Abstract
To what extent does Noether’s principle apply to quantum channels? Here, we quantify the degree to which imposing a symmetry constraint on quantum channels implies a conservation law and show that this relates to physically impossible transformations in quantum theory, such as time reversal and spin inversion. In this analysis, the convex structure and extremal points of the set of quantum channels symmetric under the action of a Lie group becomes essential. It allows us to derive bounds on the deviation from conservation laws under any symmetric quantum channel in terms of the deviation from closed dynamics as measured by the unitarity of the channel . In particular, we investigate in detail the U(1) and SU(2) symmetries related to energy and angular momentum conservation laws. In the latter case, we provide fundamental limits on how much a spin- system can be used to polarize a larger spin- system, and on how much one can invert spin polarization using a rotationally symmetric operation. Finally, we also establish novel links between unitarity, complementary channels, and purity that are of independent interest.
- Received 13 November 2019
- Revised 18 June 2020
- Accepted 21 August 2020
DOI:https://doi.org/10.1103/PhysRevX.10.041035
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Noether’s theorem is a celebrated result showing that continuous symmetries are intimately related to conservation laws. For example, the fact that no point in time is special leads to energy conservation, while rotational symmetry results in conservation of angular momentum. These important relations place symmetries at the forefront of modern physics. However, their applicability is mostly constrained to classical and quantum systems that are isolated. In this work, we ask to what degree does Noether’s principle hold for open (nonisolated) quantum systems that possess a continuous symmetry? To this end, we quantify the admissible disconnects that occur in quantum mechanics between conservation laws and symmetries of open systems and uncover highly nontrivial structure that connects to physically impossible quantum processes, such as spin inversion and time reversal.
Our analysis strongly relates to core results from entanglement theory as well as recent theories of coherent resources. In doing so, we use techniques from quantum information science that have been remarkably successful in quantifying general limitations on quantum-processing tasks. We obtain novel structural results that describe the set of all quantum operations respecting a symmetry principle. These, in turn, allow us to derive general trade-off relations between deviations from conservation laws and openness of quantum dynamics under a given symmetry.
Besides providing fundamental insights into the structure of quantum theory, our results are relevant to a range of research areas, such as open system dynamics, quantum information science, and the development of quantum technologies. Moreover, our technical results on the reversibility of quantum dynamics can find applications in randomized benchmarking of quantum information processing devices and decoherence theory.
