arXiv subjects

Nilanjana Datta

Publications and source records attributed to Nilanjana Datta.

At least 73 records · Page 4Linked to original sources

arXiv2017

Finite blocklength and moderate deviation analysis of hypothesis testing of correlated quantum states and application to classical-quantum channels with memory

Martingale concentration inequalities constitute a powerful mathematical tool in the analysis of problems in a wide variety of fields ranging from probability and statistics to information theory and machine learning. Here we apply techniques borrowed from this field to quantum hypothesis testing, which is the problem of discriminating quantum states belonging to two different sequences $\{ρ_n\}_{n}$ and $\{σ_n\}_n$. We obtain upper bounds on the finite blocklength type II Stein- and Hoeffding errors, which, for i.i.d. states, are in general tighter than the corresponding bounds obtained by Audenaert, Mosonyi and Verstraete [Journal of Mathematical Physics, 53(12), 2012]. We also derive finite blocklength bounds and moderate deviation results for pairs of sequences of correlated states satisfying a (non-homogeneous) factorization property. Examples of such sequences include Gibbs states of spin chains with translation-invariant finite range interaction, as well as finitely correlated quantum states. We apply our results to find bounds on the capacity of a certain class of classical-quantum channels with memory, which satisfy a so-called channel factorization property- both in the finite blocklength and moderate deviation regimes.

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arXiv2016

Data processing for the sandwiched Rényi divergence: a condition for equality

The $α$-sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for $α\geq 1/2$. In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the $α$-sandwiched Rényi divergence for all $α\geq 1/2$. For the range $α\in [1/2,1)$, our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace, and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb [J. Math. Phys. 54.12 (2013), p. 122201] using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a Rényi version of the Araki-Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb [Lett. Math. Phys. 101.1 (2012), pp. 1-11] about equality in the original Araki-Lieb inequality. Furthermore, we prove a general lower bound on a Rényi version of the entanglement of formation, and observe that it is attained by states saturating the Rényi version of the Araki-Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.

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arXiv2016

Strong converse theorems using Rényi entropies

We use a Rényi entropy method to prove strong converse theorems for certain information-theoretic tasks which involve local operations and quantum or classical communication between two parties. These include state redistribution, coherent state merging, quantum state splitting, measurement compression with quantum side information, randomness extraction against quantum side information, and data compression with quantum side information. The method we employ in proving these results extends ideas developed by Sharma [arXiv:1404.5940], which he used to give a new proof of the strong converse theorem for state merging. For state redistribution, we prove the strong converse property for the boundary of the entire achievable rate region in the $(e,q)$-plane, where $e$ and $q$ denote the entanglement cost and quantum communication cost, respectively. In the case of measurement compression with quantum side information, we prove a strong converse theorem for the classical communication cost, which is a new result extending the previously known weak converse. For the remaining tasks, we provide new proofs for strong converse theorems previously established using smooth entropies. For each task, we obtain the strong converse theorem from explicit bounds on the figure of merit of the task in terms of a Rényi generalization of the optimal rate. Hence, we identify candidates for the strong converse exponents for each task discussed in this paper. To prove our results, we establish various new entropic inequalities, which might be of independent interest. These involve conditional entropies and mutual information derived from the sandwiched Rényi divergence. In particular, we obtain novel bounds relating these quantities, as well as the Rényi conditional mutual information, to the fidelity of two quantum states.

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arXiv2016

Second order asymptotics of visible mixed quantum source coding via universal codes

The simplest example of a quantum information source with memory is a mixed source which emits signals entirely from one of two memoryless quantum sources with given a priori probabilities. Considering a mixed source consisting of a general one-parameter family of memoryless sources, we derive the second order asymptotic rate for fixed-length visible source coding. Furthermore, we specialize our main result to a mixed source consisting of two memoryless sources. Our results provide the first example of second order asymptotics for a quantum information-processing task employing a resource with memory. For the case of a classical mixed source (using a finite alphabet), our results reduce to those obtained by Nomura and Han [IEEE Trans. on Inf. Th. 59.1 (2013), pp. 1-16]. To prove the achievability part of our main result, we introduce universal quantum source codes achieving second order asymptotic rates. These are obtained by an extension of Hayashi's construction [IEEE Trans. on Inf. Th. 54.10 (2008), pp. 4619-4637] of their classical counterparts.

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arXiv2016

Entropy power inequalities for qudits

Shannon's entropy power inequality (EPI) can be viewed as a statement of concavity of an entropic function of a continuous random variable under a scaled addition rule: $$f(\sqrt{a}\,X + \sqrt{1-a}\,Y) \ge a f(X) + (1-a) f(Y) \quad \forall \, a \in [0,1].$$ Here, $X$ and $Y$ are continuous random variables and the function $f$ is either the differential entropy or the entropy power. König and Smith [arXiv:1205.3409] and De Palma, Mari, and Giovannetti [arXiv:1402.0404] obtained quantum analogues of these inequalities for continuous-variable quantum systems, where $X$ and $Y$ are replaced by bosonic fields and the addition rule is the action of a beamsplitter with transmissivity $a$ on those fields. In this paper, we similarly establish a class of EPI analogues for $d$-level quantum systems (i.e. qudits). The underlying addition rule for which these inequalities hold is given by a quantum channel that depends on the parameter $a \in [0,1]$ and acts like a finite-dimensional analogue of a beamsplitter with transmissivity $a$, converting a two-qudit product state into a single qudit state. We refer to this channel as a partial swap channel because of the particular way its output interpolates between the states of the two qudits in the input as $a$ is changed from zero to one. We obtain analogues of Shannon's EPI, not only for the von Neumann entropy and the entropy power for the output of such channels, but for a much larger class of functions as well. This class includes the Rényi entropies and the subentropy. We also prove a qudit analogue of the entropy photon number inequality (EPnI). Finally, for the subclass of partial swap channels for which one of the qudit states in the input is fixed, our EPIs and EPnI yield lower bounds on the minimum output entropy and upper bounds on the Holevo capacity.

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arXiv2016

Second-order asymptotics for quantum hypothesis testing in settings beyond i.i.d. - quantum lattice systems and more

Quantum Stein's Lemma is a cornerstone of quantum statistics and concerns the problem of correctly identifying a quantum state, given the knowledge that it is one of two specific states ($ρ$ or $σ$). It was originally derived in the asymptotic i.i.d. setting, in which arbitrarily many (say, $n$) identical copies of the state ($ρ^{\otimes n}$ or $σ^{\otimes n}$) are considered to be available. In this setting, the lemma states that, for any given upper bound on the probability $α_n$ of erroneously inferring the state to be $σ$, the probability $β_n$ of erroneously inferring the state to be $ρ$ decays exponentially in $n$, with the rate of decay converging to the relative entropy of the two states. The second order asymptotics for quantum hypothesis testing, which establishes the speed of convergence of this rate of decay to its limiting value, was derived in the i.i.d. setting independently by Tomamichel and Hayashi, and Li. We extend this result to settings beyond i.i.d.. Examples of these include Gibbs states of quantum spin systems (with finite-range, translation-invariant interactions) at high temperatures.

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arXiv2016

alpha-z-relative Renyi entropies

We consider a two-parameter family of Rényi relative entropies $D_{α,z}(ρ||σ)$ that are quantum generalisations of the classical Rényi divergence $D_α(p||q)$. This family includes many known relative entropies (or divergences) such as the quantum relative entropy, the recently defined quantum Rényi divergences, as well as the quantum Rényi relative entropies. All its members satisfy the quantum generalizations of Rényi's axioms for a divergence. We consider the range of the parameters $α,z$ for which the data processing inequality holds. We also investigate a variety of limiting cases for the two parameters, obtaining explicit formulas for each one of them.

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arXiv2016

On the Second-Order Asymptotics for Entanglement-Assisted Communication

The entanglement-assisted classical capacity of a quantum channel is known to provide the formal quantum generalization of Shannon's classical channel capacity theorem, in the sense that it admits a single-letter characterization in terms of the quantum mutual information and does not increase in the presence of a noiseless quantum feedback channel from receiver to sender. In this work, we investigate second-order asymptotics of the entanglement-assisted classical communication task. That is, we consider how quickly the rates of entanglement-assisted codes converge to the entanglement-assisted classical capacity of a channel as a function of the number of channel uses and the error tolerance. We define a quantum generalization of the mutual information variance of a channel in the entanglement-assisted setting. For covariant channels, we show that this quantity is equal to the channel dispersion, and thus completely characterize the convergence towards the entanglement-assisted classical capacity when the number of channel uses increases. Our results also apply to entanglement-assisted quantum communication, due to the equivalence between entanglement-assisted classical and quantum communication established by the teleportation and super-dense coding protocols.

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arXiv2016

Equivalence between divisibility and monotonic decrease of information in classical and quantum stochastic processes

The crucial feature of a memoryless stochastic process is that any information about its state can only decrease as the system evolves. Here we show that such a decrease of information is equivalent to the underlying stochastic evolution being divisible. The main result, which holds for both classical and quantum stochastic processes, rely on a quantum version of the so-called Blackwell-Sherman-Stein theorem in classical statistics.

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arXiv2015

Quantum Markov chains, sufficiency of quantum channels, and Renyi information measures

A short quantum Markov chain is a tripartite state $ρ_{ABC}$ such that system $A$ can be recovered perfectly by acting on system $C$ of the reduced state $ρ_{BC}$. Such states have conditional mutual information $I(A;B|C)$ equal to zero and are the only states with this property. A quantum channel $\mathcal{N}$ is sufficient for two states $ρ$ and $σ$ if there exists a recovery channel using which one can perfectly recover $ρ$ from $\mathcal{N}(ρ)$ and $σ$ from $\mathcal{N}(σ)$. The relative entropy difference $D(ρ\Vertσ)-D(\mathcal{N}(ρ)\Vert\mathcal{N}(σ))$ is equal to zero if and only if $\mathcal{N}$ is sufficient for $ρ$ and $σ$. In this paper, we show that these properties extend to Renyi generalizations of these information measures which were proposed in [Berta et al., J. Math. Phys. 56, 022205, (2015)] and [Seshadreesan et al., J. Phys. A 48, 395303, (2015)], thus providing an alternate characterization of short quantum Markov chains and sufficient quantum channels. These results give further support to these quantities as being legitimate Renyi generalizations of the conditional mutual information and the relative entropy difference. Along the way, we solve some open questions of Ruskai and Zhang, regarding the trace of particular matrices that arise in the study of monotonicity of relative entropy under quantum operations and strong subadditivity of the von Neumann entropy.

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arXiv2014

Game-theoretic characterization of antidegradable channels

We introduce a guessing game involving a quantum channel, three parties - the sender, the receiver and an eavesdropper, Eve - and a quantum public side channel. We prove that a necessary and sufficient condition for the quantum channel to be antidegradable, is that Eve wins the game. We thus obtain a complete operational characterization of antidegradable channels in a game-theoretic framework.

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arXiv2014

An upper bound on the second order asymptotic expansion for the quantum communication cost of state redistribution

State redistribution is the protocol in which, given an arbitrary tripartite quantum state, with two of the subsystems initially being with Alice and one being with Bob, the goal is for Alice to send one of her subsystems to Bob, possibly with the help of prior shared entanglement. We derive an upper bound on the second order asymptotic expansion for the quantum communication cost of achieving state redistribution with a given finite accuracy. In proving our result, we also obtain an upper bound on the quantum communication cost of this protocol in the one-shot setting, by using the protocol of coherent state merging as a primitive.

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arXiv2014

Bounding Polynomial Entanglement Measures for Mixed States

We generalize the notion of the best separable approximation (BSA) and best W-class approximation (BWA) to arbitrary pure state entanglement measures, defining the best zero-$E$ approximation (BEA). We show that for any polynomial entanglement measure $E$, any mixed state $ρ$ admits at least one "$S$-decomposition," i.e., a decomposition in terms of a mixed state on which $E$ is equal to zero, and a single additional pure state with (possibly) non-zero $E$. We show that the BEA is not in general the optimal $S$-decomposition from the point of view of bounding the entanglement of $ρ$, and describe an algorithm to construct the entanglement-minimizing $S$-decomposition for $ρ$ and place an upper bound on $E(ρ)$. When applied to the three-tangle, the cost of the algorithm is linear in the rank $d$ of the density matrix and has accuracy comparable to a steepest descent algorithm whose cost scales as $d^8 \log d$. We compare the upper bound to a lower bound algorithm given by Eltschka and Siewert for the three-tangle, and find that on random rank-two three-qubit density matrices, the difference between the upper and lower bounds is $0.14$ on average. We also find that the three-tangle of random full-rank three qubit density matrices is less than $0.023$ on average.

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arXiv2013

Properties of subentropy

Subentropy is an entropy-like quantity that arises in quantum information theory; for example, it provides a tight lower bound on the accessible information for pure state ensembles, dual to the von Neumann entropy upper bound in Holevo's theorem. Here we establish a series of properties of subentropy, paralleling the well-developed analogous theory for von Neumann entropy. Further, we show that subentropy is a lower bound for min-entropy. We introduce a notion of conditional subentropy and show that it can be used to provide an upper bound for the guessing probability of any classical-quantum state of two qubits; we conjecture that the bound applies also in higher dimensions. Finally we give an operational interpretation of subentropy within classical information theory.

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arXiv2013

A limit of the quantum Renyi divergence

Recently, an interesting quantity called the quantum Renyi divergence (or "sandwiched" Renyi relative entropy) was defined for pairs of positive semi-definite operators $ρ$ and $σ$. It depends on a parameter $α$ and acts as a parent quantity for other relative entropies which have important operational significances in quantum information theory: the quantum relative entropy and the min- and max-relative entropies. There is, however, another relative entropy, called the 0-relative Renyi entropy, which plays a key role in the analysis of various quantum information-processing tasks in the one-shot setting. We prove that the 0-relative Renyi entropy is obtainable from the quantum Renyi divergence only if $ρ$ and $σ$ have equal supports. This, along with existing results in the literature, suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived - the quantum Renyi divergence with parameter $α\ge 1/2$, and the $α$-relative Rényi entropy with $α\in [0,1)$.

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arXiv2013

One-shot lossy quantum data compression

We provide a framework for one-shot quantum rate distortion coding, in which the goal is to determine the minimum number of qubits required to compress quantum information as a function of the probability that the distortion incurred upon decompression exceeds some specified level. We obtain a one-shot characterization of the minimum qubit compression size for an entanglement-assisted quantum rate-distortion code in terms of the smooth max-information, a quantity previously employed in the one-shot quantum reverse Shannon theorem. Next, we show how this characterization converges to the known expression for the entanglement-assisted quantum rate distortion function for asymptotically many copies of a memoryless quantum information source. Finally, we give a tight, finite blocklength characterization for the entanglement-assisted minimum qubit compression size of a memoryless isotropic qubit source subject to an average symbol-wise distortion constraint.

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arXiv2013

Quantum rate distortion coding with auxiliary resources

We extend quantum rate distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression. The first setting we consider is that of quantum rate distortion coding with the help of a classical side channel. Our result here is that the regularized entanglement of formation characterizes the quantum rate distortion function, extending earlier work of Devetak and Berger. We also combine this bound with the entanglement-assisted bound from our prior work to obtain the best known bounds on the quantum rate distortion function for an isotropic qubit source. The second setting we consider is that of quantum rate distortion coding with quantum side information (QSI) available to the receiver. In order to prove results in this setting, we first state and prove a quantum reverse Shannon theorem with QSI (for tensor-power states), which extends the known tensor-power quantum reverse Shannon theorem. The achievability part of this theorem relies on the quantum state redistribution protocol, while the converse relies on the fact that the protocol can cause only a negligible disturbance to the joint state of the reference and the receiver's QSI. This quantum reverse Shannon theorem with QSI naturally leads to quantum rate-distortion theorems with QSI, with or without entanglement assistance.

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arXiv2013

A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels

We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Using then a relative entropy version of the Quantum Asymptotic Equipartition Property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger ($\ep$-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot $\ep$-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the $\ep$-independent version of the relative entropy-QAEP, we can recover both the Holevo-Schumacher-Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.

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