We determine the exact error and strong converse exponents of shared randomness-assisted channel simulation in worst case total-variation distance. Namely, we find that these exponents can be written as simple optimizations over the R´enyi channel mutual information. Strikingly, and in stark contrast to channel coding, there are no critical rates, allowing a tight characterization for arbitrary rates below and above the simulation capacity. We derive our results by asymptotically expanding the meta-converse for channel simulation [Caoet al., IEEE Trans. Inf. Theory (2024)], which corresponds to nonsignaling assisted codes. We prove this to be asymptotically tight by employing the approximation algorithms from [Bertaet al., Proc. IEEE ISIT (2024)], which show how to round any non-signaling assisted strategy to a strategy that only uses shared randomness. Notably, this implies that any additional quantum entanglement-assistance does not change the error or the strong converse exponents.

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We determine the exact error and strong converse exponent for entanglement-assisted classical-quantum channel simulation in worst case input purified distance. The error exponent is expressed as a single-letter formula optimized over sandwiched Rényi divergences of order $α\in [1, \infty)$, notably without the need for a critical rate--a sharp contrast to the error exponent for classical-quantum channel coding. The strong converse exponent is expressed as a single-letter formula optimized over sandwiched Rényi divergences of order $α\in [\frac{1}{2},1]$. As in the classical work [Oufkir et al., arXiv:2410.07051], we start with the goal of asymptotically expanding the meta-converse for channel simulation in the relevant regimes. However, to deal with non-commutativity issues arising from classical-quantum channels and entanglement-assistance, we critically use various properties of the quantum fidelity, additional auxiliary channel techniques, approximations via Chebyshev inequalities, and entropic continuity bounds.

We determine the exact error and strong converse exponents of shared randomness-assisted channel simulation in worst case total-variation distance. Namely, we find that these exponents can be written as simple optimizations over the Rényi channel mutual information. Strikingly, and in stark contrast to channel coding, there are no critical rates, allowing a tight characterization for arbitrary rates below and above the simulation capacity.

Understanding the noise affecting a quantum device is of fundamental importance for scaling quantum technologies. A particularly important class of noise models is that of Pauli channels, as randomized compiling techniques can effectively bring any quantum channel to this form and are significantly more structured than general quantum channels. In this paper, we show fundamental lower bounds on the sample complexity for learning Pauli channels in diamond norm. We consider strategies that may not use auxiliary systems entangled with the input to the unknown channel and have to perform a measurement before reusing the channel. For non-adaptive algorithms, we show a lower bound of Ω(23nε−2) to learn an n-qubit Pauli channel. In particular, this shows that the recently introduced learning procedure by [1] is essentially optimal. In the adaptive setting, we show a lower bound of Ω(22.5nε−2) for ε = O(2−n), and a lower bound of Ω(22nε−2) for any ε > 0. This last lower bound holds even in a stronger model where in each step, before performing the measurement, the unknown channel may be used arbitrarily many times sequentially interspersed with unital operations.

We study the effect of shared non-signaling correlations for the problem of simulating a channel using noiseless communication in the one-shot setting. For classical channels, we show how to round any non-signaling-assisted simulation strategy - which exactly corresponds to the meta-converse for channel simulation - to a strategy that only uses shared randomness. For quantum channels, we round any non-signaling-assisted simulation strategy to a strategy that only uses shared entanglement. As our main result, we prove a guarantee on the ratio of success probabilities of at least$(1-\frac{-1}{\mathbf{e}})$, for both the classical and the quantum setting. We further - show this ratio to be optimal. It can be improved to$(1-\frac{1}{t})$using$o$(ln$(t)$) additional bits (qubits) of communication.

In the problem of quantum channel certification, we have black box access to a quantum process and would like to decide if this process matches some predefined specification or is $\eps$-far from this specification. The objective is to achieve this task while minimizing the number of times the black box is used. Note that the state certification problem is a special case where the black box has no input. Here, we focus on two relevant extreme cases. The first one is when the predefined specification is a unitary channel, e.g., a gate in a quantum circuit. In this case, we show that testing whether the black box is described by a fixed unitary or $\eps$-far from it in the trace norm requires $\Theta(d/\eps^2)$ uses of the black box. The second setting we consider is when the predefined specification is a completely depolarizing channels with input dimension $\din$ and output dimension $\dout$. In this case, we prove that, in the non-adaptive setting, $\Tilde{\Theta}(\din^2\dout^{1.5}/\eps^2)$ uses of the channel are necessary and sufficient to verify whether it is equal to the depolarizing channel or $\eps$-far from it in the diamond norm. Finally, we prove a lower bound of $\Omega(\din^2\dout/\eps^2)$ for this problem in the adaptive setting. Note that the special case $\din = 1$ corresponds to the well-studied quantum identity testing problem.

How many copies of a quantum process are necessary and sufficient to construct an approximate classical description of it? We extend the result of Surawy-Stepney, Kahn, Kueng, and Guta (2022) to show that ${\tilde {\mathcal{O}}}\left( {{d^6}/{\varepsilon ^2}} \right)$ copies are sufficient to learn a d-dimensional quantum channel to within ε in the diamond norm. Moreover, we show that Ω(d6/ε2) copies are necessary for any strategy using incoherent non-adaptive measurements. This lower bound applies even for ancilla-assisted strategies.

What advantage do sequential procedures provide over batch algorithms for testing properties of unknown distributions? Focusing on the problem of testing whether two distributions $\mathcal{D}_1$ and $\mathcal{D}_2$ on $\{1,\dots, n\}$ are equal or $\epsilon$-far, we give several answers to this question. We show that for a small alphabet size $n$, there is a sequential algorithm that outperforms any batch algorithm by a factor of at least $4$ in terms sample complexity. For a general alphabet size $n$, we give a sequential algorithm that uses no more samples than its batch counterpart, and possibly fewer if the actual distance between $\mathcal{D}_1$ and $\mathcal{D}_2$ is larger than $\epsilon$. As a corollary, letting $\epsilon$ go to $0$, we obtain a sequential algorithm for testing closeness (with no a priori bound on the distance between $\mathcal{D}_1$ and $\mathcal{D}_2$) with a sample complexity $\tilde{\mathcal{O}}(\frac{n^{2/3}}{TV(\mathcal{D}_1, \mathcal{D}_2)^{4/3}})$: this improves over the $\tilde{\mathcal{O}}(\frac{n/\log n}{TV(\mathcal{D}_1, \mathcal{D}_2)^{2} })$ tester of [Daskalakis and Kawase 2017] and is optimal up to multiplicative constants. We also establish limitations of sequential algorithms for the problem of testing closeness: they can improve the worst case number of samples by at most a constant factor.