Prepare-and-Measure Dimension Witness

Updated 25 January 2026

Prepare-and-measure dimension witness is a method to certify a lower bound on the quantum system's dimension using observed input-output statistics in a PM scenario.
The approach employs both linear (QRAC-based) and nonlinear (determinant) witnesses, with tunable weak measurements enabling simultaneous classical bound violations by different observers.
These techniques underpin semi-device-independent protocols like QRNG and QKD, offering practical tools for secure quantum communications and hardware certification.

A prepare-and-measure (PM) dimension witness is a functional of the observed statistics in a PM communication scenario that certifies a lower bound on the dimension of the underlying system solely from input-output data. Such witnesses play a central role in the device-independent and semi-device-independent characterization of quantum systems, enabling certification of resources such as Hilbert space dimensionality, quantumness, or non-classicality in fully or partially untrusted apparatuses. The PM scenario can be extended to multiparty and sequential architectures, as in the three-observer protocols leveraging weak measurement, which unlock simultaneous witness violations by multiple parties—an effect unattainable in purely classical settings.

1. Prepare-and-Measure Framework and Sequential Multi-Observer Extension

In the canonical PM scenario, an uncharacterized preparer ("Alice") receives a classical input $x$ and emits a physical system, which is measured by a receiver ("Bob") with input $y$ , yielding output $b$ . The observed behavior is fully described by the conditional probabilities $p(b|x,y)$ , with no assumptions about the internal operation of either device except an (optional) upper bound on system dimension $d$ .

The three-observer extension introduces an intermediary ("Charlie") who performs a tunable weak measurement on the transmitted system before it reaches the final receiver (Bob), and subsequently measures his ancillary system. The experimental sequence is as follows (Li et al., 2017):

Preparation: Alice, on input $x\in \{00,01,10,11\}$ , emits a qubit state $\rho_x$ .
Weak measurement: Charlie couples the qubit to a two-dimensional ancilla prepared in $|+\rangle$ , applies a controlled-unitary $U$ parametrized by interaction strength $\epsilon$ , and forwards the possibly disturbed signal.
Projective measurement: Bob, on input $y \in \{0,1\}$ , performs a projective measurement; his outcome $b \in \{+1,-1\}$ is recorded.
Ancilla readout: Charlie then measures his ancilla in a basis determined by $z \in \{0,1\}$ , outputting $c \in \{+1,-1\}$ .

By judiciously tuning the weak measurement strength $\epsilon$ , the protocol enables both Bob and Charlie to individually violate classical dimension bounds.

2. Dimension-Witness Inequalities: Linear and Nonlinear Cases

Dimension witnesses in PM utilize the observed probabilities to form inequalities whose violation certifies a dimensional threshold. The principal types are:

A. Linear QRAC-Based Witness ( $W_1$ ):

Based on the $2\to1$ quantum random access code (QRAC), $W_1$ is defined by:

$W_1 = p(+1|00,0) + p(+1|01,0) + p(+1|10,1) - p(+1|11,1) - p(+1|10,0) + p(+1|11,0) - p(+1|00,1) - p(+1|01,1)$

A classical two-dimensional system must satisfy $W_1 \le 2$ , whereas quantum qubits can reach $W_1 = 2\sqrt{2}$ . In the tripartite protocol, one evaluates both Bob’s and Charlie’s statistics, yielding $W_{1,\mathrm{AB}}, W_{1,\mathrm{AC}}$ respectively.

B. Nonlinear Determinant Witness ( $W_2$ ):

The nonlinear witness is:

$W_2 = |\det M|, \; M = \begin{bmatrix} p(+1|00,0) - p(+1|01,0) & p(+1|10,0) - p(+1|11,0) \ p(+1|00,1) - p(+1|01,1) & p(+1|10,1) - p(+1|11,1) \end{bmatrix}$

Classical dimension-2 models always yield $W_2 = 0$ , whereas quantum realizations can attain $W_2 = 1$ . Both Bob’s and Charlie's marginals can be tested individually.

3. Analytical Characterization and Weak-Measurement Model

Charlie's weak measurement is formalized by coupling the system qubit to an ancilla through a controlled-unitary:

$U = W^{+}_z \otimes I + W^{-}_z \otimes e^{i\epsilon \sigma_z}$

with $W^{\pm}_z$ being projectors rotated according to $z$ . The output density matrices for Bob and Charlie after this stage are:

For Bob (after tracing out Charlie's ancilla):

$\rho'_{B|x}(z) = (1 - \cos\epsilon)(W^+_z \rho_x W^+_z + W^-_z \rho_x W^-_z) + \cos\epsilon\, \rho_x$

For Charlie's ancilla (after tracing out Bob):

$\rho'_{C|x}(z) = \operatorname{Tr}_B(W^+_z \rho_x)|+\rangle\langle+| + \operatorname{Tr}_B(W^-_z \rho_x) e^{i\epsilon \sigma_z}|+\rangle\langle+|e^{-i\epsilon \sigma_z}$

Bob and Charlie's measurement statistics are:

$p(b|x,y,z) = \operatorname{Tr}[B^b_y \rho'_{B|x}(z)], \quad p(c|x,y,z) = \operatorname{Tr}[C^c_z \rho'_{C|x}(z)]$

4. Double Violation Regimes and Thresholds

For the tripartite protocol, analytical expressions for the witness values as functions of $\epsilon$ are:

$W_{1,\mathrm{AB}} = \sqrt{2}(1 + \cos\epsilon)$
$W_{1,\mathrm{AC}} = 2\sqrt{2}\sin^2\epsilon$
$W_{2,\mathrm{AB}} = \left(\frac{1 + \cos\epsilon}{2}\right)^2$
$W_{2,\mathrm{AC}} = \sin^4\epsilon$

Double violation of the classical bounds ( $W_1 > 2$ , $W_2 > 0$ for both observers) is possible. For $W_1$ , simultaneous violation occurs iff:

$\arcsin(2^{-1/4}) < \epsilon < \arccos(\sqrt{2}-1) \implies 0.615~\text{rad} < \epsilon < 0.955~\text{rad}$

For $W_2$ , both observers always have $W_2 > 0$ for any $\epsilon \in (0, \pi)\setminus\{0, \pi\}$ .

5. Quantum vs. Classical Dimension Boundaries

Classically, in a sequential PM scenario with $d=2$ (a bit), any attempt to divide the information content between two observers (even with shared randomness) cannot yield $W_1 > 2$ or $W_2 > 0$ in both marginals. However, a single quantum system, when weakly measured and then projectively post-measured, can yield two independent witness violations. This difference concretely demonstrates the operational advantage of quantum over classical dimension even in sequential access scenarios, and quantifies how disturbance and information extraction jointly depend on measurement strength (Li et al., 2017).

6. Semi-Device-Independent Randomness Generation and Key Distribution

Dimension witness violations underpin semi-device-independent (SI) protocols such as SI quantum random number generators (SI-QRNG) and SI quantum key distribution (QKD):

QRNG: The min-entropy $H_{\min}$ of the global outcome distribution (Bob+Charlie) or local ( $b$ alone) can be bounded directly from the observed witness value: $H_{\min,1} \geq -\log_2[\max_{b,c,x,y,z} p(b,c|x,y,z)]$ , with tighter local bounds available via analytic formulas dependent on $W_1$ or $W_2$ (Li et al., 2017, Bowles et al., 2013).
QKD: In SI QKD, the dimension witness violation constrains an adversary's accessible information, enforcing security so long as all mediating signals are qubits. The multi-observer protocol allows the same quantum system to contribute to multiple correlated bitstreams, potentially enhancing key rates or randomness yield per channel use.

Dimension witnesses in PM scenarios extend beyond linear inequalities; determinant-type (nonlinear) witnesses (Batle et al., 2022, Bowles et al., 2013) offer robustness to noise, independence from shared randomness, and equality-based strictness (e.g., the witness is strictly zero for dimension $d$ , nonzero iff $d'>d$ ). Analytical lower bounds on system dimension from observed statistics can be obtained using trace-norm and Gram-matrix constructions, universally applicable across PM tasks, including communication complexity, QRACs, and state-discrimination schemes (Sikora et al., 2016, Vicente, 2018).

Recent work has explored minimality in the number of preparations and measurements required for self-testing and tight dimension certification (Drótos et al., 2024, Batle et al., 2022). Experiments on quantum devices have validated determinant-based witnesses with sub-milliarcsecond accuracy and shown sensitivity to subtle leakage or hardware inconsistencies (Białecki et al., 2023). Non-classicality, non-stabilizerness, and further resource properties have been characterized using PM-dimension-witness structures (Zamora et al., 2 Jun 2025).

Witness	Functional Form	Classical Bound (d=2)	Quantum Max (qubit)
$W_1$ (linear, QRAC)	See above definition	$W_1 \leq 2$	$W_1 = 2\sqrt{2}$
$W_2$ (nonlinear, det)	$\|\det M\|$ as described	$W_2 = 0$	$W_2 = 1$

By integrating tunable weak measurement and analyzing both linear and nonlinear witnesses, prepare-and-measure dimension-witness protocols in multipartite and sequential architectures sharply discriminate between classical and quantum resources, inform the design of robust semi-device-independent cryptographic primitives, and offer a nuanced toolset for hardware certification and quantum communications research (Li et al., 2017, Batle et al., 2022, Bowles et al., 2013, Sikora et al., 2016).