesmentioning

confidence: 99%

History states of systems and operators

Boette

Rossignoli

2018

Phys. Rev. A

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“…To begin, we observe that the path γ t traced by the state vector |ψ(t) = cos t |A + sin t |B is a geodesic path. Indeed, a simple calculation shows that |ψ(t) can be recast as a quantum geodesic line [20]:…”

Section: Time-independent Scenariomentioning

confidence: 99%

Minimum Time for the Evolution to a Nonorthogonal Quantum State and Upper Bound of the Geometric Efficiency of Quantum Evolutions

Cafaro

Alsing

2021

Quantum Reports

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We present a simple proof of the fact that the minimum time TAB for quantum evolution between two arbitrary states A and B equals TAB=ℏcos−1A|B/ΔE with ΔE being the constant energy uncertainty of the system. This proof is performed in the absence of any geometrical arguments. Then, being in the geometric framework of quantum evolutions based upon the geometry of the projective Hilbert space, we discuss the roles played by either minimum-time or maximum-energy uncertainty concepts in defining a geometric efficiency measure ε of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the inequality ε≤1 even when the system only passes through nonorthogonal quantum states.

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“…The concept of a distance between quantum states in Hilbert space [1][2][3] has found its application in different fields of quantum mechanics related to the evolution of quantum systems [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], quantum entanglement [8,[20][21][22][23][24][25][26][27][28][29][30], quantum computations [31][32][33][34][35], etc. It was shown that the distance, which the quantum system passes during the evolution in the Hilbert space, is related to the integral of the uncertainty of energy that in turn defines the speed of evolution [4].…”

Section: Introductionmentioning

confidence: 99%

Measuring distance between quantum states on a quantum computer

Kuzmak¹

2021

Preprint

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We propose protocols for determining the distances in Hilbert space between pure and mixed quantum states prepared on a quantum computer. In the case of pure quantum states the protocol is based on measuring the square of modulus of scalar product between certain states. Determination of the distance between mixed quantum states is reduced to measuring the squares of modules of scalar products between all pure states included in the mixed states. In addition, we develop a protocol which allows one to determine the speed of evolution of the spin system simulated by quantum computer. These protocols we apply to measure distances and speeds of evolution of different quantum systems implemented on the ibmq-santiago quantum computer.

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Geometric characteristics of quantum evolution: curvature and torsion

Cited by 21 publications

References 16 publications

History states of systems and operators

History states of systems and operators

Minimum Time for the Evolution to a Nonorthogonal Quantum State and Upper Bound of the Geometric Efficiency of Quantum Evolutions

Measuring distance between quantum states on a quantum computer

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