Von Neumann Entropy and Quantum Algorithmic Randomness

• URL: http://arxiv.org/abs/2412.18646v1
• Date: Tue, 24 Dec 2024 18:09:45 GMT
• Title: Von Neumann Entropy and Quantum Algorithmic Randomness
• Authors: Tejas Bhojraj,
• Abstract summary: A state $rho=(rho_n)_n=1infty$ is a sequence such that $rho_n$ is a density matrix on $n$ qubits. The von Neumann entropy $H(d)$ of a density matrix $d$ is the Shannon entropy of its eigenvalue distribution.
• Score: 0.0
• License:
• Abstract: A state $\rho=(\rho_n)_{n=1}^{\infty}$ is a sequence such that $\rho_n$ is a density matrix on $n$ qubits. It formalizes the notion of an infinite sequence of qubits. The von Neumann entropy $H(d)$ of a density matrix $d$ is the Shannon entropy of its eigenvalue distribution. We show: (1) If $\rho$ is a computable quantum Schnorr random state then $\lim_n [H(\rho_n )/n] = 1$. (2) We define quantum s-tests for $s\in [0,1]$, show that $\liminf_n [H(\rho_n)/n]\geq \{ s: \rho$ is covered by a quantum s-test $\}$ for computable $\rho$ and construct states where this inequality is an equality. (3) If $\exists c \exists^\infty n H(\rho_n)> n-c$ then $\rho$ is strong quantum random. Strong quantum randomness is a randomness notion which implies quantum Schnorr randomness relativized to any oracle. (4) A computable state $(\rho_n)_{n=1}^{\infty}$ is quantum Schnorr random iff the family of distributions of the $\rho_n$'s is uniformly integrable. We show that the implications in (1) and (3) are strict.

Related papers

• Towards Optimal Convergence Rates for the Quantum Central Limit Theorem [3.130722489512822]
  We contribute to the problem of finding the optimal rate of convergence for this quantum central limit theorem. We introduce a notion of Poincar'e inequality for quantum states and show that if $rho$ satisfies this Poincar'e inequality, then $D(rhoboxplus n| rho_G)= mathcal O(n-1)$.
  arXiv  Detail & Related papers  (2023-10-15T12:02:43Z)
• Maximal intrinsic randomness of a quantum state [1.0470286407954037]
  Quantum information science has greatly progressed in the study of intrinsic, or secret, quantum randomness in the past decade. We answer this question for three different randomness quantifiers: the conditional min-entropy, the conditional von Neumann entropy and the conditional max-entropy. For the conditional von Neumann entropy, the maximal value is $H*= log_2d-S(rho)$, with $S(rho)$ the von Neumann entropy of $rho$, while for the conditional max-entropy, we find the maximal value $
  arXiv  Detail & Related papers  (2023-07-28T17:58:13Z)
• Constructions of $k$-uniform states in heterogeneous systems [65.63939256159891]
  We present two general methods to construct $k$-uniform states in the heterogeneous systems for general $k$. We can produce many new $k$-uniform states such that the local dimension of each subsystem can be a prime power.
  arXiv  Detail & Related papers  (2023-05-22T06:58:16Z)
• A Quantum Algorithm Framework for Discrete Probability Distributions with Applications to Rényi Entropy Estimation [13.810917492304565]
  We propose a unified quantum algorithm framework for estimating properties of discrete probability distributions. Our framework estimates $alpha$-R'enyi entropy $H_alpha(p)$ to within additive error $epsilon$ with probability at least $2/3$.
  arXiv  Detail & Related papers  (2022-12-03T08:01:55Z)
• Enlarging the notion of additivity of resource quantifiers [62.997667081978825]
  Given a quantum state $varrho$ and a quantifier $cal E(varrho), it is a hard task to determine $cal E(varrhootimes N)$. We show that the one shot distillable entanglement of certain spherically symmetric states can be quantitatively approximated by such an augmented additivity.
  arXiv  Detail & Related papers  (2022-07-31T00:23:10Z)
• Random quantum circuits transform local noise into global white noise [118.18170052022323]
  We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. For local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution $p_textnoisy$ of a generic noisy circuit instance shrink exponentially. If the noise is incoherent, the output distribution approaches the uniform distribution $p_textunif$ at precisely the same rate.
  arXiv  Detail & Related papers  (2021-11-29T19:26:28Z)
• Sublinear quantum algorithms for estimating von Neumann entropy [18.30551855632791]
  We study the problem of obtaining estimates to within a multiplicative factor $gamma>1$ of the Shannon entropy of probability distributions and the von Neumann entropy of mixed quantum states. We work with the quantum purified query access model, which can handle both classical probability distributions and mixed quantum states, and is the most general input model considered in the literature.
  arXiv  Detail & Related papers  (2021-11-22T12:00:45Z)
• Random matrices in service of ML footprint: ternary random features with no performance loss [55.30329197651178]
  We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$. We propose a novel random technique, called Ternary Random Feature (TRF) The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
  arXiv  Detail & Related papers  (2021-10-05T09:33:49Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)
• The Quantum Supremacy Tsirelson Inequality [0.22843885788439797]
  For a quantum circuit $C$ on $n$ qubits and a sample $z in 0,1n$, the benchmark involves computing $|langle z|C|0n rangle|2$. We show that for any $varepsilon ge frac1mathrmpoly(n)$, outputting a sample $z$ is the optimal 1-query for $|langle z|C|0nrangle|2$ on average.
  arXiv  Detail & Related papers  (2020-08-20T01:04:32Z)
• Quantum Coupon Collector [62.58209964224025]
  We study how efficiently a $k$-element set $Ssubseteq[n]$ can be learned from a uniform superposition $|Srangle of its elements. We give tight bounds on the number of quantum samples needed for every $k$ and $n$, and we give efficient quantum learning algorithms.
  arXiv  Detail & Related papers  (2020-02-18T16:14:55Z)

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.