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arXiv:2108.13974 · Physical Review A

Time-energy uncertainty relation for quantum events

Matteo Fadel, Lorenzo Maccone

Published Nov 24, 2021• 10 citations
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Abstract

Textbook quantum mechanics treats time as a classical parameter and not as a quantum observable with an associated Hermitian operator. For this reason, to make sense of usual time-energy uncertainty relations such as $\mathrm{\ensuremath{\Delta}}t\mathrm{\ensuremath{\Delta}}E\ensuremath{\gtrsim}\ensuremath{\hbar}$, the term $\mathrm{\ensuremath{\Delta}}t$ must be interpreted as a time interval and not as a time measurement uncertainty due to quantum noise. However, quantum clocks allow for a measurement of the ``time at which an event happens'' by conditioning the system's evolution on an additional quantum degree of freedom. Within this framework we derive here two uncertainty relations that relate the uncertainty in the quantum measurement of the time at which a quantum event happens on a system to its energy uncertainty.

DOI
10.1103/physreva.104.l050204

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Textbook quantum mechanics treats time as a classical parameter and not as a quantum observable with an associated Hermitian operator. For this reason, to make sense of usual time-energy uncertainty relations such as \ensuremathΔt\ensuremathΔE\ensuremath≳\ensuremathℏ\mathrm{\ensuremath{\Delta}}t\mathrm{\ensuremath{\Delta}}E\ensuremath{\gtrsim}\ensuremath{\hbar}\ensuremathΔt\ensuremathΔE\ensuremath≳\ensuremathℏ, the term \ensuremathΔt\mathrm{\ensuremath{\Delta}}t\ensuremathΔt must be interpreted as a time interval and not as a time measurement uncertainty due to quantum noise. However, quantum clocks allow for a measurement of the ``time at which an event happens'' by conditioning the system's evolution on an additional quantum degree of freedom. Within this framework we derive here two uncertainty relations that relate the uncertainty in the quantum measurement of the time at which a quantum event happens on a system to its energy uncertainty.
Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.
Energy-time uncertainty plays an important role in quantum foundations and technologies, and it was even discussed by the founders of quantum mechanics. However, standard approaches (e.g., Robertson's uncertainty relation) do not apply to energy-time uncertainty because, in general, there is no Hermitian operator associated with time. Following previous approaches, we quantify time uncertainty by how well one can read off the time from a quantum clock. We then use entropy to quantify the information-theoretic distinguishability of the various time states of the clock. Our main result is an entropic energy-time uncertainty relation for general time-independent Hamiltonians, stated for both the discrete-time and continuous-time cases. Our uncertainty relation is strong, in the sense that it allows for a quantum memory to help reduce the uncertainty, and this formulation leads us to reinterpret it as a bound on the relative entropy of asymmetry. Because of the operational relevance of entropy, we anticipate that our uncertainty relation will have information-processing applications.
We prove the uncertainty relation σT σE≥ℏ/2σ_T \, σ_E \geq \hbar/2σT​σE​≥ℏ/2 between the time TTT of detection of a quantum particle on the surface ∂Ω\partial Ω∂Ω of a region Ω⊂R3Ω\subset \mathbb{R}^3Ω⊂R3 containing the particle's initial wave function, using the "absorbing boundary rule" for detection time, and the energy EEE of the initial wave function. Here, σσσ denotes the standard deviation of the probability distribution associated with a quantum observable and a wave function. Since TTT is associated with a POVM rather than a self-adjoint operator, the relation is not an instance of the standard version of the uncertainty relation due to Robertson and Schrödinger. We also prove that if there is nonzero probability that the particle never reaches ∂Ω\partial Ω∂Ω (in which case we write T=∞T=\inftyT=∞), and if σTσ_TσT​ denotes the standard deviation conditional on the event T<∞T<\inftyT<∞, then σT σE≥(ℏ/2)Prob(T<∞)σ_T \, σ_E \geq (\hbar/2) \sqrt{\mathrm{Prob}(T<\infty)}σT​σE​≥(ℏ/2)Prob(T<∞)​.
Abstract Dynamical evolution can be reconstructed within stationary, closed quantum systems by employing the Page-Wootters “timeless approach”. When conditioning upon the state of a “clock” subsystem, the rest of the system regains its time dependence. This mechanism, involving entanglement between the above subsystems has gained much attention during the last few years. After a brief introduction to the topic we will elaborate on a few recent results: The derivation of new time-energy uncertainty relations, emergence of non-Hermitian dynamics when utilizing non-inertial quantum clocks and dynamical nonlocality in quantum time.