The role of shared randomness in quantum state certification with
unentangled measurements
- URL: http://arxiv.org/abs/2401.09650v1
- Date: Wed, 17 Jan 2024 23:44:52 GMT
- Title: The role of shared randomness in quantum state certification with
unentangled measurements
- Authors: Yuhan Liu, Jayadev Acharya
- Abstract summary: We study quantum state certification using unentangled quantum measurements.
$Theta(d2/varepsilon2)$ copies are necessary and sufficient for state certification.
We develop a unified lower bound framework for both fixed and randomized measurements.
- Score: 36.19846254657676
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Given $n$ copies of an unknown quantum state $\rho\in\mathbb{C}^{d\times d}$,
quantum state certification is the task of determining whether $\rho=\rho_0$ or
$\|\rho-\rho_0\|_1>\varepsilon$, where $\rho_0$ is a known reference state. We
study quantum state certification using unentangled quantum measurements,
namely measurements which operate only on one copy of $\rho$ at a time. When
there is a common source of shared randomness available and the unentangled
measurements are chosen based on this randomness, prior work has shown that
$\Theta(d^{3/2}/\varepsilon^2)$ copies are necessary and sufficient. This holds
even when the measurements are allowed to be chosen adaptively. We consider
deterministic measurement schemes (as opposed to randomized) and demonstrate
that ${\Theta}(d^2/\varepsilon^2)$ copies are necessary and sufficient for
state certification. This shows a separation between algorithms with and
without shared randomness.
We develop a unified lower bound framework for both fixed and randomized
measurements, under the same theoretical framework that relates the hardness of
testing to the well-established L\"uders rule. More precisely, we obtain lower
bounds for randomized and fixed schemes as a function of the eigenvalues of the
L\"uders channel which characterizes one possible post-measurement state
transformation.
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