QRC-Eval: Protocol for Quantum Instrument Certification

• QRC-Eval is a protocol that applies sequential Quantum Random Access Codes on a single qubit to certify unsharp quantum instruments within a prepare–transform–measure scenario.
• The method analytically characterizes the optimal trade-off between information extraction (via Bob's measurement) and disturbance (affecting Charlie's outcome) to enable self-testing of measurement devices.
• QRC-Eval is applicable in quantum cryptography and randomness generation, offering robust semi-device-independent certification with minimal assumptions on state preparation and measurement systems.

QRC-Eval is an operational protocol grounded in the sequential application of Quantum Random Access Codes (QRACs) on a single qubit, designed for the semi-device-independent evaluation and certification of quantum measurement instruments, notably unsharp quantum instruments. Developed within the prepare–transform–measure framework, QRC-Eval quantifies the fundamental trade-off between information extraction and induced disturbance in sequential measurements and leverages this relationship to bound, or even uniquely identify, the underlying quantum measurement dynamics realized by an experimenter.

1. Sequential QRACs in the Prepare–Transform–Measure Scenario

QRC-Eval utilizes a three-party linear sequence—Alice, Bob, Charlie—executing two instances of the canonical 2→1 QRAC on the same qubit. The protocol proceeds as follows:

• State Preparation: Alice receives a classical input $x = (x_0, x_1) \in \{0,1\}^2$, chosen uniformly, and prepares a quantum state $\rho_x \in \mathcal{B}(\mathbb{C}^2)$. She sends this state to Bob.
• Instrument Action: Bob, given $y \in \{0,1\}$, applies a measurement instrument $\{K_{b|y}\}$ (a set of completely positive trace-preserving maps parameterized by outcome $b \in \{0,1\}$) to $\rho_x$, records the classical outcome $b$, and forwards the post-measurement state to Charlie.
• Final Measurement: Charlie receives the qubit, gets $z \in \{0,1\}$, and performs a binary POVM $\{C_{c|z}\}$, recording outcome $c$.

The sequence produces joint conditional probabilities $$p(b, c|x, y, z) = \operatorname{Tr}[K_{b|y}\, \rho_x\, K_{b|y}^\dagger\, C_{c|z}]$$, enabling extraction of two figures of merit:

• $W_{AB}$: Average success probability of Bob guessing $x_y$,

$$W_{AB} = \frac{1}{8} \sum_{x, y} \operatorname{Tr}[\rho_x\, M_{x_y|y}],$$

where $M_{b|y} = K_{b|y}^\dagger K_{b|y}$.

• $W_{AC}$: Average success probability of Charlie guessing $x_z$,

$$W_{AC} = \frac{1}{16} \sum_{x, y, b, z} \operatorname{Tr}[K_{b|y} \rho_x K_{b|y}^\dagger C_{x_z|z}].$$

Under classical strategies (diagonal $\rho_x$, non-disturbing instruments), $W_{AB}, W_{AC} \leq 3/4$; quantum strategies allow both to exceed $3/4$, subject to a disturbance trade-off enforced by Bob’s instrument (1905.06726).

2. Optimal Information–Disturbance Trade-Off

A distinguishing feature of QRC-Eval is the analytic characterization of the optimal quantum boundary—the achievable region for pairs $(W_{AB}, W_{AC})$. Fixing $W_{AB} = \alpha \in [1/2, (1 + 1/\sqrt{2})/2]$, the maximal $W_{AC}$ is

$$W_{AC}^{\alpha} = \frac{1}{8}\Big[ 4 + \sqrt{2} + \sqrt{16\alpha - 16\alpha^2 - 2} \, \Big]$$

and equivalently

$$8W_{AC} - (4 + \sqrt{2}) \leq \sqrt{16 W_{AB} - 16 W_{AB}^2 - 2}.$$

Key points on this curve include:

• For $W_{AB} = (1+1/\sqrt{2})/2 \approx 0.8536$ (optimal single QRAC), $W_{AC}^{\alpha} \approx 0.6768$.
• When $W_{AB} = W_{AC}$, the common value is $(5 + 2\sqrt{2})/10 \approx 0.7828 > 3/4$.

This boundary is derived via Bloch-sphere parametrization of input states, polar decomposition of the instrument, and convexity/convex extremality arguments for the final measurement. The maximal trade-off is realized for ensembles forming a square on a great circle of the Bloch sphere and instruments corresponding to equally unsharp Lüders measurements along orthogonal axes (1905.06726).

3. Self-Testing and Certification of Quantum Instruments

QRC-Eval provides a semi-device-independent self-testing mechanism: observing $(W_{AB}, W_{AC})$ exactly on the quantum boundary uniquely constrains the experimental realization (up to global unitaries):

• Preparation: Alice’s ensemble $\{\rho_x\}$ consists of four pure states that form a square in a plane of the Bloch sphere.
• Instrument: Bob’s measurement instrument corresponds to two Lüders measurements (POVMs along $\sigma_x$ and $\sigma_z$), with equal sharpness parameter $\eta$; the Kraus operators are $$K_{b|0} = \sqrt{[\mathbb{I} + (-1)^b \eta \sigma_x]/2}$$ and $$K_{b|1} = \sqrt{[\mathbb{I} + (-1)^b \eta \sigma_z]/2}$$.
• Measurement: Charlie measures projectively along the same orthogonal axes.

Consequently, measuring the optimal trade-off “self-tests” both the measurement bases and the quantifiable unsharpness of the intervening quantum instrument (1905.06726).

4. Sharpness Parameter Estimation and Certified Bounds

The quantum instrument’s “sharpness” $\eta$ (i.e., the length of the POVM Bloch vector $|\vec{c}_y|$) can be estimated from observed sequential QRAC statistics:

• Lower bound from $W_{AB}$:

$\eta \geq \sqrt{2}\,(2 W_{AB} - 1)$

This bound is tight when the “square strategy” is optimal.

• Upper bound from $W_{AC}$:

$$\eta \leq 2 \sqrt{(2+\sqrt{2} - 4W_{AC})(2W_{AC} - 1)}$$

This applies in the range $W_{AC} \in [(4+\sqrt{2})/8, (2+\sqrt{2})/4]$; otherwise, $0 \leq \eta \leq 1$.

When the observed pair $(W_{AB}, W_{AC})$ lies precisely on the quantum boundary, both bounds coincide, pinning $\eta$ to the self-tested value $\eta = \sqrt{2} (2\alpha - 1)$. In realistic experimental scenarios (e.g., with finite state/instrument/measurement visibilities), these bounds certify $\eta$ to a finite interval (1905.06726).

5. Practical Protocol Instantiation and Data Analysis

To instantiate QRC-Eval, an experimentalist collects the statistics required to compute $W_{AB}$ and $W_{AC}$, and locates the experimental $(W_{AB}, W_{AC})$ pair relative to the theoretically derived optimal boundary. By applying the explicit analytic bounds above, one extracts either:

• Complete self-testing of all elements of the protocol (state preparation, instrument structure, and measurement bases), if the point lies on the quantum boundary.
• Noise-robust upper and lower bounds on instrument sharpness, diagnosing and quantifying deviations from ideality caused by practical imperfections.

For example, with finite visibilities $v_a = 0.95$ (Alice), $v_b = 0.90$ (Bob), $v_c = 0.95$ (Charlie) and a target $\eta = 1/\sqrt{2}$, typical observed witnesses ($W_{AB} \approx 0.7138$, $W_{AC} \approx 0.7826$) yield $0.6047 \leq \eta \leq 0.8010$. Even moderate losses broaden the interval, necessitating high-fidelity components for tight certification (1905.06726).

6. Relevance to Quantum Information Protocols

QRC-Eval enables the semi-device-independent certification of unsharp quantum measurements with minimal assumptions: only Hilbert space dimension (qubit) and trusted inputs are assumed. This methodology directly witnesses the information–disturbance compromise and allows robust, quantitative certification of intermediate measurements without requiring full process tomography or Bell nonlocality. Applications include quantum cryptography (where measurement sharpness modulates information security), randomness generation, quantum network coding, and foundational protocols demanding certification of intermediate unsharp instruments (1905.06726).

A plausible implication is that QRC-Eval forms a general template for quantifying the informativeness and disturbance of intermediate measurements in other prepare–transform–measure scenarios where measurement back-action is nontrivial and direct process access is restricted.