arXiv subjects

Pranab Sen

Publications and source records attributed to Pranab Sen.

At least 19 recordsLinked to original sources

arXiv2025

Matrix Chernoff concentration bounds for multipartite soft covering and expander walks

We prove Chernoff style exponential concentration bounds for classical quantum soft covering generalising previous works which gave bounds only in expectation. Our first result is an exponential concentration bound for fully smooth multipartite classical quantum soft covering, extending Ahlswede-Winter's seminal result in several important directions. Next, we prove a new exponential concentration result for smooth unipartite classical quantum soft covering when the samples are taken via a random walk on an expander graph. The resulting expander matrix Chernoff bound complements the results of Garg, Lee, Song and Srivastava in important ways. We prove our new expander matrix Chernoff bound by generalising McDiarmid's method of bounded differences for functions of independent random variables to a new method of bounded excision for functions of expander walks. This new technical tool should be of independent interest. A notable feature of our new concentration bounds is that they have no explicit Hilbert space dimension factor. This is because our bounds are stated in terms of the trace distance of the sample averaged quantum state to the `ideal' quantum state. Our bounds are sensitive to certain smooth Renyi max divergences, giving a clear handle on the number of samples required to achieve a target trace distance. Using these novel features, we prove new one shot inner bounds for sending private classical information over different kinds of quantum wiretap channels with many non-interacting eavesdroppers that are independent of the Hilbert space dimensions of the eavesdroppers. Such powerful results were unknown earlier even in the fully classical setting.

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arXiv2024

Fully smooth one shot multipartite covering and decoupling of quantum states via telescoping

We prove fully smooth one shot multipartite covering, aka convex split, results as well as fully smooth multipartite decoupling results for quantum states. Fully smooth one shot results for these problems were not known earlier, though the works of Cheng, Gao and Berta (arXiv:2304.12056) for convex split, and Colomer and Winter (arXiv:2304.12114) for decoupling, had made substantial progress by introducing a technique called telescoping cum mean zero decomposition of quantum states. We show that the telescoping cum mean zero decomposition technique can in fact be simplified and further extended in order to prove fully smooth decoupling and convex split results. Our techniques allow us to prove the first fully smooth one shot inner bounds for various fundamental network quantum information theory problems like e.g. the generalised Slepian Wolf problem of Anshu, Jain and Warsi (arXiv:1703.09961). We can also prove for the first time the natural polyhedral inner bound for sending quantum information over a quantum multiple access channel with limited entanglement assistance, first conjectured in Chakraborty, Nema and Sen (arXiv::2102.02187).

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arXiv2024

Fully smooth one shot multipartite soft covering of quantum states without pairwise independence

We provide a powerful machinery to prove fully smooth one shot multipartite covering, aka convex split, type results for quantum states. In the important case of smooth multipartite convex split for classical quantum states, aka smooth multipartite soft covering, our machinery works even when certain marginals of these states do not satisfy pairwise independence. The recent paper (arXiv:2410.17893) gave the first proof of fully smooth multipartite convex split by simplifying and extending a technique called telescoping, developed originally for convex split by (arXiv:2304.12056). However, that work as well as all earlier works on convex split assumed pairwise or even more independence amongst suitable marginals of the quantum states. We develop our machinery by leveraging known results from (arXiv:1806.07278) involving tilting and augmentation smoothing of quantum states, combined with a novel observation that a natural quantum operation `flattening' quantum states actually preserves the fidelity. This machinery is powerful enough to lead to non pairwise independent results as mentioned above. As an application of our soft covering lemma without pairwise independence, we prove the `natural' one shot inner bounds for sending private classical information over a quantum wiretap interference channel, even when the classical encoders at the input lose pairwise independence in their encoding strategies to a certain extent. This result was unknown earlier even in the classical setting.

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arXiv2024

One-Shot Non-Catalytic Distributed Purity Distillation

Pure states are an important resource in many quantum information processing protocols. However, even making a fixed pure state, say $|0\rangle$, in the laboratory requires a considerable amount of effort. Often one ends up with a mixed state $\rho$ whose classical description is nevertheless known. Hence it is important to develop protocols that extract a fixed pure state from a known mixed state. In this work, we study the problem of extracting a fixed pure state $|0\rangle^{A'} |0\rangle^{B'}$ from a known pure state $\rho^{AB}$ distributed between two parties $A$ and $B$. Here, $A'$, $B'$ are subspaces of $A$, $B$ and the total amount of purity extracted is $\log |A'| + \log |B'|$. The parties can borrow local pure ancilla, apply local unitary operations and send a message from $A$ to $B$ through a dephasing channel. If local pure ancilla is borrowed, it must be subtracted in order to properly account for the purity extracted. We obtain the most efficient achievable bounds on one shot distributed purity extraction, in terms of the rate of local ancilla borrowed by the protocol, while distilling pure qubits at the best known rate. Our protocols borrow little to no local pure ancilla. Our bounds improve upon the existing bounds for this problem in both one shot as well as asymptotic iid settings. In particular they subsume all the asymptotic iid results of Devetak and Krovi-Devetak. In addition, we derive upper bounds for the rate of distillation in the one shot setting, which nearly match our achievable bounds.

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arXiv2022

Centralised multi link measurement compression with side information

We prove new one shot achievability results for measurement compression of quantum instruments with side information at the receiver. Unlike previous one shot results for this problem, our one shot bounds are nearly optimal and do not need catalytic randomness. In fact, we state a more general problem called centralised multi link measurement compression with quantum side information and provide one shot achievability results for it. As a simple corollary, we obtain one shot measurement compression results for quantum instruments with side information that we mentioned earlier. All our one shot results lead to the standard results for this problem in the asymptotic iid setting. We prove our achievability bounds by first proving a novel sequential classical quantum multipartite covering lemma, which should be of independent interest.

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arXiv2021

One-shot inner bounds for sending private classical information over a quantum MAC

We provide the first inner bounds for sending private classical information over a quantum multiple access channel. We do so by using three powerful information theoretic techniques: rate splitting, quantum simultaneous decoding for multiple access channels, and a novel smoothed distributed covering lemma for classical quantum channels. Our inner bounds are given in the one shot setting and accordingly the three techniques used are all very recent ones specifically designed to work in this setting. The last technique is new to this work and is our main technical advancement. For the asymptotic iid setting, our one shot inner bounds lead to the natural quantum analogue of the best classical inner bounds for this problem.

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arXiv2021

One-shot multi-sender decoupling and simultaneous decoding for the quantum MAC

In this work, we prove a novel one-shot multi-sender decoupling theorem generalising Dupuis result. We start off with a multipartite quantum state, say on A1 A2 R, where A1, A2 are treated as the two sender systems and R is the reference system. We apply independent Haar random unitaries in tensor product on A1 and A2 and then send the resulting systems through a quantum channel. We want the channel output B to be almost in tensor with the untouched reference R. Our main result shows that this is indeed the case if suitable entropic conditions are met. An immediate application of our main result is to obtain a one-shot simultaneous decoder for sending quantum information over a k-sender entanglement unassisted quantum multiple access channel (QMAC). The rate region achieved by this decoder is the natural one-shot quantum analogue of the pentagonal classical rate region. Assuming a simultaneous smoothing conjecture, this one-shot rate region approaches the optimal rate region of Yard, Dein the asymptotic iid limit. Our work is the first one to obtain a non-trivial simultaneous decoder for the QMAC with limited entanglement assistance in both one-shot and asymptotic iid settings; previous works used unlimited entanglement assistance.

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arXiv2021

Novel one-shot inner bounds for unassisted fully quantum channels via rate splitting

We prove the first non-trivial one-shot inner bounds for sending quantum information over an entanglement unassisted two-sender quantum multiple access channel (QMAC) and an unassisted two-sender two-receiver quantum interference channel (QIC). Previous works only studied the unassisted QMAC in the limit of many independent and identical uses of the channel also known as the asymptotic iid limit, and did not study the unassisted QIC at all. We employ two techniques, rate splitting and successive cancellation}, in order to obtain our inner bound. Rate splitting was earlier used to obtain inner bounds, avoiding time sharing, for classical channels in the asymptotic iid setting. Our main technical contribution is to extend rate splitting from the classical asymptotic iid setting to the quantum one-shot setting. In the asymptotic iid limit our one-shot inner bound for QMAC approaches the rate region of Yard, Devetak and Hayden. For the QIC we get novel non-trivial rate regions in the asymptotic iid setting. All our results also extend to the case where limited entanglement assistance is provided, in both one-shot and asymptotic iid settings. The limited entanglement results for one-setting for both QMAC and QIC are new. For the QIC the limited entanglement results are new even in the asymptotic iid setting.

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arXiv2020

High probability decoupling via approximate unitary designs and efficient relative thermalisation

We prove a new concentration result for non-catalytic decoupling by showing that, for suitably large $t$, applying a unitary chosen uniformly at random from an approximate $t$-design on a quantum system followed by a fixed quantum operation almost decouples, with high probability, the given system from another reference system to which it may initially have been correlated. Earlier works either did not obtain high decoupling probability, or used provably inefficient unitaries, or required catalytic entanglement for decoupling. In contrast, our approximate unitary designs always guarantee decoupling with exponentially high probability and, under certain conditions, lead to computationally efficient unitaries. As a result we conclude that, under suitable conditions, efficiently implementable approximate unitary designs achieve relative thermalisation in quantum thermodynamics with exponentially high probability. We also show the scrambling property of black hole, when the black hole evolution is according to pseudorandom approximate unitary $t$-design, as opposed to the Haar random evolution considered earlier by Hayden-Preskill.

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arXiv2019

Approximate unitary $n^{2/3}$-designs give rise to quantum channels with super additive classical Holevo capacity

In a breakthrough, Hastings' showed that there exist quantum channels whose classical capacity is superadditive i.e. more classical information can be transmitted by quantum encoding strategies entangled across multiple channel uses as compared to unentangled quantum encoding strategies. Hastings' proof used Haar random unitaries to exhibit superadditivity. In this paper we show that a unitary chosen uniformly at random from an approximate $n^{2/3}$-design gives rise to a quantum channel with superadditive classical Holevo capacity, where $n$ is the dimension of the unitary exhibiting the Stinespring dilation of the channel superoperator. We prove a sharp Dvoretzky-like theorem (similar to Aubrun, Szarek, Werner, 2010) stating that, with high probability under the choice of a unitary from an approximate $t$-design, random subspaces of large dimension make a Lipschitz function take almost constant value. Such theorems were known earlier only for Haar random unitaries. We obtain our result by appealing to Low's technique for proving concentration of measure for an approximate $t$-design, combined with a stratified analysis of the variational behaviour of Lipschitz functions on the unit sphere in high dimension. The stratified analysis is the main technical advance of this work. Finally we also show that for any $p>1$, approximate unitary $(n^{1.7} \log n)$-designs give rise to channels violating subadditivity of R\'{e}nyi $p$-entropy. In addition to stratified analysis, the proof of this result uses a new technique of approximating a monotonic differentiable function defined on a closed bounded interval and its derivative by moderate degree polynomials which should be of independent interest. Hence, our work can be viewed as a partial derandomisation of Hastings' result and a step towards the quest of finding an explicit quantum channel with superadditive classical Holevo capacity.

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arXiv2019

Efficiently estimating average fidelity of a quantum logic gate using few classical random bits

We give three new algorithms for efficient in-place estimation, without using ancilla qubits, of average fidelity of a quantum logic gate acting on a d-dimensional system using much fewer random bits than what was known so far. Previous approaches for efficient estimation of average gate fidelity replaced Haar random unitaries in the naive estimation algorithm by approximate unitary 2-designs, and sampled them uniformly and independently. In contrast, in our first algorithm we sample the unitaries of the approximate unitary 2-design uniformly using a limited independence pseudorandom generator, a powerful tool from derandomisation theory. This algorithm uses the same number of basic operations as previous efficient algorithms but much fewer number of random bits. Reducing the requirement of classical random bits increases the reliability of estimation as often, high quality random bits are an expensive computational resource. Our second efficient algorithm, based on a 4-quantum tensor product expander, works if the gate dimension d is large. It uses even lesser random bits than the first algorithm, and has the added advantage that it needs to implement only one unitary versus potentially all the unitaries of an approximate 2-design in the first algorithm. Our third efficient algorithm, based on an l-quantum tensor product expander for moderately large values of l, works for all values of the parameters. It uses slightly more random bits than the other algorithms but has the advantage that it needs to implement only a small number of unitaries versus potentially all the unitaries of an approximate 2-design in the first algorithm. This advantage is of great importance to experimental implementations in the near future.

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arXiv2018

Efficient quantum tensor product expanders and unitary t-designs via the zigzag product

A classical t-tensor product expander is a natural way of formalising correlated walks of t particles on a regular expander graph. A quantum t-tensor product expander is a completely positive trace preserving map that is a straightforward analogue of a classical t-tensor product expander. Interest in these maps arises from the fact that iterating a quantum t-tensor product expander gives us a unitary t-design, which has many applications to quantum computation and information. We show that the zigzag product of a high dimensional quantum expander (i.e. t = 1) of moderate degree with a moderate dimensional quantum t-tensor product expander of low degree gives us a high dimensional quantum t-tensor product expander of low degree. Previously such a result was known only for quantum expanders i.e. t = 1. Using the zigzag product we give efficient constructions of quantum t-tensor product expanders in dimension D where t = polylog(D). We then show how replacing the zigzag product by the generalised zigzag product leads to almost-Ramanujan quantum tensor product expanders i.e. having near-optimal almost quadratic tradeoff between the degree and the second largest singular value. Both the products give better tradeoffs between the degree and second largest singular value than what was previously known for efficient constructions.

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arXiv2018

A quantum Johnson-Lindenstrauss lemma via unitary t-designs

The famous Johnson-Lindenstrauss lemma states that for any set of n vectors, there is a linear transformation into a space of dimension O(log n) that approximately preserves all their lengths. In fact, a Haar random unitary transformation followed by projection onto the first O(log n) coordinates followed by a scaling works as a valid transformation with high probability. In this work, we show that the Haar random unitary can be replaced by a uniformly random unitary chosen from a finite set called an approximate unitary t-design for t = O(log n). Choosing a unitary from such a design requires only polylogarithmic random bits as opposed to exponential in dimension random bits required to choose a Haar random unitary with reasonable precision. Moreover, since such unitaries can be efficiently implemented in the superpositional setting, our result can be viewed as an efficient quantum Johnson-Lindenstrauss transform akin to efficient quantum Fourier transforms widely used in earlier work on quantum algorithms. We prove our result by leveraging a method of Low for showing concentration for approximate unitary t-designs. We discuss algorithmic advantages and limitations of our result and conclude with a toy application to private information retrieval.

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arXiv2018

Inner bounds via simultaneous decoding in quantum network information theory

We prove new inner bounds for several multiterminal channels with classical inputs and quantum outputs. Our inner bounds are all proved in the one-shot setting, and are natural analogues of the best classical inner bounds for the respective channels. For some of these channels, similar quantum inner bounds were unknown even in the asymptotic iid setting. We prove our inner bounds by appealing to a new classical-quantum joint typicality lemma proved in a companion paper ("A one-shot quantum joint typicality lemma", Pranab Sen, arXiv:1806.07278). This lemma allows us to lift to the quantum setting many inner bound proofs for classical multiterminal channels that use intersections and unions of typical sets.

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arXiv2018

Unions, intersections and a one-shot quantum joint typicality lemma

A fundamental tool to prove inner bounds in classical network information theory is the so-called conditional joint typicality lemma. In addition to the lemma, one often uses unions and intersections of typical sets in the inner bound arguments without so much as giving them a second thought. These arguments do not work in the quantum setting. This bottleneck shows up in the fact that so-called simultaneous decoders, as opposed to successive cancellation decoders, are known for very few channels in quantum network information theory. Another manifestation of this bottleneck is the lack of so-called simultaneous smoothing theorems for quantum states. In this paper, we overcome the bottleneck by proving for the first time a one-shot quantum joint typicality lemma with robust union and intersection properties. To do so, we develop two novel tools in quantum information theory which may be of independent interest. The first tool is a simple geometric idea called tilting, which increases the angles between a family of subspaces in orthogonal directions. The second tool, called smoothing and augmentation, is a way of perturbing a multipartite quantum state such that the partial trace over any subset of registers does not increase the operator norm by much. Our joint typicality lemma allows us to construct simultaneous quantum decoders for many multiterminal quantum channels. It provides a powerful tool to extend many results in classical network information theory to the one-shot quantum setting.

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arXiv2017

One-Shot Private Classical Capacity of Quantum Wiretap Channel: Based on one-shot quantum covering lemma

In this work we study the problem of communication over the quantum wiretap channel. For this channel there are three parties Alice (sender), Bob (legitimate receiver) and Eve (eavesdropper). We obtain upper and lower bounds on the amount of information Alice can communicate to Bob such that Eve gets to know as little information as possible about the transmitted messages. Our bounds are in terms of quantum hypothesis testing divergence and smooth max quantum relative entropy. To obtain our result we prove a one-shot version of the quantum covering lemma along with operator Chernoff bound for non-square matrices.

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arXiv2014

One-shot Marton inner bound for classical-quantum broadcast channel

We consider the problem of communication over a classical-quantum broadcast channel with one sender and two receivers. Generalizing the classical inner bounds shown by Marton and the recent quantum asymptotic version shown by Savov and Wilde, we obtain one-shot inner bounds in the quantum setting. Our bounds are stated in terms of smooth min and max Renyi divergences. We obtain these results using a different analysis of the random codebook argument and employ a new one-shot classical mutual covering argument based on rejection sampling. These results give a full justification of the claims of Savov and Wilde in the classical-quantum asymptotic iid setting; the techniques also yield similar bounds in the information spectrum setting.

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arXiv2013

From Low-Distortion Norm Embeddings to Explicit Uncertainty Relations and Efficient Information Locking

The existence of quantum uncertainty relations is the essential reason that some classically impossible cryptographic primitives become possible when quantum communication is allowed. One direct operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n-bit message is encoded in a quantum system using a classical key of size much smaller than n. Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message, in which case the message is said to be "locked". Furthermore, knowing the key, it is possible to recover, that is "unlock", the message. In this paper, we make the following contributions by exploiting a connection between uncertainty relations and low-distortion embeddings of L2 into L1. We introduce the notion of metric uncertainty relations and connect it to low-distortion embeddings of L2 into L1. A metric uncertainty relation also implies an entropic uncertainty relation. We prove that random bases satisfy uncertainty relations with a stronger definition and better parameters than previously known. Our proof is also considerably simpler than earlier proofs. We apply this result to show the existence of locking schemes with key size independent of the message length. We give efficient constructions of metric uncertainty relations. The bases defining these metric uncertainty relations are computable by quantum circuits of almost linear size. This leads to the first explicit construction of a strong information locking scheme. Moreover, we present a locking scheme that is close to being implementable with current technology. We apply our metric uncertainty relations to exhibit communication protocols that perform quantum equality testing.

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