Sujeet Bhalerao

In this work we improve the quantum communication rates of various quantum channels of interest using permutation-invariant quantum codes. We focus in particular on parametrized families of quantum channels and aim to improve bounds on their quantum capacity threshold, defined as the lowest noise level at which the quantum capacity of the channel family vanishes. These thresholds are important quantities as they mark the noise level up to which faithful quantum communication is theoretically possible. Our method exploits the fact that independent and identically distributed quantum channels preserve any permutation symmetry present at the input. The resulting symmetric output states can be described succinctly using the representation theory of the symmetric and general linear groups, which we use to derive an efficient algorithm for computing the channel coherent information of a permutation-invariant code. Our approach allows us to evaluate coherent information values for a large number of channel copies, e.g., at least 100 channel copies for qubit channels. We apply this method to various physically relevant channel models, including general Pauli channels, the dephrasure channel, the generalized amplitude damping channel, and the damping-dephasing channel. For each channel family we obtain improved lower bounds on their quantum capacities. For example, for the 2-Pauli and BB84 channel families we significantly improve the best known quantum capacity thresholds derived in [Fern, Whaley 2008]. These threshold improvements are achieved using a repetition code-like input state with non-orthogonal code states, which we further analyze in our representation-theoretic framework.

With sensitive information encoded in data, it is important to ensure the privacy to those sensitive information while learning useful information about the data. Towards that, quantum versions of differential privacy frameworks have been introduced. Privatizing data often comes with a cost. Meaning that, there are perfectly private mechanisms one could use that may lead to no utility depending on the application. However, privacy-utility tradeoffs in the quantum setting are not extensively studied. In this work, we study optimal privacy-utility tradeoffs for both generic and application-specific utility metrics when privacy is quantified by $(\varepsilon,\delta)$-quantum local differential privacy. As generic measures, we focus on optimizing fidelity and trace distance between the original state and the privatized state, showing that the depolarizing mechanism achieves the optimal utility while obtaining analytical expressions for utility for all privacy parameter regimes. Next, we study a specific application where one needs to learn the expectation of an observable with respect to an input state (property of quantum data), given access to only privatized states. There, we obtain a lower bound on the number of samples of privatized data required to achieve a fixed accuracy guarantee with high probability by utilizing lower bounds on private quantum hypothesis testing. We also obtain private mechanisms that achieve order optimality with respect to the privacy parameters and accuracy parameters, showcasing how the awareness of the task can be utilized to improve the utility in contrast to using mechanisms that optimize generic utility metrics. Furthermore, we show that the number of samples required to privately learn the expectation values scales as $\Theta((\varepsilon \beta)^{-2})$, where $\varepsilon \in (0,1)$ is the privacy parameter and $\beta$ is the accuracy tolerance. We also study a private version of classical shadows, which may be useful for the private estimation of properties of quantum states and processes.

We study the information-theoretic properties of the full two-parameter family of quantum channels covariant under the orthogonal group O(d), which we call Brauer channels. This family contains the depolarizing and generalized Werner-Holevo channels as one-parameter subfamilies, and its Choi matrix admits a three-block decomposition via Schur-Weyl duality for O(d). We derive closed-form expressions for the maximum output p-norms of Brauer channels and identify a region where two-copy multiplicativity of these norms fails. We prove that the minimum output Renyi 2-entropy is additive for two copies across the entire CPTP region, extending prior partial results. Using orthogonal covariance, we simplify several classical and quantum capacity upper bounds, including SDP-based, approximate entanglement-breaking, approximate covariance, Rains information, and additive-extension bounds, and compare them across the family of Brauer channels. Finally, we derive a closed-form expression for the coherent information of qudit repetition codes through multiple copies of any Brauer channel. We find that repetition codes exhibit positive coherent information beyond the hashing bound in a broad region of channels that includes the qudit depolarizing channel, demonstrating superadditivity of coherent information for Brauer channels.