Abstract We develop a resource theory of symmetric distinguishability, the fundamental objects of which are elementary quantum information sources, i.e. sources that emit one of two possible quantum states with given prior probabilities. Such a source can be represented by a classical-quantum state of a composite system XA , corresponding to an ensemble of two quantum states, with X being classical and A being quantum. We study the resource theory for two different classes of free operations: (i) CPTP A , which consists of quantum channels acting only on A , and (ii) conditional doubly stochastic maps acting on XA . We introduce the notion of symmetric distinguishability of an elementary source and prove that it is a monotone under both these classes of free operations. We study the tasks of distillation and dilution of symmetric distinguishability, both in the one-shot and asymptotic regimes. We prove that in the asymptotic regime, the optimal rate of converting one elementary source to another is equal to the ratio of their quantum Chernoff divergences, under both these classes of free operations. This imparts a new operational interpretation to the quantum Chernoff divergence. We also obtain interesting operational interpretations of the Thompson metric, in the context of the dilution of symmetric distinguishability.
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Testing symmetry on quantum computers
Symmetry is a unifying concept in physics. In quantum information and beyond, it is known that quantum states possessing symmetry are not useful for certain information-processing tasks. For example, states that commute with a Hamiltonian realizing a time evolution are not useful for timekeeping during that evolution, and bipartite states that are highly extendible are not strongly entangled and thus not useful for basic tasks like teleportation. Motivated by this perspective, this paper details several quantum algorithms that test the symmetry of quantum states and channels. For the case of testing Bose symmetry of a state, we show that there is a simple and efficient quantum algorithm, while the tests for other kinds of symmetry rely on the aid of a quantum prover. We prove that the acceptance probability of each algorithm is equal to the maximum symmetric fidelity of the state being tested, thus giving a firm operational meaning to these latter resource quantifiers. Special cases of the algorithms test for incoherence or separability of quantum states. We evaluate the performance of these algorithms on choice examples by using the variational approach to quantum algorithms, replacing the quantum prover with a parameterized circuit. We demonstrate this approach for numerous examples using the IBM quantum noiseless and noisy simulators, and we observe that the algorithms perform well in the noiseless case and exhibit noise resilience in the noisy case. We also show that the maximum symmetric fidelities can be calculated by semi-definite programs, which is useful for benchmarking the performance of these algorithms for sufficiently small examples. Finally, we establish various generalizations of the resource theory of asymmetry, with the upshot being that the acceptance probabilities of the algorithms are resource monotones and thus well motivated from the resource-theoretic perspective.
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- Award ID(s):
- 2315398
- PAR ID:
- 10534461
- Publisher / Repository:
- Quantum Journal
- Date Published:
- Journal Name:
- Quantum
- Volume:
- 7
- ISSN:
- 2521-327X
- Page Range / eLocation ID:
- 1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Published by the American Physical Society 2024 -
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Abstract Classical shadows provide a noise-resilient and sample-efficient method for learning quantum system properties, relying on a user-specified unitary ensemble. What is the weakest assumption on this ensemble that can still yield meaningful results? To address this, we focus on Pauli-invariant unitary ensembles—those invariant under multiplication by Pauli operators. For these ensembles, we present explicit formulas for the reconstruction map and sample complexity bounds and extend our results to the case when noise impacts the protocol implementation. Two applications are explored: one for locally scrambled unitary ensembles, where we present formulas for the reconstruction map and sample complexity bounds that circumvent the need to solve an exponential-sized linear system, and another for the classical shadows of quantum channels. Our results establish a unified framework for classical shadows with Pauli-invariant unitary ensembles, applicable to both noisy and noiseless scenarios for states and channels and primed for implementation on near-term quantum devices.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.22331/q-2023-09-25-1120
