arXiv subjects

Jalex Stark

Publications and source records attributed to Jalex Stark.

4 recordsLinked to original sources

arXiv2019

Robust self-testing for linear constraint system games

We study linear constraint system (LCS) games over the ring of arithmetic modulo $d$. We give a new proof that certain LCS games (the Mermin--Peres Magic Square and Magic Pentagram over binary alphabets, together with parallel repetitions of these) have unique winning strategies, where the uniqueness is robust to small perturbations. In order to prove our result, we extend the representation-theoretic framework of Cleve, Liu, and Slofstra (Journal of Mathematical Physics 58.1 (2017): 012202.) to apply to linear constraint games over $\mathbb{Z}_d$ for $d\geq 2$. We package our main argument into machinery which applies to any nonabelian finite group with a ''solution group'' presentation. We equip the $n$-qubit Pauli group for $n\geq 2$ with such a presentation; our machinery produces the Magic Square and Pentagram games from the presentation and provides robust self-testing bounds. The question of whether there exist LCS games self-testing maximally entangled states of local dimension not a power of $2$ is left open. A previous version of this paper falsely claimed to show self-testing results for a certain generalization of the Magic Square and Pentagram mod $d\neq 2$. We show instead that such a result is impossible.

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arXiv2019

Trading locality for time: certifiable randomness from low-depth circuits

The generation of certifiable randomness is the most fundamental information-theoretic task that meaningfully separates quantum devices from their classical counterparts. We propose a protocol for exponential certified randomness expansion using a single quantum device. The protocol calls for the device to implement a simple quantum circuit of constant depth on a 2D lattice of qubits. The output of the circuit can be verified classically in linear time, and is guaranteed to contain a polynomial number of certified random bits assuming that the device used to generate the output operated using a (classical or quantum) circuit of sub-logarithmic depth. This assumption contrasts with the locality assumption used for randomness certification based on Bell inequality violation or computational assumptions. To demonstrate randomness generation it is sufficient for a device to sample from the ideal output distribution within constant statistical distance. Our procedure is inspired by recent work of Bravyi et al. (Science 2018), who introduced a relational problem that can be solved by a constant-depth quantum circuit, but provably cannot be solved by any classical circuit of sub-logarithmic depth. We develop the discovery of Bravyi et al. into a framework for robust randomness expansion. Our proposal does not rest on any complexity-theoretic conjectures, but relies on the physical assumption that the adversarial device being tested implements a circuit of sub-logarithmic depth. Success on our task can be easily verified in classical linear time. Finally, our task is more noise-tolerant than most other existing proposals that can only tolerate multiplicative error, or require additional conjectures from complexity theory; in contrast, we are able to allow a small constant additive error in total variation distance between the sampled and ideal distributions.

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arXiv2018

Unconditional separation of finite and infinite-dimensional quantum correlations

Determining the relationship between quantum correlation sets is a long-standing open problem. The most well-studied part of the hierarchy is captured by the chain of inclusions $\mathcal C_q \subseteq \mathcal C_{qs} \subsetneq \mathcal C_{qa} \subseteq \mathcal C_{qc}$. The separation $\mathcal C_{qs} \neq \mathcal C_{qa}$, showing that the set of quantum spatial correlations is not closed, was proven in breakthrough work by Slofstra [arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)]. Resolving the question of $\mathcal C_{qa} = \mathcal C_{qc}$ would resolve the Connes Embedding Conjecture and would represent major progress in the mathematical field of operator algebras. In this work, we resolve the ambiguity in the first inclusion, showing that $\mathcal{C}_q \neq \mathcal{C}_{qs}$. We provide an explicit construction of a correlation that can be attained on a tensor product of infinite-dimensional Hilbert spaces but not finite-dimensional ones. This property is also conjectured to be possessed by any correlation which maximally violates the $I_{3322}$ inequality.

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arXiv2017

Separation of finite and infinite-dimensional quantum correlations, with infinite question or answer sets

Completely determining the relationship between quantum correlation sets is a long-standing open problem, known as Tsirelson's problem. Following recent progress by Slofstra [arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)] only two instances of the problem remain open. One of them is the question of whether the set of finite-dimensional quantum correlations is strictly contained in the set of infinite-dimensional ones (i.e. whether $\mathcal C_{q} \neq \mathcal C_{qs}$). The usual formulation of the question assumes finite question and answer sets. In this work, we show that, when one allows for either infinite answer sets (and finite question sets) or infinite question sets (and finite answer sets), there exist correlations that are achievable using an infinite-dimensional quantum strategy, but not a finite-dimensional one. For the former case, our proof exploits a recent result [Nat. Comm. 8, 15485 (2017)], which shows self-testing of any pure bipartite entangled state of arbitrary local dimension $d$, using question sets of size 3 and 4 and answer sets of size $d$. For the latter case, a key step in our proof is to show a novel self-test, inspired by [Nat. Comm. 8, 15485 (2017)], of all bipartite entangled states of any local dimension d, using question sets of size $O(d)$, and answer sets of size 4 and 3 respectively.

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