The entropy of a quantum system is a measure of its randomness, and has applications in measuring quantum entanglement. We study the problem of estimating the von Neumann entropy, S(ρ), and Rényi entropy, Sα(ρ) of an unknown mixed quantum state ρ in d dimensions, given access to independent copies of ρ. We provide algorithms with copy complexity O(d2/α) for estimating Sα(ρ) for α2) for estimating S(ρ), and Sα(ρ) for non-integral α > 1. These bounds are at least quadratic in d, which is the order dependence on the number of copies required for estimating the entire state ρ. For integral α > 1, on the other hand, we provide an algorithm for estimating Sα(ρ) with a sub-quadratic copy complexity of O(d2-2/α), and we show the optimality of the algorithms by providing a matching lower bound.
The entropy of a quantum system is a measure of its randomness, and has applications in measuring quantum entanglement. We study the problem of estimating the von Neumann entropy, S(ρ), and Rényi entropy, Sα(ρ) of an unknown mixed quantum state ρ in d dimensions, given access to independent copies of ρ. We provide algorithms with copy complexity O(d2/α) for estimating Sα(ρ) for α2) for estimating S(ρ), and Sα(ρ) for non-integral α > 1. These bounds are at least quadratic in d, which is the order dependence on the number of copies required for estimating the entire state ρ. For integral α > 1, on the other hand, we provide an algorithm for estimating Sα(ρ) with a sub-quadratic copy complexity of O(d2-2/α), and we show the optimality of the algorithms by providing a matching lower bound.
The entropy of a quantum system is a measure of its randomness and is useful in quantifying entanglement. We study the problem of measuring the von Neumann and Rényi entropies of an unknown mixed quantum state given access to independent copies of the state. For Rényi entropy of integral order exceeding one, we determine the order-optimal copy complexity and show that it is strictly lower than the number of copies required to learn the underlying state. The main technical innovation is a concentration result for certain polynomials that arise in the Kerov algebra of Young diagrams, which is proven using the cycle structure of compositions of certain types of permutations. For von Neumann entropy and Rényi entropy of non-integral orders, we provide upper and lower bounds on the sample complexity of the Empirical Young Diagram (EYD) algorithm, which is the analogue of the empirical plug-in estimator in classical estimation.
Entropy is a measure of the randomness of a system. Estimating the entropy of a quantum state is a basic problem in quantum information. In this paper, we introduce a time-efficient quantum approach to estimating the von Neumann entropyS(ρ) and Rényi entropySα(ρ) of anN-dimensional quantum state ρ, given access to independent samples of ρ. Specifically, we provide the following quantum estimators. • A quantum estimator forS(ρ) with time complexity Õ (N2),1improving the prior best time complexity Õ(N6) by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). • A quantum estimator forSα(ρ) with time complexity Õ (N4/α−2) for 0N4−2/α) for α > 1, improving the prior best time complexity Õ(N6/α) for 0N6) for α > 1 by Acharya, Issa, Shende, and Wagner (2020), though at a cost of a slightly larger sample complexity. Moreover, these estimators are naturally extensible to the low-rank case. We also provide a sample lower bound Ω(max{N/ε,N1/α−1/ε1/α}) for estimatingSα(ρ). Technically, our method is quite different from the previous ones that are based on weak Schur sampling and Young diagrams. At the heart of our construction, is a novel tool calledsamplizer, which can “samplize” a quantum query algorithm to a quantum algorithm with similar behavior using only samples of quantum states; this suggests a new framework for estimating quantum entropies. Specifically, when a quantum oracleUblockencodes a mixed quantum state ρ, any quantum query algorithm usingQqueries toUcan be samplized to a δ-close (in the diamond norm) quantum algorithm using Θ(Q2/δ) samples of ρ. Moreover, this samplization is proven to be optimal, up to a polylogarithmic factor.
Entropy is a measure of the randomness of a system. Estimating the entropy of a quantum state is a basic problem in quantum information. In this paper, we introduce a time-efficient quantum approach to estimating the von Neumann entropy S(ρ) and Rényi entropy S_α(ρ) of an N-dimensional quantum state ρ, given access to independent samples of ρ. Specifically, we provide the following quantum estimators. - A quantum estimator for S(ρ) with time complexity Õ(N²), improving the prior best time complexity Õ(N⁶) by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). - A quantum estimator for S_α(ρ) with time complexity Õ(N^{4/α-2}) for 0 < α < 1 and Õ(N^{4-2/α}) for α > 1, improving the prior best time complexity Õ(N^{6/α}) for 0 < α < 1 and Õ(N⁶) for α > 1 by Acharya, Issa, Shende, and Wagner (2020), though at a cost of a slightly larger sample complexity. Moreover, these estimators are naturally extensible to the low-rank case. We also provide a sample lower bound Ω(max{N/ε, N^{1/α-1}/ε^{1/α}}) for estimating S_α(ρ). Technically, our method is quite different from the previous ones that are based on weak Schur sampling and Young diagrams. At the heart of our construction, is a novel tool called samplizer, which can "samplize" a quantum query algorithm to a quantum algorithm with similar behavior using only samples of quantum states; this suggests a unified framework for estimating quantum entropies. Specifically, when a quantum oracle U block-encodes a mixed quantum state ρ, any quantum query algorithm using Q queries to U can be samplized to a δ-close (in the diamond norm) quantum algorithm using Θ~(Q²/δ) samples of ρ. Moreover, this samplization is proven to be optimal, up to a polylogarithmic factor.
The von Neumann, Tsallis, and Rényi entropies are fundamental measures of quantum information. A common estimation strategy multiplies the worst-case costs of spectral transformation and statistical readout, leaving gaps to query lower bounds. We reduce this overhead with multi-level algorithms that use local normalization and precision allocation, while variable-time estimation accounts for the probability of reaching expensive spectral tests. With controlled purified access and its inverse, we obtain bounds for additive error ε, rank upper bound R, and fixed order α. Tsallis entropy estimation has near-optimal rank-independent query complexity O(ε−1/(α−1)) for 1<α<2, when the allowed dimension and rank accommodate the lower-bound instances, and O(1/ε) for α≥2, where O suppresses logarithmic factors. The first saves a factor 1/ε; the second extends known integer-order scaling to noninteger orders. We also improve the von Neumann entropy bound from O(R/ε2) to O(R/ε) and obtain O(R/ε) for noninteger Rényi orders 1<α<3, with further bounds at other orders. Our functional-estimation theorem replaces the global product by a sum of local costs weighted by function magnitudes and spectral masses. We further estimate fixed logarithmic moments and entropy variance using O(R/ε) queries, providing efficient access to fluctuation parameters that enter finite-blocklength quantum compression and pure-state entanglement conversion. More broadly, our framework extends beyond entropy to a broad class of density-matrix functionals, offering a systematic approach toward optimal query complexity in quantum spectral estimation.
We describe a quantum algorithm to estimate the α-Renyi entropy of an unknown density matrix ρ∈Cd×d for α≠1 by combining the recent technique of quantum singular value transformations with the method of estimating normalized traces in the one clean qubit model. We consider an oracular input model where the input state is prepared via a quantum oracle that outputs a purified version of the state, assumed to be nonsingular. Our method outputs an estimate of the α-Renyi entropy to additive precision ϵ, using an expected total number O(1(xϵ)2) of independent applications of a quantum circuit which coherently queries the input unitary O(1δlogdϵ) times, in each case measuring a single output qubit. Here δ is a lower cutoff on the smallest eigenvalue of ρ and x=1dTr(ρα). The expected number of measurements made in this method can be compared to results in the sample complexity model that generally require Θ(d2ϵ2) samples. Furthermore, we also show that multiplicative approximations can be obtained by iteratively using additive approximations, with an overhead logarithmic in the dimension d.
Estimation of Shannon and Rényi entropies of unknown discrete distributions is a fundamental problem in statistical property testing. In this paper, we give the first quantum algorithms for estimating α-Rényi entropies (Shannon entropy being 1-Rényi entropy). In particular, we demonstrate a quadratic quantum speedup for Shannon entropy estimation and a generic quantum speedup for α-Rényi entropy estimation for all α ≥ 0 values, including tight bounds for the Shannon entropy, the Hartley entropy (α = 0), and the collision entropy (α = 2). We also provide quantum upper bounds for estimating min-entropy (α = +∞) as well as the Kullback-Leibler divergence. We complement our results with quantum lower bounds on α-Rényi entropy estimation for all α ≥ 0 values. Our approach is inspired by the pioneering work of Bravyi, Harrow, and Hassidim (BHH); however, with many new technical ingredients: 1) we improve the error dependence of the BHH framework by a fine-tuned error analysis together with Montanaro's approach to estimating the expected output of quantum subroutines for α = 0, 1; 2) we develop a procedure, similar to cooling schedules in simulated annealing, for general α ≥ 0, and 3) in the cases of integer α ≥ 2 and α = +∞, we reduce the entropy estimation problem to the α-distinctness and [log n]-distinctness problems, respectively.