We present a constructive example of the violation of the additivity of minimum output Rényi entropy for each p > 2. The example is provided by an antisymmetric subspace of a suitable dimension. We discuss the possibility of extension of the result to go beyond p > 2 and obtain additivity for p = 0 for a class of entanglement breaking channels.

arXiv:0911.2515Journal of Physics A Mathematical and TheoreticalOct 4, 201028 citations

Towards Optimal Quantum Estimators for State Frame Potential

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arXiv:0906.1877 · Advances in Mathematics

Random quantum channels II: Entanglement of random subspaces, Rényi entropy estimates and additivity problems

Benoı̂t Collins, Ion Nechita

Published Aug 22, 2010• 39 citations

arXiv HTML PDF

DOI: 10.1016/j.aim.2010.08.002

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The state frame potential is a standard diagnostic of how closely a quantum state ensemble approximates Haar randomness. In this work, we study the problem of estimating the state frame potential of order $t$ to within additive error $\varepsilon$ under three progressively weaker access models: (i) query access to a multi-state-preparation oracle, (ii) general sample access, and (iii) single-copy sample access. In the query model, we establish a near-optimal query complexity of $\widetildeΘ(\sqrt{t}/\varepsilon)$ , yielding a quadratic improvement in the dependence on $t$ over the previous best result of Nakata, Takeuchi, Kliesch, and Darmawan (PRX Quantum 2025). In the general sample model, we establish the optimal sample complexity $Θ(t/\varepsilon^2)$ . In the single-copy sample model, we present a store-and-estimate approach whose sample complexity depends on the Rényi entropy of the ensemble weights. As an application, we use the single-copy algorithm to assess the randomness of projected state ensembles, where the entropy term becomes the observational Rényi entropy associated with measuring one subsystem.The state frame potential is a standard diagnostic of how closely a quantum state ensemble approximates Haar randomness. In this work, we study the problem of estimating the state frame potential of order $t$ to within additive error $\varepsilon$ under three progressively weaker access models: (i) query access to a multi-state-preparation oracle, (ii) general sample access, and (iii) single-copy sample access. In the query model, we establish a near-optimal query complexity of $\widetildeΘ(\sqrt{t}/\varepsilon)$, yielding a quadratic improvement in the dependence on $t$ over the previous best result of Nakata, Takeuchi, Kliesch, and Darmawan (PRX Quantum 2025). In the general sample model, we establish the optimal sample complexity $Θ(t/\varepsilon^2)$. In the single-copy sample model, we present a store-and-estimate approach whose sample complexity depends on the Rényi entropy of the ensemble weights. As an application, we use the single-copy algorithm to assess the randomness of projected state ensembles, where the entropy term becomes the observational Rényi entropy associated with measuring one subsystem.

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Counterexamples to additivity of minimum output

p

-Rényi entropy of quantum channels for all

p\ge 0

The additivity of minimum output entropies is a central problem in quantum information theory. Nonadditivity is known for every Rényi order

p>1

, at the von Neumann point

p=1

, and for sufficiently small positive

p

, while much of the interval

0<p<1

has remained open. In this work, we prove that for every

p>0

there exist finite-dimensional quantum channels whose minimum output

p

-Rényi entropies violate additivity. Our proof combines two constructions: random projection-induced channels yield nonadditivity for

p>3/4

, and antisymmetric postprocessing extends the violation to all positive Rényi orders. Our estimates also improve the output-dimension bound obtained by Belinschi, Collins, and Nechita for additivity violation of minimum output von Neumann entropy.

t

to within additive error

\varepsilon

under three progressively weaker access models: (i) query access to a multi-state-preparation oracle, (ii) general sample access, and (iii) single-copy sample access. In the query model, we establish a near-optimal query complexity of

\widetildeΘ(\sqrt{t}/\varepsilon)

, yielding a quadratic improvement in the dependence on

t

over the previous best result of Nakata, Takeuchi, Kliesch, and Darmawan (PRX Quantum 2025). In the general sample model, we establish the optimal sample complexity

Θ(t/\varepsilon^2)

. In the single-copy sample model, we present a store-and-estimate approach whose sample complexity depends on the Rényi entropy of the ensemble weights. As an application, we use the single-copy algorithm to assess the randomness of projected state ensembles, where the entropy term becomes the observational Rényi entropy associated with measuring one subsystem.