Time and Energy

Jeff Tollaksen

This paper focuses on "time-extended" properties that quantum mechanics represents as existing at a given moment, but which cannot be measured at a given moment. Several examples are presented in which the energy cannot be measured in a short time. The essence of these examples is an attempt to measure the momenta (or the energy) of an ideal quantum clock by having an interaction that lasts only a short time, where this short time is defined with respect to the internal time which is conjugate to this momenta. However, this momenta and this time cannot both be definite at once, even though quantum mechanics claims that a definite energy should be definite at a definite time. However, the momenta can be definite at a definite external parameter time, rather than this internal time. From the internal perspective, however, it is shown that the energy cannot be defined at a given internal time and therefore, this aspect of "time-extension" is completely quantum in origin, unlike the classical aspect of "energy is frequency." An additional consequence of time extended properties is that the connection between the internal time before and after the energy measurement is made uncertain. This suggests that the complementarity between energy and time is deeper than the notion that precise measurements of energy take a long time. Going beyond the "negative" statements about what cannot be measured (which characterize most of the discussions of the energy-time uncertainty relation), a positive aspect of ΔEΔt > 1 is demonstrated in a closed system based on causality. In this example, if one were to argue that energy really existed at a definite moment in time, then causality could be called into question. The standard understanding of the relationship between energy and time is that if the energy is conserved then we can calculate what the energy is at any point in time and thus we should be able to speak about energy as actually existing at that definite moment in time. This section challenges that assertion and is motivated by the question: "is there an example in physics of a property that the formalism tells us exists at a given moment, but which cannot be checked at a given time?"

Lut Pil

Yuxia Zhang, Jian Zou, Bin Shao

In this paper, analogous to the no-signaling-in-time (NSIT) conditions, a series of equalities for the change of conditional and average energy of a quantum system are given to test macrorealism. These equalities are named no-signaling-in-time conditions for conditional energy (CNSIT) and no-signaling-in-time conditions for average energy (ANSIT), respectively. Then, we investigate the violations of the NSIT conditions, the CNSIT conditions and the ANSIT conditions for a qubit in the following scenarios: pure coherent dynamics, dynamics with drive, dynamics under dissipation and dephasing. For the pure qubit, when the NSIT conditions or the CNSIT conditions are not violated, the ANSIT conditions can not be violated, and a suitable conjunction of the CNSIT conditions and the NSIT conditions may be better for testing macrorealism. While for the driven qubit, the non-violation of the CNSIT conditions implies the non-violation of the NSIT conditions, which in turn implies the non-violation of the ANSIT conditions. For dephasing and dissipative qubits, the relationships among the NSIT conditions, the CNSIT conditions and the ANSIT conditions are similar to those of the pure and driven qubits, respectively. While the degree of violations of the NSIT conditions, the CNSIT conditions and the ANSIT conditions is decreased with the increasing time interval between measurements; and if this time interval tends to a very large number, all three kinds of conditions are satisfied.

Chi‐Hang Fred Fung, H. F. Chau

Quantum mechanics sets limits on how fast quantum processes can run given some system energy through time-energy uncertainty relations, and they imply that time and energy are tradeoffs against each other. Thus, we propose to measure the time energy as a single unit for quantum channels. We consider a time-energy measure for quantum channels and compute lower and upper bounds of it using the channel Kraus operators. For a special class of channels (which includes the depolarizing channel), we can obtain the exact value of the time-energy measure. One consequence of our result is that erasing quantum information requires $\sqrt{(n+1)/n}$ times more time-energy resource than erasing classical information, where $n$ is the system dimension.

Holger Lyre