arXiv subjects

Olivier Lalonde

Publications and source records attributed to Olivier Lalonde.

7 recordsLinked to original sources

arXiv2026

An exponential separation between entanglement-assisted and unassisted one-way quantum communication

A longstanding question in quantum communication complexity is whether some task can be accomplished with a small amount of communication in the presence of entanglement, yet require much more quantum communication in the absence of entanglement. Separations of this nature were previously known for relational problems and, in the simultaneous message passing model, for partial functions. But it has remained unresolved whether any such separation exists for a total Boolean function. We resolve this question with an exponential separation in the one-way setting: we exhibit a family of total Boolean functions $f_n\colon \{0,1\}^n \times \{0,1\}^n \to \{0,1\}$ that can be computed with $O(\log n)$ bits of one-way classical communication given prior entanglement, but that require $Ω(n^{1/3})$ qubits of one-way quantum communication without entanglement. Our function is a special case of the subgroup membership problem, first studied in the communication setting by Aaronson, Le Gall, Russell, and Tani.

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arXiv2026

Non-local games and communication complexity with noisy entanglement

We study the impact of noise on the theories of quantum nonlocality and entanglement-assisted communication complexity. We consider non-local games and entanglement-assisted communication complexity in a model where Alice and Bob may share arbitrarily many noisy EPR pairs. We study four noise models: depolarizing noise, unital noise, biased reset noise, and erasure noise. Our results are as follows: 1. We upper bound the value of the CHSH game under all these noise models in terms of the noise parameter, without any assumptions on the measurements used in the strategy. 2. We prove a parallel repetition theorem for general non-local games under all noise models except biased reset noise; we prove an improved parallel repetition theorem for unique games. Our parallel repetition rate is smaller than the quantum parallel repetition rate for CHSH in a nontrivial noise regime. 3. Using our unique-game parallel repetition theorem and a relation defined by the CHSH game, we prove a separation between communication complexity with noisy vs noiseless entanglement. This implies an $Ω(n)$ two-way communication lower bound for distilling $n$ EPR pairs from noisy EPR pairs, in the same nontrivial noise regime. 4. We show that any interactive entanglement-assisted communication protocol can be simulated by an SMP communication protocol with noisy shared randomness, with an exponential blowup in communication. This generalizes a known result on the simulation of noiseless shared randomness with noisy shared randomness. 5. We show a polynomial lower bound on the number of copies of noisy EPR pairs required to compute the Equality function with constant communication, under all four noise models. The previous result gives a matching upper bound, and moreover, this answers an open question in the literature on whether logarithmically many noisy shared bits suffice for communication.

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arXiv2026

Maximally entangled states are not complete for pseudo-telepathy

One of the longstanding open problems in quantum nonlocality is to determine if maximally entangled states are complete for bipartite pseudo-telepathic games: namely, if every nonlocal game which admits a perfect entangled strategy admits such a strategy which uses a maximally entangled state. We exhibit a counterexample to this in the form of a bipartite nonlocal game with input sets of size 4 and 3 and output sets both of size 6. This game is part of a new class of nonlocal games, which we call inner product games, which could be of independent interest.

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arXiv2026

Quantum Colorings of Spheres

Cameron, Montanaro, Newman, Severini and Winter gave a construction which shows that, for $n \in \{2,4,8\}$, any graph $G$ which admits a real $n$-dimensional orthogonal representation is quantumly $n$-colorable. This result can be recast as the statement that the real sphere $S^{n-1}$ is quantumly $n$-colorable for these values of $n$. We investigate possible extensions of their construction. We first show that their hypothesis that the orthogonal representation be real-valued is required by proving that there is no analogue of this for the complex spheres, which all have quantum chromatic number strictly bigger than the dimension except in two dimensions. We also provide candidate finitary witnesses of this and show for the first time that the real and complex orthogonal ranks are distinct as a byproduct. For the real case, we show that if $S^{n-1}$ is quantumly $n$-colorable, then either $n=2$ or $n$ is a multiple of 4, and show that the converse holds whenever a Hadamard matrix of order $n$ exists. Hence, assuming the Hadamard conjecture, this completely classifies the dimensions to which the CMNSW construction can be extended. Our method of proof involves showing the equivalence between the existence of such a construction and the existence of a maximal code space for Clifford-algebraic errors given a clean ancilla, and we believe that the representation-theoretic techniques we use for tackling the latter problem could be of independent interest. It also follows from this equivalence that $S^{n-1}$ admits a rank-one quantum $n$-coloring if and only if $n \in \{2,4,8\}$, thereby settling a conjecture of Zeng and Zhang, as does the fact that for all $m \geq 1$, there exists a catalytic zero-error remote state preparation protocol for real $m$-qubit states with $m$ bits of communication and which consumes $m$ ebits.

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arXiv2026

Near-optimal entanglement-communication tradeoffs for remote state preparation

We study the following task: Alice is given a classical description of a rank-$k$ projector $P$ on $\mathbb{C}^d$, and Alice and Bob want to prepare the quantum state $P/k$ on Bob's side using shared entanglement and classical communication. The general form of this task is known as remote state preparation (RSP). We give nearly-matching lower and upper bounds for the entanglement cost and communication cost for RSP of the states $P/k$. Ours are the first nearly matching upper and lower bounds for RSP of mixed states, and in the special case of pure states, our lower bound outperforms the best previously known lower bound. Our results show that any pure entangled state that can be used to do RSP of these states with $o(d)$ bits of communication, can distill $\log d$ ebits of entanglement, and conversely, any state that can distill $\log d$ ebits of entanglement can be used to do RSP of these states efficiently. As applications of our results, we rederive a previously-known incompressibility result for states of the form $P/k$, and give a new entanglement-assisted communication protocol for the equality function that uses $\frac{1}{2}\log n + O(1)$ many ebits, and $O(1)$ communication.

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arXiv2023

On the Quantum Chromatic Numbers of Small Graphs

We make two contributions pertaining to the study of the quantum chromatic numbers of small graphs. Firstly, in an elegant paper, Mančinska and Roberson [\textit{Baltic Journal on Modern Computing}, 4(4), 846-859, 2016] gave an example of a graph $G_{14}$ on 14 vertices with quantum chromatic number 4 and classical chromatic number 5, and conjectured that this is the smallest graph exhibiting a separation between the two parameters. We describe a computer-assisted proof of this conjecture, thereby resolving a longstanding open problem in quantum graph theory. Our second contribution pertains to the study of the rank-$r$ quantum chromatic numbers. While it can now be shown that for every $r$, $χ_q$ and $χ^{(r)}_q$ are distinct, few small examples of separations between these parameters are known. We give the smallest known example of such a separation in the form of a graph $G_{21}$ on 21 vertices with $χ_q(G_{21}) = χ^{(2)}_q(G_{21}) = 4$ and $ ξ(G_{21}) = χ^{(1)}_q(G_{21}) = χ(G_{21}) = 5$. The previous record was held by a graph $G_{msg}$ on 57 vertices that was first considered in the aforementioned paper of Mančinska and Roberson and which satisfies $χ_q(G_{msg}) = 3$ and $χ^{(1)}_q(G_{msg}) = 4$. In addition, $G_{21}$ provides the first provable separation between the parameters $χ^{(1)}_q$ and $χ^{(2)}_q$. We believe that our techniques for constructing $G_{21}$ and lower bounding its orthogonal rank could be of independent interest.

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arXiv2023

Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error

We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability $ε$, getting optimal constant factors in the leading terms in a number of different models. In the randomized model, 1) we give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error, at a small additive cost. This is an improvement over Newman's theorem [Inf. Proc. Let.'91] in the dependence on the error parameter. 2) Using this we obtain a $(\log(n/ε^2)+4)$-cost private-coin communication protocol that computes the $n$-bit Equality function, to error $ε$. This improves upon the $\log(n/ε^3)+O(1)$ upper bound implied by Newman's theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.'09], up to an additive $\log\log(1/ε)+O(1)$. In the quantum model, 1) we exhibit a one-way protocol of cost $\log(n/ε)+4$, that uses only pure states and computes the $n$-bit Equality function to error $ε$. This bound was implicitly already shown by Nayak [PhD thesis'99]. 2) We show that any $ε$-error one-way protocol for $n$-bit Equality that uses only pure states communicates at least $\log(n/ε)-\log\log(1/ε)-O(1)$ qubits. 3) We exhibit a one-way protocol of cost $\log(\sqrt{n}/ε)+3$, that uses mixed states and computes the $n$-bit Equality function to error $ε$. This is also tight up to an additive $\log\log(1/ε)+O(1)$, which follows from Alon's result. 4) We study the number of EPR pairs required to be shared in an entanglement-assisted one-way protocol. Our upper bounds also yield upper bounds on the approximate rank and related measures of the Identity matrix.

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