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Quantum Nonlocality and Device-Independent Randomness Robust to Relaxations of Bell Assumptions

Ravishankar Ramanathan, Yuan Liu

TL;DR

The notion of measure-dependent parameter-dependent locality as the set of input-output behaviors under simultaneous relaxations of measurement and parameter independence is introduced and a rigorous characterisation of the vertices of the polytope of joint input-output behaviors that obey a $\mu$-relaxation of MI and $\epsilon$-relaxation of PI is provided.

Abstract

The question of certifying quantum nonlocality under a relaxation of the assumptions in the Bell theorem has gained traction, with potential for device-independent applications under weak seeds and cross-talk. Recently, it was shown that quantum nonlocality can be certified even under a simultaneous arbitrary (but not full) relaxation of the assumptions of Measurement Independence (MI) and Parameter Independence (PI), using states of local dimension $d = poly((1-ε)^{-1})$ for an $ε\in [0,1)$-relaxation. Here, we derive three results strengthening the state-of-art. Firstly, we show that states of constant local dimension $d$ are already sufficient to certify quantum nonlocality under arbitrary MI and PI relaxation, albeit in a non-robust manner. Secondly, and as a theoretical paradigm to derive the above, we introduce the notion of \textit{measurement-dependent parameter-dependent locality} as the set of input-output behaviors under simultaneous relaxations of measurement and parameter independence. We provide a rigorous characterisation of the vertices of the polytope of joint input-output behaviors that obey a $μ$-relaxation of MI and $ε$-relaxation of PI. We highlight a relation between nonlocality certification under PI relaxation and that under detection inefficiencies by pointing out alternative extremal correlations to the Eberhard correlations that also allow to achieve detection efficiency of $η= 2/3$ in the two-input scenario. Finally, we study the implication of the relaxed assumptions for device-independent randomness certification. We analytically derive the quantum guessing probability for one player's outcomes in the CHSH Bell test, as a function of the noise in the test as well as of a leakage of an average amount of $I(X:B) < 1$ bits of input information per measurement round.

Quantum Nonlocality and Device-Independent Randomness Robust to Relaxations of Bell Assumptions

TL;DR

The notion of measure-dependent parameter-dependent locality as the set of input-output behaviors under simultaneous relaxations of measurement and parameter independence is introduced and a rigorous characterisation of the vertices of the polytope of joint input-output behaviors that obey a -relaxation of MI and -relaxation of PI is provided.

Abstract

The question of certifying quantum nonlocality under a relaxation of the assumptions in the Bell theorem has gained traction, with potential for device-independent applications under weak seeds and cross-talk. Recently, it was shown that quantum nonlocality can be certified even under a simultaneous arbitrary (but not full) relaxation of the assumptions of Measurement Independence (MI) and Parameter Independence (PI), using states of local dimension for an -relaxation. Here, we derive three results strengthening the state-of-art. Firstly, we show that states of constant local dimension are already sufficient to certify quantum nonlocality under arbitrary MI and PI relaxation, albeit in a non-robust manner. Secondly, and as a theoretical paradigm to derive the above, we introduce the notion of \textit{measurement-dependent parameter-dependent locality} as the set of input-output behaviors under simultaneous relaxations of measurement and parameter independence. We provide a rigorous characterisation of the vertices of the polytope of joint input-output behaviors that obey a -relaxation of MI and -relaxation of PI. We highlight a relation between nonlocality certification under PI relaxation and that under detection inefficiencies by pointing out alternative extremal correlations to the Eberhard correlations that also allow to achieve detection efficiency of in the two-input scenario. Finally, we study the implication of the relaxed assumptions for device-independent randomness certification. We analytically derive the quantum guessing probability for one player's outcomes in the CHSH Bell test, as a function of the noise in the test as well as of a leakage of an average amount of bits of input information per measurement round.

Paper Structure

This paper contains 9 sections, 8 theorems, 88 equations, 2 figures, 2 tables.

Table of Contents

  1. Characterization of the Measurement-Dependent Parameter-Dependent Local Set
  2. MDPDL is a Strict Superset of MDL
  3. Facet inequality of the $\epsilon$-Parameter dependent polytope $\mathcal{P}^{AB,(\epsilon, \epsilon)}_2$ in the $(2,2;2,2)$ Bell scenario and its quantum violation
  4. Facet inequality of the $\epsilon$-Parameter dependent polytope $\mathcal{P}^{AB,(\epsilon, \epsilon)}_2$ in the $(2,2;2,2)$ Bell scenario
  5. Quantum violation of the Facet Inequality Eq. \ref{['eq:pd_ineq']}
  6. Supporting hyperplane of the MDPDL set in the $(2,2;2,2)$ Bell scenario and its quantum violation
  7. Guessing probability in the CHSH Bell test under leakage of input information
  8. CHSH Bell test under partial leakage of Alice's input information
  9. Eve's guessing probability in the CHSH Bell test under partial leakage of Alice's input information

Key Result

Theorem 1

Consider the Bell scenario $(|\mathcal{X}|,|\mathcal{A}|; |\mathcal{Y}|,|\mathcal{B}|)$ for arbitrary $|\mathcal{X}|, |\mathcal{Y}| \geq 2$, with $|\mathcal{A}|= |\mathcal{B}| = 2$. For any extreme point (vertex) $p^{ext}_{AB|XY}$ of the set $\mathcal{P}^{AB,(\epsilon_A, \epsilon_B)}_2$ given in eq:

Figures (2)

  • Figure 1: (a) Solid lines: Lower bound of the guessing probability ${P}_{g}^{(q)}(S,\kappa,x^*=0)$ as a function of the observed CHSH value $S$ under $\kappa$ bits of input information leakage, as derived in Eq. \ref{['eq:pg_boud2']}, for $\kappa = 0, 0.25, 0.5, 0.75$. Black dots: Upper bounds on the guessing probability obtained using the level-3 NPA hierarchy NPA1NPA2. (b) Corresponding min-entropy $H_{\min}(S,\kappa,x^*=0) = -\log_2({P}_{g}^{(q)}(S,\kappa,x^*=0))$ as a function of $S$.
  • Figure 2: The guessing probability as a function of $\kappa$, which quantifies the amount of input information leakage, for a fixed observed CHSH value $S = 2\sqrt{2}$. The black line corresponds to the one-way information-leakage scenario, where only Alice's input information is leaked, i.e. $\kappa_A=\kappa$, $\kappa_B=0$, and the guessing probability is ${P}_{g}^{(q)}(S=2\sqrt{2},\kappa,x^*=0)$ derived above. The blue line corresponds to the two-way information-leakage scenario, where both Alice's and Bob's input information are leaked, with $\kappa_A=\kappa_B=\kappa$; the associated guessing probability curve is obtained numerically using the NPA hierarchy.

Theorems & Definitions (15)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Lemma 1
  • proof
  • ...and 5 more