arXiv subjects

Nilanjana Datta

Publications and source records attributed to Nilanjana Datta.

At least 37 records · Page 2Linked to original sources

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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arXiv2024

Robustness of Fixed Points of Quantum Channels and Application to Approximate Quantum Markov Chains

Given a quantum channel and a state which satisfy a fixed point equation approximately (say, up to an error $\varepsilon$), can one find a new channel and a state, which are respectively close to the original ones, such that they satisfy an exact fixed point equation? It is interesting to ask this question for different choices of constraints on the structures of the original channel and state, and requiring that these are also satisfied by the new channel and state. We affirmatively answer the above question, under fairly general assumptions on these structures, through a compactness argument. Additionally, for channels and states satisfying certain specific structures, we find explicit upper bounds on the distances between the pairs of channels (and states) in question. When these distances decay quickly (in a particular, desirable manner) as $\varepsilon\to 0$, we say that the original approximate fixed point equation is rapidly fixable. We establish rapid fixability, not only for general quantum channels, but also when the original and new channels are both required to be unitary, mixed unitary or unital. In contrast, for the case of bipartite quantum systems with channels acting trivially on one subsystem, we prove that approximate fixed point equations are not rapidly fixable. In this case, the distance to the closest channel (and state) which satisfy an exact fixed point equation can depend on the dimension of the quantum system in an undesirable way. We apply our results on approximate fixed point equations to the question of robustness of quantum Markov chains (QMC) and establish the following: For any tripartite quantum state, there exists a dimension-dependent upper bound on its distance to the set of QMCs, which decays to zero as the conditional mutual information of the state vanishes.

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arXiv2024

Parallelization of Adaptive Quantum Channel Discrimination in the Non-Asymptotic Regime

We investigate the performance of parallel and adaptive quantum channel discrimination strategies for a finite number of channel uses. It has recently been shown that, in the asymmetric setting with asymptotically vanishing type I error probability, adaptive strategies are asymptotically not more powerful than parallel ones. We extend this result to the non-asymptotic regime with finitely many channel uses, by explicitly constructing a parallel strategy for any given adaptive strategy, and bounding the difference in their performances, measured in terms of the decay rate of the type II error probability per channel use. We further show that all parallel strategies can be optimized over in time polynomial in the number of channel uses, and hence our result can also be used to obtain a poly-time-computable asymptotically tight upper bound on the performance of general adaptive strategies.

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arXiv2023

Infinite Dimensional Asymmetric Quantum Channel Discrimination

We study asymmetric binary channel discrimination, for qantum channels acting on separable Hilbert spaces. We establish quantum Stein's lemma for channels for both adaptive and parallel strategies, and show that under finiteness of the geometric Rényi divergence between the two channels for some $α> 1$, adaptive strategies offer no asymptotic advantage over parallel ones. One major step in our argument is to demonstrate that the geometric Rényi divergence satisfies a chain rule and is additive for channels also in infinite dimensions. These results may be of independent interest. Furthermore, we not only show asymptotic equivalence of parallel and adaptive strategies, but explicitly construct a parallel strategy which approximates a given adaptive $n$-shot strategy, and give an explicit bound on the difference between the discrimination errors for these two strategies. This extends the finite dimensional result from [B. Bergh et al., arxiv:2206.08350]. Finally, this also allows us to conclude, that the chain rule for the Umegaki relative entropy in infinite dimensions, recently shown in [O. Fawzi, L. Gao, and M. Rahaman, arxiv:2212.14700v2] given finiteness of the max divergence between the two channels, also holds under the weaker condition of finiteness of the geometric Rényi divergence. We give explicit examples of channels which show that these two finiteness conditions are not equivalent.

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arXiv2023

Classical shadow tomography for continuous variables quantum systems

In this article we develop a continuous variable (CV) shadow tomography scheme with wide ranging applications in quantum optics. Our work is motivated by the increasing experimental and technological relevance of CV systems in quantum information, quantum communication, quantum sensing, quantum simulations, quantum computing and error correction. We introduce two experimentally realisable schemes for obtaining classical shadows of CV (possibly non-Gaussian) quantum states using only randomised Gaussian unitaries and easily implementable Gaussian measurements such as homodyne and heterodyne detection. For both schemes, we show that $N=O\big(\operatorname{poly}\big(\frac{1}ε,\log\big(\frac{1}δ\big),M_n^{r+α},\log(m)\big)\big)$ samples of an unknown $m$-mode state $ρ$ suffice to learn the expected value of any $r$-local polynomial in the canonical observables of degree $α$, both with high probability $1-δ$ and accuracy $ε$, as long as the state $ρ$ has moments of order $n>α$ bounded by $M_n$. By simultaneously truncating states and operators in energy and phase space, we are able to overcome new mathematical challenges that arise due to the infinite-dimensionality of CV systems. We also provide a scheme to learn nonlinear functionals of the state, such as entropies over any small number of modes, by leveraging recent energy-constrained entropic continuity bounds. Finally, we provide numerical evidence of the efficiency of our protocols in the case of CV states of relevance in quantum information theory, including ground states of quadratic Hamiltonians of many-body systems and cat qubit states. We expect our scheme to provide good recovery in learning relevant states of 2D materials and photonic crystals.

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arXiv2022

Coherent information of a quantum channel or its complement is generically positive

The task of determining whether a given quantum channel has positive capacity to transmit quantum information is a fundamental open problem in quantum information theory. In general, the coherent information needs to be computed for an unbounded number of copies of a channel in order to detect a positive value of its quantum capacity. However, in this Letter, we show that the coherent information of a single copy of a randomly selected channel is positive almost surely if the channel's output space is larger than its environment. Hence, in this case, a single copy of the channel typically suffices to determine positivity of its quantum capacity. Put differently, channels with zero coherent information have measure zero in the subset of channels for which the output space is larger than the environment. On the other hand, if the environment is larger than the channel's output space, identical results hold for the channel's complement.

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arXiv2022

Ergodic theory of diagonal orthogonal covariant quantum channels

We analyze the ergodic properties of quantum channels that are covariant with respect to diagonal orthogonal transformations. We prove that the ergodic behaviour of a channel in this class is essentially governed by a classical stochastic matrix. This allows us to exploit tools from classical ergodic theory to study quantum ergodicity of such channels. As an application of our analysis, we study dual unitary brickwork circuits which have recently been proposed as minimal models of quantum chaos in many-body systems. Upon imposing a local diagonal orthogonal invariance symmetry on these circuits, the long-term behaviour of spatio-temporal correlations between local observables in such circuits is completely determined by the ergodic properties of a channel that is covariant under diagonal orthogonal transformations. We utilize this fact to show that such symmetric dual unitary circuits exhibit a rich variety of ergodic behaviours, thus emphasizing their importance.

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arXiv2022

A tight uniform continuity bound for the Arimoto-Rényi conditional entropy and its extension to classical-quantum states

We prove a tight uniform continuity bound for Arimoto's version of the conditional $α$-Rényi entropy, for the range $α\in [0, 1)$. This definition of the conditional Rényi entropy is the most natural one among the multiple forms which exist in the literature, since it satisfies two desirable properties of a conditional entropy, namely, the fact that conditioning reduces entropy, and that the associated reduction in uncertainty cannot exceed the information gained by conditioning. Furthermore, it has found interesting applications in various information theoretic tasks such as guessing with side information and sequential decoding. This conditional entropy reduces to the conditional Shannon entropy in the limit $α\to 1$, and this in turn allows us to recover the recently obtained tight uniform continuity bound for the latter from our result. Finally, we apply our result to obtain a tight uniform continuity bound for the conditional $α$-Rényi entropy of a classical-quantum state, for $α$ in the same range as above. This again yields the corresponding known bound for the conditional entropy of the state in the limit $α\to 1$.

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arXiv2022

Total insecurity of communication via strong converse for quantum privacy amplification

Quantum privacy amplification is a central task in quantum cryptography. Given shared randomness, which is initially correlated with a quantum system held by an eavesdropper, the goal is to extract uniform randomness which is decoupled from the latter. The optimal rate for this task is known to satisfy the strong converse property and we provide a lower bound on the corresponding strong converse exponent. In the strong converse region, the distance of the final state of the protocol from the desired decoupled state converges exponentially fast to its maximal value, in the asymptotic limit. We show that this necessarily leads to totally insecure communication by establishing that the eavesdropper can infer any sent messages with certainty, when given very limited extra information. In fact, we prove that in the strong converse region, the eavesdropper has an exponential advantage in inferring the sent message correctly, compared to the achievability region. Additionally we establish the following technical result, which is central to our proofs, and is of independent interest: the smoothing parameter for the smoothed max-relative entropy satisfies the strong converse property.

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arXiv2021

Guesswork with Quantum Side Information

What is the minimum number of guesses needed on average to correctly guess a realization of a random variable? The answer to this question led to the introduction of the notion of a quantity called guesswork by Massey in 1994, which can be viewed as an alternate security criterion to entropy. In this paper, we consider the guesswork in the presence of quantum side information, and show that a general sequential guessing strategy is equivalent to performing a single measurement and choosing a guessing strategy from the outcome. We use this result to deduce entropic one-shot and asymptotic bounds on the guesswork in the presence of quantum side information, and to formulate a semi-definite program (SDP) to calculate the quantity. We evaluate the guesswork for a simple example involving the BB84 states, both numerically and analytically, and prove a continuity result that certifies the security of slightly imperfect key states when the guesswork is used as the security criterion.

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arXiv2021

Second-order asymptotics for source coding, dense coding and pure-state entanglement conversions

We introduce two variants of the information spectrum relative entropy defined by Tomamichel and Hayashi which have the particular advantage of satisfying the data-processing inequality, i.e. monotonicity under quantum operations. This property allows us to obtain one-shot bounds for various information-processing tasks in terms of these quantities. Moreover, these relative entropies have a second order asymptotic expansion, which in turn yields tight second order asymptotics for optimal rates of these tasks in the i.i.d. setting. The tasks studied in this paper are fixed-length quantum source coding, noisy dense coding, entanglement concentration, pure-state entanglement dilution, and transmission of information through a classical-quantum channel. In the latter case, we retrieve the second order asymptotics obtained by Tomamichel and Tan. Our results also yield the known second order asymptotics of fixed-length classical source coding derived by Hayashi. The second order asymptotics of entanglement concentration and dilution provide a refinement of the inefficiency of these protocols - a quantity which, in the case of entanglement dilution, was studied by Harrow and Lo. We prove how the discrepancy between the optimal rates of these two processes in the second order implies the irreversibility of entanglement concentration established by Kumagai and Hayashi. In addition, the spectral divergence rates of the Information Spectrum Approach (ISA) can be retrieved from our relative entropies in the asymptotic limit. This enables us to directly obtain the more general results of the ISA from our one-shot bounds.

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arXiv2021

Symmetric distinguishability as a quantum resource

We develop a resource theory of symmetric distinguishability, the fundamental objects of which are elementary quantum information sources, i.e., sources that emit one of two possible quantum states with given prior probabilities. Such a source can be represented by a classical-quantum state of a composite system $XA$, corresponding to an ensemble of two quantum states, with $X$ being classical and $A$ being quantum. We study the resource theory for two different classes of free operations: $(i)$ ${\rm{CPTP}}_A$, which consists of quantum channels acting only on $A$, and $(ii)$ conditional doubly stochastic (CDS) maps acting on $XA$. We introduce the notion of symmetric distinguishability of an elementary source and prove that it is a monotone under both these classes of free operations. We study the tasks of distillation and dilution of symmetric distinguishability, both in the one-shot and asymptotic regimes. We prove that in the asymptotic regime, the optimal rate of converting one elementary source to another is equal to the ratio of their quantum Chernoff divergences, under both these classes of free operations. This imparts a new operational interpretation to the quantum Chernoff divergence. We also obtain interesting operational interpretations of the Thompson metric, in the context of the dilution of symmetric distinguishability.

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arXiv2021

Universal proofs of entropic continuity bounds via majorization flow

We introduce a notion of majorization flow, and demonstrate it to be a powerful tool for deriving simple and universal proofs of continuity bounds for entropic functions relevant in information theory. In particular, for the case of the alpha-Rényi entropy, whose connections to thermodynamics are discussed in this article, majorization flow yields a Lipschitz continuity bound for the case alpha > 1, thus resolving an open problem and providing a substantial improvement over previously known bounds.

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arXiv2021

Energy-constrained discrimination of unitaries, quantum speed limits and a Gaussian Solovay-Kitaev theorem

We investigate the energy-constrained (EC) diamond norm distance between unitary channels acting on possibly infinite-dimensional quantum systems, and establish a number of results. Firstly, we prove that optimal EC discrimination between two unitary channels does not require the use of any entanglement. Extending a result by Acín, we also show that a finite number of parallel queries suffices to achieve zero error discrimination even in this EC setting. Secondly, we employ EC diamond norms to study a novel type of quantum speed limits, which apply to pairs of quantum dynamical semigroups. We expect these results to be relevant for benchmarking internal dynamics of quantum devices. Thirdly, we establish a version of the Solovay--Kitaev theorem that applies to the group of Gaussian unitaries over a finite number of modes, with the approximation error being measured with respect to the EC diamond norm relative to the photon number Hamiltonian.

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arXiv2021

Interpolating between symmetric and asymmetric hypothesis testing

The task of binary quantum hypothesis testing is to determine the state of a quantum system via measurements on it, given the side information that it is in one of two possible states, say $ρ$ and $σ$. This task is generally studied in either the symmetric setting, in which the two possible errors incurred in the task (the so-called type I and type II errors) are treated on an equal footing, or the asymmetric setting in which one minimizes the type II error probability under the constraint that the corresponding type I error probability is below a given threshold. Here we define a one-parameter family of binary quantum hypothesis testing tasks, which we call $s$-hypothesis testing, and in which the relative significance of the two errors are weighted by a parameter $s$. In particular, $s$-hypothesis testing interpolates continuously between the regimes of symmetric and asymmetric hypothesis testing. Moreover, if arbitrarily many identical copies of the system are assumed to be available, then the minimal error probability of $s$-hypothesis testing is shown to decay exponentially in the number of copies, with a decay rate given by a quantum divergence which we denote as $ξ_s(ρ\|σ)$, and which satisfies a host of interesting properties. Moreover, this one-parameter family of divergences interpolates continuously between the corresponding decay rates for symmetric hypothesis testing (the quantum Chernoff divergence) for $s = 1$, and asymmetric hypothesis testing (the Umegaki relative entropy) for $s = 0$.

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arXiv2021

The $α\to 1$ Limit of the Sharp Quantum Rényi Divergence

Fawzi and Fawzi recently defined the sharp Rényi divergence, $D_α^\#$, for $α\in (1, \infty)$, as an additional quantum Rényi divergence with nice mathematical properties and applications in quantum channel discrimination and quantum communication. One of their open questions was the limit $α \to 1$ of this divergence. By finding a new expression of the sharp divergence in terms of a minimization of the geometric Rényi divergence, we show that this limit is equal to the Belavkin-Staszewski relative entropy. Analogous minimizations of arbitrary generalized divergences lead to a new family of generalized divergences that we call kringel divergences, and for which we prove various properties including the data-processing inequality.

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arXiv2020

Quantum Zeno effect for open quantum systems

We prove the quantum Zeno effect in open quantum systems whose evolution, governed by quantum dynamical semigroups, is repeatedly and frequently interrupted by the action of a quantum operation. For the case of a quantum dynamical semigroup with a bounded generator, our analysis leads to a refinement of existing results and extends them to a larger class of quantum operations. We also prove the existence of a novel strong quantum Zeno limit for quantum operations for which a certain spectral gap assumption, which all previous results relied on, is lifted. The quantum operations are instead required to satisfy a weaker property of strong power-convergence. In addition, we establish, for the first time, the existence of a quantum Zeno limit for the case of unbounded generators. We also provide a variety of physically interesting examples of quantum operations to which our results apply.

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