Universal and Efficient Quantum State Verification via Schmidt Decomposition and Mutually Unbiased Bases

Abstract

Efficient verification of multipartite quantum states is crucial to many applications in quantum information processing. By virtue of Schmidt decomposition and mutually unbiased bases, here we propose a universal protocol to verify arbitrary multipartite pure quantum states using adaptive local projective measurements. Moreover, we establish a universal upper bound on the sample complexity that is independent of the local dimensions. Numerical calculations further indicate that Haar-random pure states can be verified with a constant sample cost, irrespective of the qudit number and local dimensions, even in the adversarial scenario in which the source cannot be trusted. As alternatives, we provide several simpler variants that can achieve similar high efficiencies without using Schmidt decomposition. The simplest variant consists of only two distinct tests.

Figures (16)

• Figure 1: Schematic of the SD protocol for verifying an $n$-qudit state $|\Psi\rangle$. The first party performs a projective measurement in either the Schmidt basis $(m_1=0)$ or a MUB $(m_1=1)$ with respect to the Schmidt basis. Then party $k$ for each $k=2,3,\ldots,n-1$ performs a projective measurement in succession in either the Schmidt basis $(m_k=0)$ or a MUB $(m_k=1)$ associated with the conditional reduced state of $|\Psi\rangle$ for parties $k,k+1,\ldots, n$ depending on the previous measurement choices and outcomes. Finally, party $n$ performs the projective measurement $\{|\psi_n\rangle\langle\psi_n|,1-|\psi_n\rangle\langle\psi_n|\}$, where $|\psi_n\rangle$ is the conditional reduced state of $|\Psi\rangle$ for party $n$, and the first outcome means passing the test.
• Figure 2: Average spectral gaps achieved by the SD and CSD protocols for $n$-qudit Haar-random pure states (see Table \ref{['tab:SampleNumHaar']} in Appendix \ref{['app:SampleNumHaar']} for the numbers of states sampled). Solid lines with circles (dashed lines with squares) represent the SD (CSD) protocol under the uniform probability distribution, while dash-dot lines with diamonds represent the SD protocol under the optimized probability distribution determined by SDP. The gray dashed lines correspond to the spectral gap of $1/5$.
• Figure 3: Schematic of the CSD protocol as a probabilistic mixture of $n$ SD protocols tied to $n$ different orders of SD. The first party is chosen randomly, which determines the order of SD and the corresponding SD protocol according to a cyclic sequence.
• Figure 4: Average spectral gaps achieved by eight variants of the MUB protocol for $n$-qudit Haar-random pure states with $d=2,3,4,5$. The color encoding of the local dimension $d$ follows the lower plot in Fig. \ref{['fig:HaarSD']} (blue, orange, green, and purple mean $d=2,3,4,5$, respectively).
• Figure 5: Schematic of the SMUB protocol as a probabilistic mixture of $n$ MUB protocols. Each MUB protocol featured in the SMUB protocol is specified by the choice of the last party (here $k=4$), the party that performs the projective measurement at last conditioned on the measurement choices [specified by the string $\mathbf{m}=(1,0,0,1)$] and outcomes of the other $n-1$ parties. The projective measurement of the last party is determined by the conditional reduced state $|\psi_k\rangle=|\psi_{k,\mathbf{o}, \mathbf{m}}\rangle$ of the target state.

Theorems & Definitions (2)

• Theorem 1
• proof : Proof of Theorem \ref{['thm:QSVSDgap']}