On families of anticommuting matrices

Hrubeš, Pavel

doi:10.1016/j.laa.2015.12.015

Cited by 15 publications

(6 citation statements)

References 9 publications

(14 reference statements)

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“…Proof. We note that this lemma is well-known; it follows for example from Proposition 9 in [5], and is also proved in [6]. Here we give an elegant and simpler proof of this fact.…”

Section: Setmentioning

confidence: 63%

On sets of commuting and anticommuting Paulis

Sarkar¹,

Berg²

2019

Preprint

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In this work we study the structure and cardinality of maximal sets of commuting and anticommuting Paulis in the setting of the abelian Pauli group. We provide necessary and sufficient conditions for anticommuting sets to be maximal, and present an efficient algorithm for generating anticommuting sets of maximum size. As a theoretical tool, we introduce commutativity maps, and study properties of maps associated with elements in the cosets with respect to anticommuting minimal generating sets. We also derive expressions for the number of distinct sets of commuting and anticommuting abelian Paulis of a given size.

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“…Proof. We note that this lemma is well-known; it follows for example from Proposition 9 in [5], and is also proved in [6]. Here we give an elegant and simpler proof of this fact.…”

Section: Setmentioning

confidence: 63%

On sets of commuting and anticommuting Paulis

Sarkar¹,

Berg²

2019

Preprint

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“…It is known (c.f. [8]) that the maximum possible number of complex invertible anticommuting N by N matrices is 2 log 2 N + 1. In our case N = 2 n and thus we cannot have more than 2n + 1 anticommuting matrices corresponding to edge signing (note that each such matrix is invertible, as its square is the identity).…”

Section: Proposition 52 ([3]mentioning

confidence: 99%

Unitary signings and induced subgraphs of Cayley graphs of $\mathbb{Z}_2^{n}$

Alon

Zheng

2020

AiC

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Boolean functions play an important role in many different areas of computer science. The _local sensitivity_ of a Boolean function $f:\{0,1\}^n\to \{0,1\}$ on an input $x\in\{0,1\}^n$ is the number of coordinates whose flip changes the value of $f(x)$, i.e., the number of i's such that $f(x)\not=f(x+e_i)$, where $e_i$ is the $i$-th unit vector. The _sensitivity_ of a Boolean function is its maximum local sensitivity. In other words, the sensitivity measures the robustness of a Boolean function with respect to a perturbation of its input. Another notion that measures the robustness is block sensitivity. The _local block sensitivity_ of a Boolean function $f:\{0,1\}^n\to \{0,1\}$ on an input $x\in\{0,1\}^n$ is the number of disjoint subsets $I$ of $\{1,..,n\}$ such that flipping the coordinates indexed by $I$ changes the value of $f(x)$, and the _block sensitivity_ of $f$ is its maximum local block sensitivity. Since the local block sensitivity is at least the local sensitivity for any input $x$, the block sensitivity of $f$ is at least the sensitivity of $f$. The next example demonstrates that the block sensitivity of a Boolean function is not linearly bounded by its sensitivity. Fix an integer $k\ge 2$ and define a Boolean function $f:\{0,1\}^{2k^2}\to\{0,1\}$ as follows: the coordinates of $x\in\{0,1\}^{2k^2}$ are split into $k$ blocks of size $2k$ each and $f(x)=1$ if and only if at least one of the blocks contains exactly two entries equal to one and these entries are consecutive. While the sensitivity of the function $f$ is $2k$, its block sensitivity is $k^2$. The Sensitivity Conjecture, made by Nisan and Szegedy in 1992, asserts that the block sensitivity of a Boolean function is polynomially bounded by its sensivity. The example above shows that the degree of such a polynomial must be at least two. The Sensitivity Conjecture has been recently proven by Huang in [Annals of Mathematics 190 (2019), 949-955](https://doi.org/10.4007/annals.2019.190.3.6). He proved the following combinatorial statement that implies the conjecture (with the degree of the polynomial equal to four): any subset of more than half of the vertices of the $n$-dimensional cube $\{0,1\}^n$ induces a subgraph that contains a vertex with degree at least $\sqrt{n}$. The present article extends this result as follows: every Cayley graph with the vertex set $\{0,1\}^n$ and any generating set of size $d$ (the vertex set is viewed as a vector space over the binary field) satisfies that any subset of more than half of its vertices induces a subgraph that contains a vertex of degree at least $\sqrt{d}$. In particular, when the generating set consists of the $n$ unit vectors, the Cayley graph is the $n$-dimensional hypercube.

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“…Here, σ X , σ Y and σ Z are the Pauli matrices. The dimension dependence of this construction is essentially optimal [New32,Hru16]. See also [PSS18] and Section 8.2 of [BN18].…”

Section: Inclusions Constants For the Complex Matrix Cubementioning

confidence: 99%

Maximal violation of steering inequalities and the matrix cube

Bluhm,

Nechita

2021

Preprint

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In this work, we characterize the amount of steerability present in quantum theory by connecting the maximal violation of a steering inequality to an inclusion problem of free spectrahedra. In particular, we show that the maximal violation of an arbitrary unbiased dichotomic steering inequality is given by the inclusion constants of the matrix cube, which is a well-studied object in convex optimization theory. This allows us to find new upper bounds on the maximal violation of steering inequalities and to show that previously obtained violations are optimal. In order to do this, we prove lower bounds on the inclusion constants of the complex matrix cube, which might be of independent interest. Finally, we show that the inclusion constants of the matrix cube and the matrix diamond are the same. This allows us to derive new bounds on the amount of incompatibility available in dichotomic quantum measurements in fixed dimension.

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On families of anticommuting matrices

Abstract: Let e1, . . . , e k be complex n × n matrices such that eiej = −ejei whenever i = j. We conjecture that

Cited by 15 publications

References 9 publications

On sets of commuting and anticommuting Paulis

On sets of commuting and anticommuting Paulis

Unitary signings and induced subgraphs of Cayley graphs of $\mathbb{Z}_2^{n}$

Maximal violation of steering inequalities and the matrix cube

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