Satvik Singh

Satvik Singh

We establish a single-letter and efficiently computable strong converse bound on the classical identification capacity of quantum channels. By equipping the $n$-fold channel output space with a product state-weighted $σ$-Euclidean geometry, we allow trace-distance separation constraints for identification codes to be controlled by Euclidean covering estimates. Using Sudakov's inequality, we bound the covering numbers of the $n$-fold channel outputs via their Gaussian mean widths in the weighted geometry, whose exponential growth in $n$ is governed by the operator norm of a single-letter positive operator. Upon optimizing over all weighing states $σ$, this yields a strong converse bound on the identification capacity of the channel, which also admits a semidefinite representation. Our method improves the best known converse bounds on the identification capacity of several important examples, such as depolarizing, Pauli, erasure, and amplitude damping channels. We also discuss extensions of this method to more general Euclidean geometries on the output space.

Satvik Singh, Nilanjana Datta

We study the information transmission capacities of quantum Markov semigroups $(Ψ^t)_{t\in \mathbb{N}}$ acting on $d-$dimensional quantum systems. We show that, in the limit of $t\to \infty$, the capacities can be efficiently computed in terms of the structure of the peripheral space of $Ψ$, are strongly additive, and satisfy the strong converse property. We also establish convergence bounds to show that the infinite-time capacities are reached after time $t\gtrsim d^2\ln (d)$. From a data storage perspective, our analysis provides tight bounds on the number of bits or qubits that can be reliably stored for long times in a quantum memory device that is experiencing Markovian noise. From a practical standpoint, we show that typically, an $n-$qubit quantum memory, with Markovian noise acting independently and identically on all qubits and a fixed time-independent global error correction mechanism, becomes useless for storage after time $t\gtrsim n2^{2n}$. In contrast, if the error correction is local, we prove that the memory becomes useless much more quickly, i.e., after time $t\gtrsim \ln(n)$. In the setting of point-to-point communication between two spatially separated parties, our analysis provides efficiently computable bounds on the optimal rate at which bits or qubits can be reliably transmitted via long Markovian communication channels $(Ψ^l)_{l\in \mathbb{N}}$ of length $l\gtrsim d^2 \ln(d)$, both in the finite block-length and asymptotic regimes.

Satvik Singh, Bjarne Bergh

We study binary discrimination of idempotent quantum channels. When the two channels share a common full-rank invariant state, we show that a simple image inclusion condition completely determines the asymptotic behavior: when it holds, a broad family of channel divergences collapse to a closed-form, single-letter expression, regularization is unnecessary, and all error exponents (Stein/Chernoff/strong-converse) are explicitly computable with no adaptive advantage. Crucially, this yields the strong converse property for this channel family, which is an important open problem for general channels. When the inclusion fails, asymmetric exponents become infinite, implying perfect asymptotic discrimination. We apply the results to GNS-symmetric channels, showing discrimination rates for large number of self iterations converge exponentially fast to those of the corresponding idempotent peripheral projections. If the two channels do not share a common invariant state, we provide a single-letter converse bound on the regularized sandwiched Rényi cb-divergence, which suffices to establish a strong converse upper bound on the Stein exponents.