scholarly journals The communication complexity of non-signaling distributions

2011 ◽  
Vol 11 (7&8) ◽  
pp. 649-676
Author(s):  
Julien Degorre ◽  
Marc Kaplan ◽  
Sophie Laplante ◽  
J\'er\'emie Roland
Keyword(s):  
Lower Bounds ◽  
Boolean Functions ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Optimization Problems ◽  
Complexity Measures ◽  
Quantum Distribution ◽  
General Technique ◽  
Classical Communication ◽  
Information Theoretic

We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input $x$, Bob gets an input $y$, and their goal is to each produce an output $a,b$ distributed according to some pre-specified joint distribution $p(a,b|x,y)$. Our results apply to any non-signaling distribution, that is, those where Alice's marginal distribution does not depend on Bob's input, and vice versa.~~~By taking a geometric view of the non-signaling distributions, we introduce a simple new technique based on affine combinations of lower-complexity distributions, and we give the first general technique to apply to all these settings, with elementary proofs and very intuitive interpretations. Specifically, we introduce two complexity measures, one which gives lower bounds on classical communication, and one for quantum communication. These measures can be expressed as convex optimization problems. We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. Despite their apparent simplicity, these lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions. We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distribution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. It also allows us to show that for some distributions, information theoretic methods are necessary to prove strong lower bounds. ~~~ Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution. One consequence of this is a simple proof that any quantum distribution can be approximated with a constant number of bits of communication.

scholarly journals A new exponential separation between quantum and classical one-way communication complexity

2011 ◽  
Vol 11 (7&8) ◽  
pp. 574-591
Author(s):  
Ashley Montanaro
Keyword(s):  
Fourier Analysis ◽  
Boolean Function ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Membership Problem ◽  
Classical Communication ◽  
Exponential Separation ◽  
Quantum Communication Complexity ◽  
Natural Generalisation ◽  
Subgroup Membership Problem

We present a new example of a partial boolean function whose one-way quantum communication complexity is exponentially lower than its one-way classical communication complexity. The problem is a natural generalisation of the previously studied Subgroup Membership problem: Alice receives a bit string $x$, Bob receives a permutation matrix $M$, and their task is to determine whether $Mx=x$ or $Mx$ is far from $x$. The proof uses Fourier analysis and an inequality of Kahn, Kalai and Linial.

scholarly journals Quantum communication complexity advantage implies violation of a Bell inequality

2016 ◽  
Vol 113 (12) ◽  
pp. 3191-3196 ◽  
Author(s):  
Harry Buhrman ◽  
Łukasz Czekaj ◽  
Andrzej Grudka ◽  
Michał Horodecki ◽  
Paweł Horodecki ◽  
...  
Keyword(s):  
Communication Complexity ◽  
Quantum Communication ◽  
Bell Inequality ◽  
Main Tool ◽  
Classical Communication ◽  
Quantum Communication Complexity ◽  
Bell Nonlocality ◽  
Large Quantum ◽  
General Connection ◽  
Inputs And Outputs

We obtain a general connection between a large quantum advantage in communication complexity and Bell nonlocality. We show that given any protocol offering a sufficiently large quantum advantage in communication complexity, there exists a way of obtaining measurement statistics that violate some Bell inequality. Our main tool is port-based teleportation. If the gap between quantum and classical communication complexity can grow arbitrarily large, the ratio of the quantum value to the classical value of the Bell quantity becomes unbounded with the increase in the number of inputs and outputs.

scholarly journals Informationally restricted quantum correlations

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 332 ◽  
Author(s):  
Armin Tavakoli ◽  
Emmanuel Zambrini Cruzeiro ◽  
Jonatan Bohr Brask ◽  
Nicolas Gisin ◽  
Nicolas Brunner
Keyword(s):  
Hilbert Space ◽  
Lower Bounds ◽  
Quantum Communication ◽  
Space Dimension ◽  
Quantum Correlations ◽  
Strong Correlations ◽  
Classical Communication ◽  
Accessible Information ◽  
Classical Correlations ◽  
The One

Quantum communication leads to strong correlations, that can outperform classical ones. Complementary to previous works in this area, we investigate correlations in prepare-and-measure scenarios assuming a bound on the information content of the quantum communication, rather than on its Hilbert-space dimension. Specifically, we explore the extent of classical and quantum correlations given an upper bound on the one-shot accessible information. We provide a characterisation of the set of classical correlations and show that quantum correlations are stronger than classical ones. We also show that limiting information rather than dimension leads to stronger quantum correlations. Moreover, we present device-independent tests for placing lower bounds on the information given observed correlations. Finally, we show that quantum communication carrying log⁡d bits of information is at least as strong a resource as d-dimensional classical communication assisted by pre-shared entanglement.

scholarly journals Generalizations of the distributed Deutsch–Jozsa promise problem

2015 ◽  
Vol 27 (3) ◽  
pp. 311-331 ◽  
Author(s):  
JOZEF GRUSKA ◽  
DAOWEN QIU ◽  
SHENGGEN ZHENG
Keyword(s):  
Communication Complexity ◽  
Quantum Communication ◽  
Hamming Distance ◽  
Finite Automata ◽  
Classical Communication ◽  
Deterministic Finite Automata ◽  
Exact Quantum ◽  
Quantum Communication Complexity

In the distributed Deutsch–Jozsa promise problem, two parties are to determine whether their respective strings x, y ∈ {0,1}n are at the Hamming distanceH(x, y) = 0 or H(x, y) = $\frac{n}{2}$. Buhrman et al. (STOC' 98) proved that the exact quantum communication complexity of this problem is O(log n) while the deterministic communication complexity is Ω(n). This was the first impressive (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch–Jozsa promise problem to determine, for any fixed $\frac{n}{2}$ ⩽ k ⩽ n, whether H(x, y) = 0 or H(x, y) = k, and show that an exponential gap between exact quantum and deterministic communication complexity still holds if k is an even such that $\frac{1}{2}$n ⩽ k < (1 − λ)n, where 0 < λ < $\frac{1}{2}$ is given. We also deal with a promise version of the well-known disjointness problem and show also that for this promise problem there exists an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the promise version of such a disjointness problem. Finally, some applications to quantum, probabilistic and deterministic finite automata of the results obtained are demonstrated.

scholarly journals Quantum communication complexity of distribution testing

2021 ◽  
Vol 21 (15&16) ◽  
pp. 1261-1273
Author(s):  
Aleksandrs Belovs ◽  
Arturo Castellanos ◽  
Francois Le Gall ◽  
Guillaume Malod ◽  
Alexander A. Sherstov
Keyword(s):  
Lower Bound ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Discrete Distributions ◽  
Classical Communication ◽  
Testing Problem ◽  
Quantum Protocols ◽  
Quantum Communication Complexity ◽  
Pattern Matrix ◽  
Distribution Testing

The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP'19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the goal is to decide whether their two distributions are equal, or are $\epsilon$-far apart in the $l_1$-distance. In the present paper we show that the quantum communication complexity of this problem is $\tilde{O}(n/(t\epsilon^2))$ qubits when the distributions have low $l_2$-norm, which gives a quadratic improvement over the classical communication complexity obtained by Andoni, Malkin and Nosatzki. We also obtain a matching lower bound by using the pattern matrix method. Let us stress that the samples received by each of the parties are classical, and it is only communication between them that is quantum. Our results thus give one setting where quantum protocols overcome classical protocols for a testing problem with purely classical samples.

scholarly journals Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates

2021 ◽  
Vol 13 (4) ◽  
pp. 1-37
Author(s):  
Valentine Kabanets ◽  
Sajin Koroth ◽  
Zhenjian Lu ◽  
Dimitrios Myrisiotis ◽  
Igor C. Oliveira
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Communication Complexity ◽  
Inner Product ◽  
Threshold Function ◽  
Seed Length ◽  
Average Case ◽  
Linear Threshold ◽  
Product Function

The class FORMULA[s]∘G consists of Boolean functions computable by size- s De Morgan formulas whose leaves are any Boolean functions from a class G. We give lower bounds and (SAT, Learning, and pseudorandom generators ( PRG s )) algorithms for FORMULA[n 1.99 ]∘G, for classes G of functions with low communication complexity . Let R (k) G be the maximum k -party number-on-forehead randomized communication complexity of a function in G. Among other results, we show the following: • The Generalized Inner Product function GIP k n cannot be computed in FORMULA[s]° G on more than 1/2+ε fraction of inputs for s=o(n 2 /k⋅4 k ⋅R (k) (G)⋅log⁡(n/ε)⋅log⁡(1/ε)) 2 ). This significantly extends the lower bounds against bipartite formulas obtained by [62]. As a corollary, we get an average-case lower bound for GIP k n against FORMULA[n 1.99 ]∘PTF k −1 , i.e., sub-quadratic-size De Morgan formulas with degree-k-1) PTF ( polynomial threshold function ) gates at the bottom. Previously, it was open whether a super-linear lower bound holds for AND of PTFs. • There is a PRG of seed length n/2+O(s⋅R (2) (G)⋅log⁡(s/ε)⋅log⁡(1/ε)) that ε-fools FORMULA[s]∘G. For the special case of FORMULA[s]∘LTF, i.e., size- s formulas with LTF ( linear threshold function ) gates at the bottom, we get the better seed length O(n 1/2 ⋅s 1/4 ⋅log⁡(n)⋅log⁡(n/ε)). In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections of n halfspaces in the regime where ε≤1/n, complementing a recent result of [45]. • There exists a randomized 2 n-t #SAT algorithm for FORMULA[s]∘G, where t=Ω(n\√s⋅log 2 ⁡(s)⋅R (2) (G))/1/2. In particular, this implies a nontrivial #SAT algorithm for FORMULA[n 1.99 ]∘LTF. • The Minimum Circuit Size Problem is not in FORMULA[n 1.99 ]∘XOR; thereby making progress on hardness magnification, in connection with results from [14, 46]. On the algorithmic side, we show that the concept class FORMULA[n 1.99 ]∘XOR can be PAC-learned in time 2 O(n/log n) .

scholarly journals Robust Bell inequalities from communication complexity

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 72 ◽  
Author(s):  
Sophie Laplante ◽  
Mathieu Laurière ◽  
Alexandre Nolin ◽  
Jérémie Roland ◽  
Gabriel Senno
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Bell Inequality ◽  
Bell Inequalities ◽  
Local Distribution ◽  
Classical Communication ◽  
Quantum Communication Complexity ◽  
Complexity Lower Bounds ◽  
Quantum Distributions

The question of how large Bell inequality violations can be, for quantum distributions, has been the object of much work in the past several years. We say that a Bell inequality is normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution. Upper and (almost) tight lower bounds have been given for the quantum violation of these Bell inequalities in terms of number of outputs of the distribution, number of inputs, and the dimension of the shared quantum states. In this work, we revisit normalized Bell inequalities together with another family: inefficiency-resistant Bell inequalities. To be inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including those that can abort. This makes the Bell inequality resistant to the detection loophole, while a normalized Bell inequality is resistant to general local noise. Both these families of Bell inequalities are closely related to communication complexity lower bounds. We show how to derive large violations from any gap between classical and quantum communication complexity, provided the lower bound on classical communication is proven using these lower bound techniques. This leads to inefficiency-resistant violations that can be exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency for these Bell inequalities.

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