arXiv subjects

Srijita Kundu

Publications and source records attributed to Srijita Kundu.

24 records · Page 2Linked to original sources

arXiv2019

Fourier Entropy-Influence Conjecture for Random Linear Threshold Functions

The Fourier-Entropy Influence (FEI) Conjecture states that for any Boolean function $f:\{+1,-1\}^n \to \{+1,-1\}$, the Fourier entropy of $f$ is at most its influence up to a universal constant factor. While the FEI conjecture has been proved for many classes of Boolean functions, it is still not known whether it holds for the class of Linear Threshold Functions. A natural question is: Does the FEI conjecture hold for a `random' linear threshold function? In this paper, we answer this question in the affirmative. We consider two natural distributions on the weights defining a linear threshold function, namely uniform distribution on $[-1,1]$ and Normal distribution.

cs.CC↗

arXiv2018

Lifting randomized query complexity to randomized communication complexity

We show that for a relation $f\subseteq \{0,1\}^n\times \mathcal{O}$ and a function $g:\{0,1\}^{m}\times \{0,1\}^{m} \rightarrow \{0,1\}$ (with $m= O(\log n)$), $$\mathrm{R}_{1/3}(f\circ g^n) = Ω\left(\mathrm{R}_{1/3}(f) \cdot \left(\log\frac{1}{\mathrm{disc}(M_g)} - O(\log n)\right)\right),$$ where $f\circ g^n$ represents the composition of $f$ and $g^n$, $M_g$ is the sign matrix for $g$, $\mathrm{disc}(M_g)$ is the discrepancy of $M_g$ under the uniform distribution and $\mathrm{R}_{1/3}(f)$ ($\mathrm{R}_{1/3}(f\circ g^n)$) denotes the randomized query complexity of $f$ (randomized communication complexity of $f\circ g^n$) with worst case error $\frac{1}{3}$. In particular, this implies that for a relation $f\subseteq \{0,1\}^n\times \mathcal{O}$, $$\mathrm{R}_{1/3}(f\circ \mathrm{IP}_m^n) = Ω\left(\mathrm{R}_{1/3}(f) \cdot m\right),$$ where $\mathrm{IP}_m:\{0,1\}^m\times \{0,1\}^m\rightarrow \{0,1\}$ is the Inner Product (modulo $2$) function and $m= O(\log(n))$.

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arXiv2017

Quadratically Tight Relations for Randomized Query Complexity

Let $f:\{0,1\}^n \rightarrow \{0,1\}$ be a Boolean function. The certificate complexity $C(f)$ is a complexity measure that is quadratically tight for the zero-error randomized query complexity $R_0(f)$: $C(f) \leq R_0(f) \leq C(f)^2$. In this paper we study a new complexity measure that we call expectational certificate complexity $EC(f)$, which is also a quadratically tight bound on $R_0(f)$: $EC(f) \leq R_0(f) = O(EC(f)^2)$. We prove that $EC(f) \leq C(f) \leq EC(f)^2$ and show that there is a quadratic separation between the two, thus $EC(f)$ gives a tighter upper bound for $R_0(f)$. The measure is also related to the fractional certificate complexity $FC(f)$ as follows: $FC(f) \leq EC(f) = O(FC(f)^{3/2})$. This also connects to an open question by Aaronson whether $FC(f)$ is a quadratically tight bound for $R_0(f)$, as $EC(f)$ is in fact a relaxation of $FC(f)$. In the second part of the work, we upper bound the distributed query complexity $D^μ_ε(f)$ for product distributions $μ$ by the square of the query corruption bound ($\mathrm{corr}_ε(f)$) which improves upon a result of Harsha, Jain and Radhakrishnan [2015]. A similar statement for communication complexity is open.

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arXiv2017

A Composition Theorem for Randomized Query Complexity

Let the randomized query complexity of a relation for error probability $ε$ be denoted by $R_ε(\cdot)$. We prove that for any relation $f \subseteq \{0,1\}^n \times \mathcal{R}$ and Boolean function $g:\{0,1\}^m \rightarrow \{0,1\}$, $R_{1/3}(f\circ g^n) = Ω(R_{4/9}(f)\cdot R_{1/2-1/n^4}(g))$, where $f \circ g^n$ is the relation obtained by composing $f$ and $g$. We also show that $R_{1/3}\left(f \circ \left(g^\oplus_{O(\log n)}\right)^n\right)=Ω(\log n \cdot R_{4/9}(f) \cdot R_{1/3}(g))$, where $g^\oplus_{O(\log n)}$ is the function obtained by composing the xor function on $O(\log n)$ bits and $g^t$.

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arXiv2016

Optimal bounds for parity-oblivious random access codes

Random access coding is an information task that has been extensively studied and found many applications in quantum information. In this scenario, Alice receives an $n$-bit string $x$, and wishes to encode $x$ into a quantum state $ρ_x$, such that Bob, when receiving the state $ρ_x$, can choose any bit $i \in [n]$ and recover the input bit $x_i$ with high probability. Here we study two variants: parity-oblivious random access codes, where we impose the cryptographic property that Bob cannot infer any information about the parity of any subset of bits of the input apart from the single bits $x_i$; and even-parity-oblivious random access codes, where Bob cannot infer any information about the parity of any even-size subset of bits of the input. In this paper, we provide the optimal bounds for parity-oblivious quantum random access codes and show that they are asymptotically better than the optimal classical ones. Our results provide a large non-contextuality inequality violation and resolve the main open problem in a work of Spekkens, Buzacott, Keehn, Toner, and Pryde (2009). Second, we provide the optimal bounds for even-parity-oblivious random access codes by proving their equivalence to a non-local game and by providing tight bounds for the success probability of the non-local game via semidefinite programming. In the case of even-parity-oblivious random access codes, the cryptographic property holds also in the device-independent model.

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arXiv2015

Quantifying incompatibility beyond entropic uncertainty

We study two operational approaches to quantifying incompatibility that depart significantly from the well known entropic uncertainty relation (EUR) formalism. Both approaches result in incompatibility measures that yield non-zero values even when the pair of incompatible observables commute over a subspace, unlike EURs which give a zero lower bound in such cases. Here, we explicitly show how these measures go beyond EURs in quantifying incompatibility: For any set of quantum observables, we show that both incompatibility measures are bounded from below by the corresponding EURs for the Tsallis ($T_{2}$) entropy. We explicitly evaluate the incompatibility of a pair of qubit observables in both operational scenarios. We also obtain an efficiently computable lower bound for the mutually incompatibility of a general set of observables.

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