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Operational Characterization of Coherent Measurements with Steering and Randomness

Chellasamy Jebarathinam, Huan-Yu Ku, Hsi-Sheng Goan

Abstract

Here, we demonstrate that a set of coherent measurements leverages semi-device-independent (SDI) steering and local randomness generation. To this end, we show that coherent measurements are a necessary resource for demonstrating SDI steering. Conversely, through one-to-one mapping of an SDI steerable correlation to a set of measurements, the coherence of the measurements, in turn, is a necessary and sufficient criterion for SDI steering. This result aligns with the relationship between the standard steering scenario and the measurement incompatibility. Then a nonconvex resource theory for SDI steering is formulated and a nonconvex monotone of the resource is obtained in the two-setting scenario using the above-mentioned one-to-one mapping. Finally, we apply this monotone to the quantification of local randomness from two-qubit states without requiring entanglement to be certified.

Operational Characterization of Coherent Measurements with Steering and Randomness

Abstract

Here, we demonstrate that a set of coherent measurements leverages semi-device-independent (SDI) steering and local randomness generation. To this end, we show that coherent measurements are a necessary resource for demonstrating SDI steering. Conversely, through one-to-one mapping of an SDI steerable correlation to a set of measurements, the coherence of the measurements, in turn, is a necessary and sufficient criterion for SDI steering. This result aligns with the relationship between the standard steering scenario and the measurement incompatibility. Then a nonconvex resource theory for SDI steering is formulated and a nonconvex monotone of the resource is obtained in the two-setting scenario using the above-mentioned one-to-one mapping. Finally, we apply this monotone to the quantification of local randomness from two-qubit states without requiring entanglement to be certified.

Paper Structure

This paper contains 5 sections, 6 theorems, 49 equations, 3 figures.

Table of Contents

  1. Proof of Theorem 1
  2. Proof of Theorem 2
  3. Proof of monotonicity of the measure in Definition 2
  4. Proof of Theorem 3
  5. Proof of the inequality of quantum random number generation protocol

Key Result

Theorem 1

Any given SEO $\mathcal{SO}^{n_a}_{n_x}$ is incoherent if and only if the state assemblage admits a decomposition of dimensionally-restricted LHS model as in Eq. (LHSdl), with $d_\lambda \le d_A$.

Figures (3)

  • Figure 1: Operational incompatibility or coherence of measurements: In the 1SDI context, any incompatible measurements on Alice's side can be used to demonstrate steering, implying Bob observes coherent SEO. On the other hand, any coherent measurements on Alice's side can be used to demonstrate SDI steering, implying Bob observes coherent SEO.
  • Figure 2: Quantification of SDI local randomness from two-qubit isotropic state (\ref{['iso']}) for one of the two measurement settings on Alice's side.
  • Figure :

Theorems & Definitions (10)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • proof
  • Theorem 2
  • Definition 2
  • Theorem 3
  • Lemma 1
  • proof