Quantum entropies: measuring chaos with almost no extra copies ✛ ✎
Context
Estimating entropy is a basic problem in information theory: how much randomness is in a source, how much information does a signal carry. For classical distributions, the complexity of estimating Shannon entropy has been known for a long time, but quantum analogs—von Neumann, Rényi, and Tsallis entropies—have proven to be a tough nut to crack. Previous work either gave loose upper bounds or lower bounds that did not match, leaving a gap of uncertainty. A whole research front has built up around this topic, from quantum algorithms for entropy estimation to verifying the accuracy of quantum computations and entropy-based methods for analyzing complex systems. This work joins that line and closes almost all remaining gaps for Rényi and Tsallis entropies.
Methods
The authors construct two different approaches depending on the entropy order α. For 0 < α < 1, they use quantum tomography with accuracy guarantees in terms of the Bures χ² divergence. First, from n copies of the state they obtain a rough estimate ρ̂, then they shift it by a regularizing term to get σ. After that, they estimate the trace of ρ^α through a combination of a computable part and measuring the observable σ^(α−1) on m additional copies. For non-integer α > 1, they apply Hayashi's algorithm for estimating pure states combined with random purification: the state is purified to a pure state in a doubled space, then repeatedly measured with a covariant POVM, and the estimates are combined with weights chosen to cancel terms in the bias expansion.
Results
For 0 < α < 1, the sample complexity for Rényi entropy is O(d^(1+1/α)/ε^(1/α) + d^(1/α−1)/ε²), and for Tsallis entropy it is O(d^(1+1/α)/ε^(1/α) + d^(2−2α)/ε²). In particular, for 0 < α ≤ 1/2 both expressions simplify to O(d^(1+1/α)/ε^(1/α)). For non-integer α > 1, the complexity for Rényi entropy is O(d²/ε^(1/α) + d^(1−1/α)/ε²). All these upper bounds match the lower bounds recently established by Wang, up to polylogarithmic factors in d and 1/ε. This means the proposed algorithms are essentially unimprovable in terms of the number of copies of the state required.
Implications
This work stands out from neighboring studies by achieving nearly optimal bounds for a wide range of entropy orders for the first time, using new inequalities with the Bures χ² divergence and a new analysis technique for Hayashi's algorithm. Previously, upper and lower bounds diverged, and it was unclear what the true optimum was. Now, for Rényi entropies (non-integer orders) and Tsallis entropies (0 < α < 1), the gap is closed, providing a precise benchmark for practical implementations. It also demonstrates the power of combining tomography and purification as a universal tool in quantum state property estimation.
Future development
A natural direction is to try to remove the polylogarithmic factors and obtain exact constants, as well as to extend the methods to other state functionals, such as trace distances or Holevo information. The Bures χ² divergence technique might find applications in quantum machine learning and quantum device verification, where efficient estimation of entropic quantities is important. It is also interesting to explore adaptive strategies that could reduce constants in practice.
Impact
The results directly affect quantum cryptography, where entropies are used to assess key secrecy, and quantum thermodynamics, where entropy is a key quantity.
Next steps
Test the algorithms on real quantum processors with noise and finite gate accuracy, and develop efficient classical post-processing to speed up estimation.
Key open problems
In quantum information theory, the question of optimal complexity for estimating von Neumann entropy at α = 1 remains open: known bounds have a gap of polylogarithmic factors, and this work does not address that case. It is also unclear whether the estimates for integer α ≥ 2 can be improved, where current algorithms are based on the generalized SWAP test and may be suboptimal.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
The paper presents nearly sample-optimal algorithms for estimating Rényi and Tsallis entropies of orders α in (0,1) and non-integer α>1. For α<1, it uses quantum tomography with regularization followed by estimating the trace of a power via measurement of an observable, while for α>1, it employs the Hayashi algorithm with random cleaning. The obtained upper bounds on sample complexity match the lower bounds up to polylogarithmic factors.
Where its strength is
It closes the gap between upper and lower bounds for almost all orders of Rényi and Tsallis entropies, providing a precise benchmark for optimal complexity. The key strength is the combination of tomography, new inequalities involving χ²-divergence, and random cleaning, which together achieve near-optimality.
Where it could go next
- It might be interesting to see how the entropy estimation methods developed here can be adapted for verifying the accuracy of quantum computations, where it is important to quickly catch rare failures. There is related work on a quantum sieve that catches failures with probability one in a million—perhaps entropy estimates could help improve the sensitivity of such methods.based on: Quantum sieve catches one-in-a-million glitches arXiv:2606.06316
- The idea of random cleaning and estimating pure states could find applications in tasks where one needs to distinguish entangled states from separable ones. There is work in the archive on distinguishing a tangle from a thread—perhaps the cleaning technique could offer a new angle for such problems.based on: How to Tell a Tangled Ball from a Thread arXiv:2607.23773
- Since the proposed algorithms are nearly optimal, they can serve as a benchmark for developing practical entropy estimation protocols in experiments. Work on a recipe for quantum experiments might suggest how to integrate such estimates into real setups without losing accuracy.based on: A Recipe for Quantum Experiments: Don't Forget a Thing arXiv:2607.03982
Worth trying
- Apply the χ²-divergence inequalities to estimate other nonlinear functionals of the state, such as traces of powers for purity testing.
- Use random cleaning in combination with covariant measurements to estimate entropy in systems with limited access to copies.
- Investigate whether the constants in the polylogarithmic factors can be improved using finer bias estimates in the Hayashi algorithm.
Why this may matter
Knowing the exact optimal sample complexity for entropy estimation is crucial for quantum tomography, verification of quantum devices, and understanding fundamental limits of quantum information processing. Closing the gap for almost all orders provides a solid foundation for further research and practical implementations.
Nearby works in our archive
- Quantum sieve catches one-in-a-million glitchessimilarity 0.61
- How to Tell a Tangled Ball from a Threadsimilarity 0.59
- A Recipe for Quantum Experiments: Don't Forget a Thingsimilarity 0.59
- How Entropy Predicts Alloy Structuressimilarity 0.59
- How a Seven-Metal Alloy Develops a Memorysimilarity 0.59
- The Suit That Fixes Its Own Mistakessimilarity 0.58
The closest work in the archive sits at 0.61 by meaning — the usual level for our corpus.
The Rényi entropy of order 2 equals minus the logarithm of the purity of the state, and it can be measured directly using a SWAP test on two copies, but for arbitrary order α one has to devise more sophisticated schemes.
In Ted Chiang's story "Story of Your Life," aliens perceive time nonlinearly, which can be compared to quantum entanglement, where information about a system is distributed nonlocally, and entanglement entropy serves as a measure of that nonlocality.
Key numbers
- entropy order α: 0 < α < 1 and non-integer α > 1
- sample complexity for α ≤ 1/2: O(d^(1+1/α)/ε^(1/α))
- sample complexity for non-integer α > 1: O(d²/ε^(1/α) + d^(1−1/α)/ε²)
- estimation accuracy: additive error ε
- algorithm success: probability at least 0.98