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arXiv:1305.5776 · Physical Review A

Time-energy measure for quantum processes

Chi‐Hang Fred Fung, H. F. Chau

Published Jul 9, 2013• 16 citations
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Abstract

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.

DOI
10.1103/physreva.88.012307

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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 (n+1)/n\sqrt{(n+1)/n}(n+1)/n​ times more time-energy resource than erasing classical information, where nnn is the system dimension.
The speed of any quantum process is limited by quantum mechanics via time-energy uncertainty relations and they imply that time and energy are tradeoff with each other. As such, 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 obtain the exact value of the timeenergy measure. Our result can be used to compare the time-energy resources of similar quantum processes. In particular, we show that erasing quantum information requires √ (n+ 1)/n times more time-energy resource than erasing classical information, where n is the system dimension. This work is published in [1].
Time and energy of quantum processes are a tradeoff against each other. We propose to ascribe to any given quantum process a time-energy cost to quantify how much computation it performs. Here, we analyze the time-energy costs for general quantum measurements, along a similar line as our previous work for quantum channels, and prove exact and lower bound formulas for the costs. We use these formulas to evaluate the efficiencies of actual measurement implementations. We find that one implementation for a Bell measurement is optimal in time energy. We also analyze the time-energy cost for unambiguous state discrimination and find evidence that only a finite time-energy cost is needed to distinguish any number of states.
Several quantum channels may fail to admit an exact joint realization, a property known as incompatibility. Channel incompatibility unifies various quantum limitations, such as measurement uncertainty, the no-information-without-disturbance principle, and the no-cloning and no-broadcasting theorems. We address how accurately incompatible channels can be approximated by the marginals of a single joint channel, quantified by the minimum total diamond-norm error over all joint realizations. We establish two-sided bounds relating it to the generalized robustness of incompatibility (RoI), a noise-based resource quantifier of incompatibility. For identity channels, we determine both the RoI and the minimum total error exactly. The upper bound is attained by the optimal symmetric universal cloner, while the prefactor in the lower bound is asymptotically optimal as the dimension tends to infinity. Applying this result to measurement channels provides a unified, model-independent framework encompassing error-error and information-error-disturbance tradeoffs. Furthermore, our robustness-based evaluation of disturbance outperforms an algebraic bound for every non-trivial POVM in dimensions up to six.
The quantum measurement problem the tension between unitary Schrödinger evolution and the projection postulate has been with us since the early days of quantum theory. Here we derive a lower bound on the time required for any quantum measurement, treating the measurement as quantum channel discrimination carried out by the environment. The resulting bound, \tau_{\mathcal{M}} \ge \frac{\ln N}{2\pi\gamma}, relates the measurement time \tau_{\mathcal{M}} to the number of distinguishable outcomes N and the system-environment coupling strength \gamma. It follows from combining a finite channel-capacity limit on information transfer with the thermodynamic cost of erasing information, as set by Landauer's principle. The bound explains why measurements look instantaneous at macroscopic scales while still being bounded away from zero, gives a timescale for the onset of classical objectivity under quantum Darwinism, and yields concrete, testable predictions for weak-measurement experiments in cavity quantum electrodynamics and superconducting circuits.