Optimal Verification of Entangled States with Local Measurements

Abstract

Consider the task of verifying that a given quantum device, designed to produce a particular entangled state, does indeed produce that state. One natural approach would be to characterize the output state by quantum state tomography, or alternatively, to perform some kind of Bell test, tailored to the state of interest. We show here that neither approach is optimal among local verification strategies for 2-qubit states. We find the optimal strategy in this case and show that quadratically fewer total measurements are needed to verify to within a given fidelity than in published results for quantum state tomography, Bell test, or fidelity estimation protocols. We also give efficient verification protocols for any stabilizer state. Additionally, we show that requiring that the strategy be constructed from local, nonadaptive, and noncollective measurements only incurs a constant-factor penalty over a strategy without these restrictions.

Efficient verification of bosonic quantum channels via benchmarking

“…Verification, along with validation that determines whether an implementation is qualified to accomplish a certain task, is important for assessing the credibility of a product or a system. Here quantum-state verification [18][19][20][21][22][23] aims to check whether an implementation of certain quantum state meets the specifications of a target quantum state or not. While [18,19,21] use 'certification' to refer to the process of verification, in this paper, we use the phrase 'quantum-state verification' rather than 'certification'.…”

Efficient Verification of Pure Quantum States in the Adversarial Scenario

“…Even popular alternatives, including compressed sensing [3] and direct fidelity estimation (DFE) [4], cannot avoid this scaling behavior in general. Recently, a powerful approach known as quantum state verification [5][6][7] has attracted increasing attention. This approach has led to efficient protocols for verifying bipartite pure states [5,6,[8][9][10][11], stabilizer states (including graph states) [7,[12][13][14][15], hypergraph states [16], weighted graph states [17], and Dicke states [18].…”

Efficient verification of quantum gates with local operations

“…Then we can verify whether the output Λ(ρ j ) is sufficiently close to the target state U(ρ j ) = U ρ j U † . To this end, we perform two-outcome tests {E l|j , 1 − E l|j } from a set of accessible tests depending on the input state ρ j [17][18][19]. The test operator E l|j corresponds to passing the test and satisfies the condition E l|j U(ρ j ) = U(ρ j ), so that the target output state U(ρ j ) can always pass the test.…”

“…The test operator E l|j corresponds to passing the test and satisfies the condition E l|j U(ρ j ) = U(ρ j ), so that the target output state U(ρ j ) can always pass the test. Suppose the test E l|j is performed with probability p l|j given the test state ρ j , then the passing probability of Λ(ρ j ) reads tr[Ω j Λ(ρ j )], where Ω j = l p l|j E l|j is a verification operator for U(ρ j ) [17][18][19]. Note that Ω j ≥ U(ρ j ) since U(ρ j ) is supported in the eigenspace of Ω j with the largest eigenvalue 1.…”

“…Under the action of the Clifford unitary, each test state is turned into a stabilizer state, which can be verified efficiently with Pauli measurements. More precisely, we can measure all nontrivial stabilizer operators with an equal probability, and the resulting measurement spectral gap is 2 n−1 /(2 n − 1) ≥ 1/2 [17,19], where n is the number of qubits. These results apply to both individual gates and gate sets, including whole Clifford circuits.…”