arXiv subjects

Nilanjana Datta

Publications and source records attributed to Nilanjana Datta.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Shannon theory made robust: a tale of three protocols for almost i.i.d. sources

The asymptotic rates of information-theoretic protocols - including error exponents, data-compression rates, and channel capacities - are traditionally derived under the idealised assumption that the underlying resources are independent and identically distributed (i.i.d.). Somewhat surprisingly, even slight departures from the exact i.i.d. structure can drastically alter the asymptotic behaviour predicted by the i.i.d. theory. If the precise nature of the perturbation is known, for instance in the case of a pointwise defect, one can design a bespoke protocol that compensates for it, e.g. by discarding the corrupted subsystem. In realistic physical settings, however, exact i.i.d. behaviour cannot be guaranteed, and deviations from the ideal regime cannot generally be identified precisely. This raises a fundamental question: which notions of almost i.i.d. structure are sufficiently robust to preserve the asymptotic predictions of quantum Shannon theory? We investigate this question for three central information-theoretic tasks: asymmetric hypothesis testing, classical and quantum data compression, and classical communication through quantum channels. Rather than designing protocols tailored to specific defects, we seek robust protocols that remain asymptotically optimal and that are universal within a broad class of almost i.i.d. resources whose precise deviations from the ideal regime are unknown. To this end, we study three inequivalent notions of almost i.i.d. structure, and determine which of them preserve the asymptotic rates and error exponents predicted by the i.i.d. theory. Along the way, we introduce the notion of an almost i.i.d. process and a new distance measure between quantum channels - the club distance - designed to capture stability under local perturbations. These notions may be of independent interest.

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arXiv2026

Robustness of Entanglement Manipulation for almost i.i.d. sources

We study the robustness of asymptotic entanglement manipulation beyond the exact i.i.d. regime, focusing on Mazzola--Sutter--Renner (MSR) almost i.i.d. sources, which allow a sublinear number of deviations from a tensor-power structure. For pure MSR sources along a bipartite reference state $|ϕ\rangle_{AB}$, we prove that the entanglement concentration rate is robust: every rate below the entropy of entanglement $S(ϕ_A)$ remains achievable. Moreover, this can be done by a single Schur--Weyl concentration protocol that is universal within the MSR class, depending only on the reference state and not on the particular source sequence. For mixed MSR sources along a reference state $ρ_{AB}$, we prove a source-dependent entanglement-distillation achievability result: every rate below the coherent information $I(A\rangle B)_ρ$ of the reference state is achievable, although the entanglement distillation protocol may depend on the particular MSR source sequence. For the reverse task of entanglement dilution, we prove a rate-robustness theorem: the asymptotic entanglement cost of any MSR target sequence along $ρ_{AB}$ is at most $E_F^\infty(ρ_{AB})$, the regularized entanglement of formation of the reference state. To establish these results, we prove structural and entropic properties of MSR almost i.i.d. sequences which may be useful in other information-theoretic settings. Thus, for the achievability statements considered here, MSR almost i.i.d. perturbations exhibit the same asymptotic behaviour as their i.i.d. reference states, despite allowing sublinear deviations from a tensor-power structure.

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arXiv2026

Entropy Concentration and Universal Typicality for Weakly Almost i.i.d. Quantum Sources

Weakly almost i.i.d. quantum sources are sequences of multipartite states whose fixed-size marginals converge, on average, to tensor powers of a reference state, while allowing arbitrary global correlations and entanglement. We establish two concentration principles for such sources: a noncommutative weak law of large numbers for empirical observables, and a universal entropy-concentration principle showing asymptotic concentration on subspaces of exponential dimension governed by the von Neumann entropy of the reference state. These concentration principles provide a unified and conceptually transparent approach to several information-theoretic applications beyond the i.i.d. setting, including direct proofs of universal compression within classes of weakly almost i.i.d. sources sharing a common reference state, asymmetric quantum hypothesis-testing bounds, concentration results for macroscopic observables in quantum many-body systems including generalized Gibbs ensembles and for repeated local measurement statistics, as well as bounds on smooth- and spectral entropy quantities.

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arXiv2026

Onset of superactivation of quantum capacity

Superactivation of quantum capacity is the phenomenon whereby two quantum channels, each with zero quantum capacity, can exhibit a strictly positive capacity when used in tandem. In this work, we explore superactivation in the previously unexplored non-asymptotic regime of finitely many channel uses. We give a definition of finite-blocklength superactivation and propose numerical methods that can certify it. Then, focusing on the 50% erasure and positive-partial-transpose channels considered in the original work on superactivation, we show that as few as 17 uses of the joint channel already enable qubit transmission with a fidelity unattainable by any number of uses of either channel alone, demonstrating a strong finite-blocklength form of superactivation and opening the door to experimental demonstration.

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arXiv2026

Tight relations and equivalences between smooth relative entropies

The precise one-shot characterisation of operational tasks in classical and quantum information theory relies on different forms of smooth entropic quantities. A particularly important connection is between the hypothesis testing relative entropy and the smooth max-relative entropy, which together govern many operational settings. We first strengthen this connection into a type of equivalence: we show that the hypothesis testing relative entropy is equivalent to a variant of the smooth max-relative entropy based on the information spectrum divergence, which can be alternatively understood as a measured smooth max-relative entropy. Furthermore, we improve a fundamental lemma due to Datta and Renner that connects the different variants of the smooth max-relative entropy, introducing a modified proof technique based on matrix geometric means and a tightened gentle measurement lemma. We use the unveiled connections and tools to strictly improve on previously known one-shot bounds and duality relations between the smooth max-relative entropy and the hypothesis testing relative entropy, establishing provably tight bounds between them. The results then allow us to refine other divergence inequalities, in particular sharpening bounds that connect the max-relative entropy with Rényi divergences.

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arXiv2026

Strong converse bounds on the classical identification capacity of the qubit depolarizing channel

A strong converse bound for the classical identification capacity of a quantum channel is an upper bound on the asymptotic identification rate of classical messages sent through the channel, such that, above this rate, the probability of an identification error necessarily converges to one. Converse bounds for identification are notoriously difficult to obtain for fully quantum channels. The only previously known converse bound, due to Atif, Pradhan and Winter [Int.~J.~Quantum Inf.~22(5):2440013, 2024], has the unsatisfactory feature of remaining strictly positive even for a completely noisy channel, for which identification is clearly impossible. We derive strong (and hence also weak) converse bounds, for the qubit depolarizing channel with noise parameter $p$, that vanish as $p\to 1$, thereby yielding the correct behavior in the completely noisy limit. Moreover, in the setting of simultaneous classical identification under the constraint of complete product measurements, our converse bound matches the corresponding achievability bound, and establishes that in this case the identification capacity equals the classical capacity of the channel.

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arXiv2025

Zero-error communication under discrete-time Markovian dynamics

Consider an open quantum system with (discrete-time) Markovian dynamics. Our task is to store information in the system in such a way that it can be retrieved perfectly, even after the system is left to evolve for an arbitrarily long time. We show that this is impossible for classical (resp. quantum) information precisely when the dynamics is mixing (resp. asymptotically entanglement breaking). Furthermore, we provide tight universal upper bounds on the minimum time after which any such dynamics 'scrambles' the encoded information beyond the point of perfect retrieval. On the other hand, for dynamics that are not of this kind, we show that information must be encoded inside the peripheral space associated with the dynamics in order for it to be perfectly recoverable at any time in the future. This allows us to derive explicit formulas for the maximum amount of information that can be protected from noise in terms of the structure of the peripheral space of the dynamics.

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arXiv2025

Information storage and transmission under Markovian noise

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.

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arXiv2025

Composite Classical and Quantum Channel Discrimination

We study the problem of binary composite channel discrimination in the asymmetric setting, where the hypotheses are given by fairly arbitrary sets of channels, and samples do not have to be identically distributed. In the case of quantum channels we prove: (i) a characterization of the Stein exponent for parallel channel discrimination strategies and (ii) an upper bound on the Stein exponent for adaptive channel discrimination strategies. We further show that already for classical channels this upper bound can sometimes be achieved and be strictly larger than what is possible with parallel strategies. Hence, there can be an advantage of adaptive channel discrimination strategies with composite hypotheses for classical channels, unlike in the case of simple hypotheses. Moreover, we show that classically this advantage can only exist if the sets of channels corresponding to the hypotheses are non-convex. As a consequence of our more general treatment, which is not limited to the composite i.i.d. setting, we also obtain a generalization of previous composite state discrimination results.

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arXiv2025

Generalized Quantum Stein's Lemma and Reversibility of Quantum Resource Theories for Classical-Quantum Channels

We extend the recent proof of the Generalized Quantum Stein's Lemma by Hayashi and Yamasaki [arXiv:2408.02722] to classical-quantum (c-q) channels. We analyze the composite hypothesis testing problem of testing a c-q channel $\mathcal{E}^{\otimes n}$ against a sequence of sets of c-q channels $(\mathcal{S}_n)_n$ (satisfying certain natural assumptions), under parallel strategies. We prove that the optimal asymptotic asymmetric error exponent is given by the regularization of Umegaki channel divergence, minimized over $\mathcal{S}_n$. This allows us to prove the reversibility of resource theories of classical-quantum channels in a natural framework, where the distance between channels (and hence also the notion of approximate interconvertibility of channels) is measured in diamond norm, and the set of free operations is the set of all asymptotically resource non-generating superchannels. The results we obtain are similar to the ones in the concurrent and independent work by Hayashi and Yamasaki [arXiv:2509.07271]. However the proof of the direct part of the GQSL uses different arguments and techniques to deal with the challenges that arise from dealing with c-q channels.

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arXiv2025

Continuity bounds for quantum entropies arising from a fundamental entropic inequality

We establish a tight upper bound for the difference in von Neumann entropies between two quantum states, $ρ_1$ and $ρ_2$. This bound is expressed in terms of the von Neumann entropies of the mutually orthogonal states derived from the Jordan-Hahn decomposition of the difference operator $(ρ_1 - ρ_2)$. This yields a novel entropic inequality that implies the well-known Audenaert-Fannes (AF) inequality. In fact, it also leads to a refinement of the AF inequality. We employ this inequality to obtain a uniform continuity bound for the quantum conditional entropy of two states whose marginals on the conditioning system coincide. We additionally use it to derive a continuity bound for the quantum relative entropy in both variables. Interestingly, the fundamental entropic inequality is also valid in infinite dimensions.

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arXiv2025

An invitation to the sample complexity of quantum hypothesis testing

Quantum hypothesis testing (QHT) has been traditionally studied from the information-theoretic perspective, wherein one is interested in the optimal decay rate of error probabilities as a function of the number of samples of an unknown state. In this paper, we study the sample complexity of QHT, wherein the goal is to determine the minimum number of samples needed to reach a desired error probability. By making use of the wealth of knowledge that already exists in the literature on QHT, we characterize the sample complexity of binary QHT in the symmetric and asymmetric settings, and we provide bounds on the sample complexity of multiple QHT. In more detail, we prove that the sample complexity of symmetric binary QHT depends logarithmically on the inverse error probability and inversely on the negative logarithm of the fidelity. As a counterpart of the quantum Stein's lemma, we also find that the sample complexity of asymmetric binary QHT depends logarithmically on the inverse type II error probability and inversely on the quantum relative entropy, provided that the type II error probability is sufficiently small. We then provide lower and upper bounds on the sample complexity of multiple QHT, with it remaining an intriguing open question to improve these bounds. The final part of our paper outlines and reviews how sample complexity of QHT is relevant to a broad swathe of research areas and can enhance understanding of many fundamental concepts, including quantum algorithms for simulation and search, quantum learning and classification, and foundations of quantum mechanics. As such, we view our paper as an invitation to researchers coming from different communities to study and contribute to the problem of sample complexity of QHT, and we outline a number of open directions for future research.

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arXiv2025

Tightening continuity bounds for entropies and bounds on quantum capacities

Uniform continuity bounds on entropies are generally expressed in terms of a single distance measure between a pair of probability distributions or quantum states, typically, the total variation distance or trace distance. However, if an additional distance measure between the probability distributions or states is known, then the continuity bounds can be significantly strengthened. Here, we prove a tight uniform continuity bound for the Shannon entropy in terms of both the local- and total variation distances, sharpening an inequality proven in [I. Sason, IEEE Trans. Inf. Th., 59, 7118 (2013)]. We also obtain a uniform continuity bound for the von Neumann entropy in terms of both the operator norm- and trace distances. The bound is tight when the quotient of the trace distance by the operator norm distance is an integer. We then apply our results to compute upper bounds on the quantum- and private classical capacities of channels. We begin by refining the concept of approximate degradable channels, namely, $\varepsilon$-degradable channels, which are, by definition, $\varepsilon$-close in diamond norm to their complementary channel when composed with a degrading channel. To this end, we introduce the notion of $(\varepsilon,ν)$-degradable channels; these are $\varepsilon$-degradable channels that are, in addition, $ν$-close in completely bounded spectral norm to their complementary channel, when composed with the same degrading channel. This allows us to derive improved upper bounds to the quantum- and private classical capacities of such channels. Moreover, these bounds can be further improved by considering certain unstabilized versions of the above norms. We show that upper bounds on the latter can be efficiently expressed as semidefinite programs. We illustrate our results by obtaining a new upper bound on the quantum capacity of the qubit depolarizing channel.

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arXiv2024

Decoding quantum information via the Petz recovery map

We obtain a lower bound on the maximum number of qubits, $Q^{n, ε}(\mathcal{N})$, which can be transmitted over $n$ uses of a quantum channel $\mathcal{N}$, for a given non-zero error threshold $ε$. To obtain our result, we first derive a bound on the one-shot entanglement transmission capacity of the channel, and then compute its asymptotic expansion up to the second order. In our method to prove this achievability bound, the decoding map, used by the receiver on the output of the channel, is chosen to be the \emph{Petz recovery map} (also known as the \emph{transpose channel}). Our result, in particular, shows that this choice of the decoder can be used to establish the coherent information as an achievable rate for quantum information transmission. Applying our achievability bound to the 50-50 erasure channel (which has zero quantum capacity), we find that there is a sharp error threshold above which $Q^{n, ε}(\mathcal{N})$ scales as $\sqrt{n}$.

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arXiv2024

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

We prove a variety of new and refined uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality and the notion of maximal coupling of random variables. We then employ this classical result to derive the first tight energy-constrained continuity bound for the von Neumann entropy of states of infinite-dimensional quantum systems, when the Hamiltonian is the number operator, which is arguably the most relevant Hamiltonian in the study of infinite-dimensional quantum systems in the context of quantum information theory. The above scheme works only for Shannon- and von Neumann entropies. Hence, to deal with more general entropies, e.g. $α$-Rényi and $α$-Tsallis entropies, with $α\in (0,1)$, for which continuity bounds are known only for finite-dimensional systems, we develop a novel approximation scheme which relies on recent results on operator Hölder continuous functions and the equivalence of all Schatten norms in special spectral subspaces of the Hamiltonian. This approach is, as we show, motivated by continuity bounds for $α$-Rényi and $α$-Tsallis entropies of random variables that follow from the Hölder continuity of the entropy functionals. Bounds for $α>1$ are provided, too. Finally, we settle an open problem on related approximation questions posed in the recent works by Shirokov on the so-called Finite-dimensional Approximation (FA) property.

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arXiv2024

Information transmission under Markovian noise

We consider an open quantum system undergoing Markovian dynamics, the latter being modelled by a discrete-time quantum Markov semigroup $(Φ^n)_{n \in {\mathbb{N}}}$, resulting from the action of sequential uses of a quantum channel $Φ$, with $n \in {\mathbb{N}}$ being the discrete time parameter. We find upper and lower bounds on the one-shot $ε$-error information transmission capacities of $Φ^n$ for a finite time $n\in \mathbb{N}$ and $ε\in [0,1)$ in terms of the structure of the peripheral space of the channel $Φ$. We consider transmission of $(i)$ classical information (both in the unassisted and entanglement-assisted settings); $(ii)$ quantum information and $(iii)$ private classical information.

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arXiv2024

Detecting positive quantum capacities of quantum channels

Determining whether a noisy quantum channel can be used to reliably transmit quantum information at a non-zero rate is a challenging problem in quantum information theory. This is because it requires computation of the channel's coherent information for an unbounded number of copies of the channel. In this paper, we devise an elementary perturbative technique to solve this problem in a wide variety of circumstances. Our analysis reveals that a channel's ability to transmit information is intimately connected to the relative sizes of its input, output, and environment spaces. We exploit this link to develop easy tests which can be used to detect positivity of quantum channel capacities simply by comparing the channels' input, output, and environment dimensions. Several noteworthy examples, such as the depolarizing and transpose-depolarizing channels (including the Werner-Holevo channel), dephasing channels, generalized Pauli channels, multi-level amplitude damping channels, and (conjugate) diagonal unitary covariant channels, serve to aptly exhibit the utility of our method. Notably, in all these examples, the coherent information of a single copy of the channel turns out to be positive.

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arXiv2024

Fully undistillable quantum states are separable

Assume that Alice, Bob, and Charlie share a tripartite pure state $|ψ_{ABC}\rangle$. We prove that if Alice cannot distill entanglement with either Bob or Charlie using $|ψ_{ABC}\rangle$ and local operations with any one of the following configurations for classical communication: $(A\to B, A\leftrightarrow C), (A\leftrightarrow B, A\to C),$ and $(A\leftrightarrow B, A\leftrightarrow C)$, then the same is also true for the other two configurations. Moreover, this happens precisely when the state is such that both its reductions on systems $AB$ and $AC$ are separable, which is further equivalent to the reductions being PPT. This, in particular, implies that any NPT bipartite state is such that either the state itself or its complement is 2-way distillable. To prove these results, we first obtain an explicit lower bound on the 2-way distillable entanglement of low rank bipartite states. Furthermore, we show that even though not all low rank states are 1-way distillable, a randomly sampled low rank state will almost surely be 1-way distillable.

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