Entropic proofs of Singleton bounds for quantum error-correcting codes

• URL: http://arxiv.org/abs/2010.07902v4
• Date: Tue, 31 May 2022 10:18:33 GMT
• Title: Entropic proofs of Singleton bounds for quantum error-correcting codes
• Authors: Markus Grassl, Felix Huber, Andreas Winter
• Abstract summary: We show that a relatively simple reasoning using von Neumann entropy inequalities yields a robust proof of the quantum Singleton bound for quantum error-correcting codes (QECC) We also prove information-theoretically tight bounds on the entanglement-communication tradeoff for EAQECC.
• Score: 0.5156484100374058
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show that a relatively simple reasoning using von Neumann entropy inequalities yields a robust proof of the quantum Singleton bound for quantum error-correcting codes (QECC). For entanglement-assisted quantum error-correcting codes (EAQECC) and catalytic codes (CQECC), a type of generalized quantum Singleton bound [Brun et al., IEEE Trans. Inf. Theory 60(6):3073--3089 (2014)] was believed to hold for many years until recently one of us found a counterexample [MG, Phys. Rev. A 103, 020601 (2021)]. Here, we rectify this state of affairs by proving the correct generalized quantum Singleton bound, extending the above-mentioned proof method for QECC; we also prove information-theoretically tight bounds on the entanglement-communication tradeoff for EAQECC. All of the bounds relate block length $n$ and code length $k$ for given minimum distance $d$ and we show that they are robust, in the sense that they hold with small perturbations for codes which only correct most of the erasure errors of less than $d$ letters. In contrast to the classical case, the bounds take on qualitatively different forms depending on whether the minimum distance is smaller or larger than half the block length. We also provide a propagation rule: any pure QECC yields an EAQECC with the same distance and dimension, but of shorter block length.

Related papers

• The multimode conditional quantum Entropy Power Inequality and the squashed entanglement of the extreme multimode bosonic Gaussian channels [53.253900735220796]
  Inequality determines the minimum conditional von Neumann entropy of the output of the most general linear mixing of bosonic quantum modes. Bosonic quantum systems constitute the mathematical model for the electromagnetic radiation in the quantum regime.
  arXiv  Detail & Related papers  (2024-10-18T13:59:50Z)
• Hidden-State Proofs of Quantumness [1.0878040851638]
  An experimental cryptographic proof of quantumness will be a vital milestone in the progress of quantum information science. Error tolerance is a persistent challenge for implementing such tests. We present a proof of quantumness which maintains the same circuit structure as (Brakerski et al.)
  arXiv  Detail & Related papers  (2024-10-08T21:04:53Z)
• Approximate quantum error correcting codes from conformal field theory [0.0]
  We consider generic 1+1D CFT codes under extensive local dephasing channels. We show that a CFT code with continuous symmetry saturates a bound on the recovery fidelity for covariant codes.
  arXiv  Detail & Related papers  (2024-06-13T19:40:36Z)
• A simple formulation of no-cloning and no-hiding that admits efficient and robust verification [0.0]
  Incompatibility is a feature of quantum theory that sets it apart from classical theory. The no-hiding theorem is another such instance that arises in the context of the black-hole information paradox. We formulate both of these fundamental features of quantum theory in a single form that is amenable to efficient verification.
  arXiv  Detail & Related papers  (2023-03-05T12:48:11Z)
• Simple Tests of Quantumness Also Certify Qubits [69.96668065491183]
  A test of quantumness is a protocol that allows a classical verifier to certify (only) that a prover is not classical. We show that tests of quantumness that follow a certain template, which captures recent proposals such as (Kalai et al., 2022) can in fact do much more. Namely, the same protocols can be used for certifying a qubit, a building-block that stands at the heart of applications such as certifiable randomness and classical delegation of quantum computation.
  arXiv  Detail & Related papers  (2023-03-02T14:18:17Z)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• A lower bound on the space overhead of fault-tolerant quantum computation [51.723084600243716]
  The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation. We prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude noise.
  arXiv  Detail & Related papers  (2022-01-31T22:19:49Z)
• Depth-efficient proofs of quantumness [77.34726150561087]
  A proof of quantumness is a type of challenge-response protocol in which a classical verifier can efficiently certify quantum advantage of an untrusted prover. In this paper, we give two proof of quantumness constructions in which the prover need only perform constant-depth quantum circuits.
  arXiv  Detail & Related papers  (2021-07-05T17:45:41Z)
• Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes [3.6192409729339223]
  We develop the theory of quasi-exact fault-tolerant quantum computation, which uses qubits encoded into quasi-exact quantum error-correction codes ("quasi codes") The model of QEQ lies in between the two well-known ones: the usual noisy quantum universality without error correction and the usual fault-tolerant quantum computation, but closer to the later.
  arXiv  Detail & Related papers  (2021-05-31T08:17:30Z)
• Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem [77.34726150561087]
  We present a proof of the approximate Eastin-Knill theorem, which connects the quality of a quantum error-correcting code with its ability to achieve a universal set of logical gates. Our derivation employs powerful bounds on the quantum Fisher information in generic quantum metrological protocols.
  arXiv  Detail & Related papers  (2020-04-24T17:58:10Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.