Beating the Optimal Verification of Entangled States via Collective Strategies

TL;DR

A new verification scheme using collective strategies is proposed, showcasing arbitrarily high efficiency that beats the optimal verification with global measurements and provides additional insight into the specific types of noise affecting the system, thereby facilitating potential targeted improvements.

Abstract

In the realm of quantum information processing, the efficient characterization of entangled states poses an overwhelming challenge, rendering the traditional methods including quantum tomography unfeasible and impractical. To tackle this problem, we propose a new verification scheme using collective strategies, showcasing arbitrarily high efficiency that beats the optimal verification with global measurements. Our collective scheme can be implemented in various experimental platforms and scalable for large systems with a linear scaling on hardware requirement, and distributed operations are allowed. Notably, larger ensembles can always improve the efficiency further, but without increasing the quantum memory. More importantly, the approach consumes only a few copies of the entangled states, while ensuring the preservation of unmeasured ones, and even boosting their fidelity for any subsequent tasks. Furthermore, our protocol provides additional insight into the specific types of noise affecting the system, thereby facilitating potential targeted improvements. These advancements hold promise for a wide range of applications, offering a pathway towards more robust and efficient quantum information processing.

Figures (10)

• Figure 1: Sketch of the verification scheme using collective strategies. We assume that a source prepares multiple copies of the target state. Then a SWAP projection followed by a random permutation is applied on all the copies. Next, standard QSV is conducted on a random subset of the copies, while the remaining are retained for any subsequent tasks. If all the measured copies pass the test, the collective verification scheme is called successful. By repeating the procedure several rounds, the target state can be verified within a certain infidelity and confidence level.
• Figure 2: (a) Construction of a SWAP projection by performing controlled SWAP operations $cS_k$ on four input states. (b) Construction of a controlled SWAP operation by performing controlled qubit-qubit SWAPs, i.e., Fredkin gates on two bipartite states.
• Figure 3: Sample complexity for verifying a $100$-qubit Dicke state using the collective strategies under independent noise with schemes $\Pi_{2,1}$ (blue) and $\Pi_{10,1}$ (orange), and under correlated noise with the scheme $\Pi_{k,1}$ (green), as compared to the best known local scheme Li.etal2020a (black dot) and the optimal global scheme (red star). The confidence level is set to $1-\delta=99\%$.
• Figure 4: Tensor diagram for solving the (partial) trace of $\bigl(S_k \cdot \sigma_1 \otimes\cdots\otimes \sigma_k \bigr)$. The permutation $S_k$ and its conjugate $S_k^\dagger$ are represented by the cyclic permutations of the indices (legs) of the $\sigma$s.
• Figure 5: Sample complexity for verifying (a) a Bell state and (b) a $100$-qubit Dicke state using the collective strategies under independent white noise with schemes $\Pi_{2,1}$ (blue) and $\Pi_{10,1}$ (orange), under global white noise with the scheme $\Pi_{k,1}$ (green), and under global unitary control with the scheme $\Pi_{k,t}$ (brown) as compared to the best known local scheme Pallister.etal2018Li.etal2020a (black dot) and the optimal global scheme (red star). The confidence level is set to $1-\delta=99\%$.

Theorems & Definitions (11)

• Proposition 1
• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1
• proof
• proof
• Theorem 3
• proof