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Semi-device-independent framework based on natural physical assumptions

Thomas Van Himbeeck, Erik Woodhead, Nicolas J. Cerf, Raúl García-Patrón, Stefano Pironio

arXiv:1612.06828v3quant-ph

TL;DR

Device-independent protocols face practical constraints, while semi-device-independent approaches commonly rely on idealized dimension bounds. This paper instead constrains states through a physical observable and finds quantum correlations beyond deterministic classical behavior under the stated constraints.

  • Problem

    Semi-device-independent protocols commonly bound message dimension, an idealization for quantum optical carriers that can occupy infinite-dimensional spaces.

    Source: §1 · p. 2, §2.1 · p. 4

  • Method

    The paper constrains prepared states using a physical observable's mean value and analyzes the simplest two-preparation, binary-outcome scenario.

    Source: §2.1 · p. 4, §2.2 · p. 5

  • Results

    Under the max-peak assumption, correlations with |E1 − E2| > 0 exclude a deterministic measurement explanation, and the optical schemes generate non-classical correlations.

    Source: §5.3 · p. 18

  • Takeaways & Limitations

    For OOK, a deterministic explanation possible under the max-average assumption is no longer possible when peak values are constrained.

    Source: §5.3 · p. 18

    The 2ASK reference-state constraint is less natural and harder to verify, requiring displacement with a coherent beam followed by photon counting.

    Source: §2.3.2 · p. 8

Abstract

from arXiv · show

The semi-device-independent approach provides a framework for prepare-and-measure quantum protocols using devices whose behavior must not be characterized nor trusted, except for a single assumption on the dimension of the Hilbert space characterizing the quantum carriers. Here, we propose instead to constrain the quantum carriers through a bound on the mean value of a well-chosen observable. This modified assumption is physically better motivated than a dimension bound and closer to the description of actual experiments. In particular, we consider quantum optical schemes where the source emits quantum states described in an infinite-dimensional Fock space and model our assumption as an upper bound on the average photon number in the emitted states. We characterize the set of correlations that may be exhibited in the simplest possible scenario compatible with our new framework, based on two energy-constrained state preparations and a two-outcome measurement. Interestingly, we uncover the existence of quantum correlations exceeding the set of classical correlations that can be produced by devices behaving in a purely pre-determined fashion (possibly including shared randomness). This feature suggests immediate applications to certified randomness generation. Along this line, we analyze the achievable correlations in several prepare-and-measure optical schemes with a mean photon number constraint and demonstrate that they allow for the generation of certified randomness. Our simplest optical scheme works by the on-off keying of an attenuated laser source followed by photocounting. It opens the path to more sophisticated energy-constrained semi-device-independent quantum cryptography protocols, such as quantum key distribution.

1 Introduction

Standard device-independent protocols face a practical obstacle: no-communication schemes require loophole-free entanglement, while semi-device-independent schemes rely on idealized dimension bounds. This paper replaces that bound with a physical observable constraint and characterizes correlations in a simple scenario.

1 Introduction

  • Loophole-free entanglement is a serious obstacle to practical device-independent protocols that forbid communication between devices.The no-communication constraint has clear operational significance, but useful non-classical correlations in that setting require entanglement.Source: §1 · p. 2
  • Semi-device-independent protocols allow communication but typically bound message dimension, an idealization for carriers such as photons in infinite-dimensional spaces.These protocols have supported randomness-generation and quantum-key-distribution proposals [12], [13] [14].Source: §1 · p. 2
  • The proposed alternative constrains mean values of a physical observable, such as photon number, rather than the dimension of the carrier space.For quantum optics, the average photon number is a natural constraint on states that may occupy an infinite-dimensional Fock space.Source: §1 · p. 2
  • The paper analytically characterizes available correlations in the simplest compatible scenario and identifies features relevant to randomness generation.The paper also describes optical implementations and outlines the analysis of classical correlations and certified randomness.Source: §1 · p. 3

2 Semi-DI setting with a physical constraint

The framework leaves device behavior uncharacterized while constraining prepared states through an observable's mean value. Its simplest setting uses two preparations and a binary measurement, with state overlap restricting classical communication.

2.1 Definition of the general model

  • The source and measurement remain uncharacterized, while a specified observable H constrains the prepared states through their mean values.The observable is defined on the Hilbert space describing the emitted systems, which may be infinite-dimensional.Source: §2.1 · pp. 3–4
  • The max-average assumption bounds mean values for each input, whereas max-peak also bounds them for each hidden parameter value λ.Max-average can be checked on average emitted states without knowing source internals; max-peak is more constraining but typically requires source modeling.Source: §2.1 · pp. 4–5

2.2 The simplest setting: two inputs and two outputs

  • The simplest non-trivial scenario has two source inputs, one measurement choice, and two possible outputs.With fewer choices, quantum messages are unnecessary and the behavior can be classically simulated.Source: §2.2 · p. 5
  • Unlike dimension-bounded schemes, the observable constraint can force the two emitted states to overlap, limiting classical channel capacity to below one bit.The two output biases E1 and E2 characterize the behavior; |E1 − E2| > 0 indicates that the measurement partly distinguishes the states.Source: §2.2 · pp. 5–6

1. H has a non-degenerate ground state,

The model applies to observables with a non-degenerate ground state and finite gap. The paper illustrates its optical use with BPSK, 2ASK, and OOK preparations and measurements.

1. H has a non-degenerate ground state,

  • The observable H is assumed to have a non-degenerate ground state and a finite gap, normalized to ground-state eigenvalue 0 and gap 1.The subsequent results apply regardless of the observable's physical meaning.Source: §1 · p. 6

2.3.1 Binary Phase-Shift Keying (BPSK)

  • BPSK sends coherent states |±ξ⟩ and uses homodyne detection of sign(X), while the semi-DI analysis bounds their non-vacuum component.The states have nonzero overlap e^−2|ξ|2; the non-vacuum component is chosen instead of photon number because it gives a more constraining bound for fixed ξ.Source: §2.3.1 · pp. 6, 8

2.3.2 2-level Amplitude-Shift Keying (2ASK)

  • 2ASK sends coherent states |ξ ± ϵ⟩ to a photon detector and constrains their distance from the reference state |ξ⟩.This constraint is less natural and harder to verify because it requires displacement using a coherent beam followed by photon counting.Source: §2.3.2 · p. 8

2.3.3 On-Off Keying (OOK)

  • In OOK, photocounting partially unambiguously distinguishes a coherent state from vacuum: a click identifies the coherent state, but no click is inconclusive.The analysis bounds the non-vacuum component of the coherent-state preparation.Source: §2.3.3 · p. 9

3 Pure-state quantum correlations

For pure-state preparations, the paper characterizes the possible two-output correlations using the states' overlap and the observable expectations. It derives a tight lower bound on that overlap from the expectations.

3 Pure-state quantum correlations

  • For pure states with expectations H1 and H2, the allowed correlations satisfy |Ex| ≤ 1 and g(E1, E2) ≥ h(H1, H2).The characterization is established for H1 + H2 ≤ 1; the physically allowed region can also be represented by the convex hull of two corner points and an ellipse.Source: §3 · pp. 10–11, Figure 4 · p. 12
  • At fixed state overlap γ, the correlation region grows as γ decreases, from E1 = E2 for indistinguishable states to the full square for orthogonal states.Figure 5 depicts the regions for several values of γ.Source: Figure 5 · p. 12, §3 · p. 13

3.2 Lower bound on the scalar product

  • The observable expectations impose the tight lower bound ⟨φ1|φ2⟩ ≥ h(H1, H2), and states attaining the bound make the correlation characterization achievable.The bound expresses how small expectations keep both states close to the non-degenerate ground state and therefore constrain their overlap.Source: §3.2 · p. 13

4 General quantum correlations

The quantum correlations allowed by fixed expectation values remain achievable with pure-state realizations even when the devices share randomness. With upper bounds on the observable’s expectation values, the max-average and max-peak assumptions yield the same correlation set, although their realizations can differ in underlying quantum properties.

4.1 Shared randomness

  • Shared randomness does not enlarge the set of pure-state quantum correlations in this scenario.The result is stated to differ from semi-device-independent scenarios based on dimension bounds.Source: §4.1 · p. 14

4.2 Upper bounds on the expectation values of H

  • Under either upper-bound assumption, every allowed correlation has an equivalent pure-state realization that saturates the expectation-value bounds.The proof uses that the allowed sets for fixed expectation values increase as either expectation value increases.Source: §4.2 · p. 15
  • The max-average and max-peak assumptions define the same set of correlations, but can correspond to realizations with different underlying quantum properties.The equivalence concerns the correlations, not the distinct quantum realizations that may produce them.Source: §4.2 · p. 15

5 Classical correlations

The paper defines classical behavior through shared-randomness mixtures of deterministic measurement outputs, then derives bounds separating those correlations from quantum ones. The resulting nonclassical correlations occur in the optical implementations discussed, with their deterministic explanations depending on the energy constraint.

5.1 Definition

  • Classical correlations are convex combinations of deterministic correlations using shared randomness; outputs are predetermined for both preparations at each shared-randomness value.Correlations outside the classical set therefore require a non-predetermined output for at least one preparation.Source: §5.1 · p. 16
  • At ω1 = ω2 = 0.15, classical systems satisfy |E−| ≤ 0.6, while quantum systems can reach |E−| ≃ 1.43.The E− axis serves as a measure of how quantum the behavior is in this comparison.Source: Figure 6 · p. 17

5.2 Bell inequality analogues

  • The classical bounds play the role of Bell inequalities by separating the quantum region from convex combinations of deterministic correlations.The paper presents these bounds as analogues of the CHSH and Tsirelson inequalities.Source: §5.2 · p. 17

5.3 Implementations discussion

  • BPSK and 2ASK lack deterministic explanations under the max-average assumption, whereas OOK admits one; constraining peak values also rules out OOK’s deterministic explanation.The max-average case allows arbitrary individual expectation values as long as their averages meet the thresholds.Source: §5.3 · p. 18
  • Under the max-peak assumption, any nonzero |E1 − E2| witnesses a nontrivial measurement on non-orthogonal states; under max-average, the threshold is 2(ω1 + ω2).For max-average, the passage states that exceeding this finite value witnesses the same kind of quantum feature.Source: §5.3 · p. 18

6 Correlations exhibiting certified randomness

The paper identifies correlation regions whose violation certifies randomness, including randomness for a chosen preparation input. It characterizes the relevant regions and shows that BPSK, 2ASK, and OOK implementations can certify randomness under the stated energy assumptions, with OOK requiring the max-peak assumption.

6 Correlations exhibiting certified randomness

  • A point outside Dx certifies some randomness for input x, while a point outside the convex hull of D1 ∪ D2 certifies randomness for either input.The corresponding max-peak regions bDx give the analogous certification criteria.Source: §6 · p. 19
  • Quantum correlations outside D1 or bD1 support a semi-device-independent randomness-generation protocol using x = 1 for the random string and both inputs to estimate correlations.The paper notes that statistical corrections are required and defers precise randomness quantification to.Source: §6 · p. 19

6.1 Characterization of the set Dx

  • The paper gives a characterization of Dx and states that the inequalities defining its region are tight, so every point in that region is attainable.The derivation is presented for D1, with the same derivation applying to D2.Source: §6.1 · pp. 19–20

6.2 Characterization of the set bDx

  • The paper also characterizes bDx, and notes that when either ω1 or ω2 is zero, D1, D2, and C coincide.The max-average and max-peak regions are compared in Figure 8.Source: §6.2 · p. 20

6.3 Implementation examples

  • BPSK and 2ASK certify randomness under max-average constraints, while OOK requires max-peak constraints.For ideal OOK without other imperfections, the paper states certification is possible at any nonzero detector efficiency.Source: §6.3 · pp. 20, 22
  • OOK certification is robust to limited detector efficiency: in the idealized case with no other imperfections, it holds for any η > 0.The example assumes a coherent-state source and a detector with efficiency η < 1.Source: §6.3 · pp. 20, 22

7 Conclusion

The paper replaces dimension bounds with bounds on physical observables and analytically characterizes correlations in a minimal prepare-and-measure scenario. It identifies quantum correlations and simple optical implementations that certify randomness, while outlining further semi-device-independent applications.

7 Conclusion

  • The framework bounds mean values of physical observables rather than carrier Hilbert-space dimension, and analytically characterizes correlations in the simplest compatible prepare-and-measure scenario.In quantum optics, a natural choice is an upper bound on emitted photon number or energy in a range of frequency modes.Source: §7 · p. 22
  • The minimal scenario uses two inputs, one measurement setting, and two outputs, and the observable bounds imply sub-unit communication capacity.The approach requires no entanglement manipulation and combines prepare-and-measure practicality with physically motivated assumptions.Source: §7 · p. 22
  • Continuous thresholds, such as bounds on average photon number, allow security margins to be introduced by analyzing with higher thresholds than the source’s target values.The paper presents this as a way to account for device imperfections.Source: §7 · pp. 22, 24
  • For BPSK, certified randomness is possible for ξ ≲ 0.55 under max-average and ξ ≲ 0.63 under max-peak assumptions.The comparison uses the range of ξ for which 0 ≤ ω1, ω2 ≤ 0.5.Source: Figure 10 · p. 23
  • The characterized correlations can provide lower bounds on observable values, including photon-number witnesses, and may support self-testing of quantum properties.The paper presents self-testing as a potential application.Source: §7 · p. 24
  • The paper identifies semi-device-independent randomness generation and quantum key distribution as motivations, with detailed randomness bounds and protocol analysis deferred to.The forthcoming analysis is described as focusing especially on OOK under realistic experimental conditions.Source: §7 · p. 24

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