Robust certification of quantum instruments through a sequential communication game

1. Problem Statement

The paper addresses the challenge of certifying quantum instruments (specifically non-projective and unsharp measurements) in a semi-device-independent (SDI) framework.

• Context: Standard certification schemes often rely on device-independent (Bell test) or fully device-dependent assumptions. While SDI protocols exist for states and projective measurements, certifying the instruments (the specific mathematical description of the measurement process, including the post-measurement state update) is difficult in standard "prepare-and-measure" scenarios.
• Gap: Previous sequential Quantum Random Access Code (QRAC) protocols allowed for instrument certification but lacked robustness against noise and did not utilize collaboration between receivers effectively.
• Goal: The authors propose a new communication game involving one sender and two sequential receivers with restricted communication to achieve a more robust certification of the sender's state preparation, the first receiver's measurement instrument, and the second receiver's measurement device.

2. Methodology

A. The Proposed Communication Game

The authors extend the standard $2 \to 1$ QRAC to a tripartite sequential scenario involving:

1. Sender (Aparna): Encodes a 2-bit message $x = (x_0, x_1)$ into a single qubit state $\rho_x$.
2. First Receiver (Barun): Receives a random input $y \in \{0, 1\}$ and attempts to guess the bit $x_y$. He performs a measurement (instrument) and passes the post-measurement state to the next party. Crucially, he does not leak his output to the second receiver.
3. Second Receiver (Chhanda): Receives the post-measurement state and the same input $y$. Her task is to guess the complementary bit $x_{\bar{y}}$.
4. Constraint: Unlike previous sequential protocols where receivers were independent, here the receivers are linked by the physical state transfer, but Chhanda has no access to Barun's classical output.

B. Theoretical Framework

• Scenario: Prepare-Transform-Measure (PTM).
• Mathematical Tools:
  • Kraus Operators: Barun's measurement is modeled by Kraus operators $K_{b|y}$, transforming the state $\rho_x \to \rho'_{x,y,b}$.
  • Lüders Rule: Used to define the state update after Barun's unsharp measurement.
  • Semidefinite Programming (SDP): Used to find the optimal trade-off between the success probabilities of Barun ($P_{AB}$) and Chhanda ($P_{AC}$).
  • See-Saw Algorithm: Used for numerical optimization in higher-dimensional systems ($d$-dimensional qudits).

C. Certification Strategy

The core of the methodology relies on identifying a non-trivial trade-off between $P_{AB}$ and $P_{AC}$.

• In classical strategies, these probabilities are independent.
• In quantum strategies, increasing Barun's sharpness (and thus $P_{AB}$) degrades the information available to Chhanda (lowering $P_{AC}$), and vice versa.
• By observing a specific pair of success probabilities $(P_{AB}, P_{AC})$ that lies on the quantum boundary, the authors can uniquely identify (self-test) the underlying quantum objects.

3. Key Contributions

1. Novel Communication Game: Introduction of a sequential game with restricted collaboration where the second receiver must guess the remaining bit without knowing the first receiver's outcome. This differs from prior work where receivers acted independently.
2. Robust Instrument Certification: The protocol enables the self-testing of quantum instruments (specifically the sharpness parameter $\eta$ of unsharp measurements) in an SDI manner.
3. Enhanced Robustness: The authors demonstrate that their protocol provides a narrower estimation gap for the sharpness parameter $\eta$ in the presence of noise compared to previous sequential QRAC protocols (e.g., Ref. [67]).
4. Higher-Dimensional Generalization: The game is generalized to $d$-dimensional systems ($2d-1 \to 1$ QRAC), showing that quantum advantage increases with dimension.

4. Key Results

A. Quantum Advantage and Trade-off

• Classical Bound: The maximum joint success probability for classical strategies is $P^C = \frac{1}{2d}(1 + \frac{1}{d})$.
• Quantum Boundary: The authors derived an optimal trade-off curve between $P_{AB}$ and $P_{AC}$.
  • For a 2-bit message ($d=2$), the optimal quantum strategy involves Aparna preparing states forming a square on the Bloch sphere, Barun performing an unsharp measurement with sharpness $\eta$, and Chhanda performing a projective measurement.
  • The relationship is given by:
    $$P_{AB} = \frac{1}{2}\left(1 + \frac{\eta}{\sqrt{2}}\right), \quad P_{AC} = \frac{1}{4}\left(2 + \sqrt{2 - 2\eta^2}\right)$$
  • This curve defines the boundary of the quantum set; any point on this curve certifies the specific quantum strategy used.

B. Robustness Analysis (Noise Tolerance)

• The authors modeled noise as a visibility parameter ($p$) for the state preparation and measurement devices.
• Result: For a given set of noisy success probabilities, the protocol estimates the sharpness parameter $\eta$ within a range $[\eta_{lower}, \eta_{upper}]$.
• Comparison: The gap $\delta = \eta_{upper} - \eta_{lower}$ in the proposed protocol is significantly smaller than the gap $\delta^*$ in the non-collaborative protocol (Ref. [67]).
  • Example: With visibilities $p_1=0.95, p_2=0.9, p_3=0.95$, the proposed protocol yields an error gap of 0.1133, whereas the previous method yields 0.1964. This indicates a ~43% improvement in certification precision.

C. Higher Dimensions

• Using the see-saw algorithm, the authors computed quantum advantages for dimensions up to $d=6$.
• Finding: While individual success probabilities decrease as dimension $d$ increases, the quantum advantage (the gap between quantum and classical success probabilities) increases with dimension.
• Unsharp measurements were found to be more resourceful than sharp measurements in these sequential tasks.

5. Significance

• Advancement in Quantum Certification: This work moves beyond certifying just states and projective measurements to certifying the dynamics of measurement (instruments). This is crucial for verifying quantum hardware where unsharp measurements are common (e.g., in quantum error correction or weak measurements).
• Practicality: The "semi-device-independent" nature makes the protocol experimentally feasible, as it does not require the strict isolation assumptions of device-independent protocols but still offers strong security guarantees.
• Noise Resilience: The demonstrated robustness against noise makes this protocol suitable for near-term quantum devices (NISQ era), where perfect state preparation and measurement are unattainable.
• Foundational Insight: The results highlight the fundamental role of unsharp measurements and sequential correlations in extracting quantum advantages, suggesting new directions for quantum communication networks involving multiple nodes.

In summary, the paper presents a theoretically rigorous and experimentally viable protocol that significantly improves the precision of certifying quantum measurement instruments in noisy environments, leveraging a novel sequential communication game with restricted collaboration.