Extended abstract for Certifying entanglement dimensionality by the moment method

Entanglement is a fundamental feature of quantum mechanics. The detection and certification of entanglement on quantum technology platforms have both practical and theoretical meanings [guhne2009entanglement]. Entanglement dimensionality is a specific quantification of entanglement. For a pure state $|\Psi\rangle$ defined on Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, its entanglement dimensionality is its Schmidt rank:

For a mixed state $\rho$, its entanglement dimensionality is defined by the convex-roof construction [terhal2000schmidt]:

where $\mathbb{D}(\rho)$ is a pure state decomposition of $\rho$. The definition implies that a mixed state $\rho$ with Schmidt number $r$ can be prepared from rank-$r$ maximally entangled states but not less. Hence, $\text{SN}(\rho)$ can be regarded as the minimal dimension of the Hilbert space needed for state preparation.

The certification of entanglement has been a very fruitful field of research [guhne2009entanglement], while less is known about the certification of entanglement dimensionality. The most common Schmidt number certification protocol is based on the fidelity-based criterion [terhal2000schmidt]. That is, the maximal fidelity between $\rho$ and a maximally entangled state is no larger than $\text{SN}(\rho)/\min\{d_{A},d_{B}\}$, where $d_{A},d_{B}$ are the dimensions of $\mathcal{H}_{A},\mathcal{H}_{B}$ separately. This maximal fidelity can be estimated by measuring the target states in the mutually unbiased bases [bavaresco2018measurements]. However, fidelity-based methods only apply to a limited type of state named faithful states [weilenmann2020entanglement]. Even pure state with depolarized noise can be unfaithful. This disadvantage is overcame by a recent protocol that combines the correlation matrix criterion with the moments of randomized measurements [wyderka2023probing, liu2023characterizing]. The key idea is to relate the Bloch decomposition of $\rho$ to the Schmidt number. However, this protocol requires at least unitary-4 design to perform, which is a hard task for large-scale quantum systems.

In this work, we present a new entanglement dimensionality certification protocol that only requires unitary-3 design. The idea originates from several recent works which exploit the moments of partial transposed states [elben2020mixed, neven2021symmetry, PhysRevLett.127.060504] for entanglement detection. Our protocol is based on the k-positive map. The $k$-positive map is a Hermicity-preserving map that preserves positivity for all pure states with Schmidt number no larger than $k$. A well-known example of $k$-positive map is the k-reduction map [tomiyama1985geometry], which is defined as

Similar to the positive partial transpose criterion, if a quantum state $\rho$ has Schmidt number no larger than $k$, then it must stay positive under the k-positive map. Conversely, [terhal2000schmidt]

Thus, we can certify Schmidt number lower bound by testing the positivity of $\mathcal{I}_{A}\otimes\mathcal{R}_{k}(\rho)$. However, we cannot directly measure the smallest eigenvalue of $\mathcal{I}_{A}\otimes\mathcal{R}_{k}(\rho)$. To turn this criterion into a practical protocol, we use known results in the moment problem [schmudgen2017moment].

In mathematical literature, the moment problem studies the relation between a Rodan measurement $d\mu$ and the moments of this measure $s_{n}=\int_{K}d\mu x^{n}$. It is possible to infer information about the $d\mu$ from the first few order of moments [schmudgen2017moment]. In our situation, we want to determine whether the spectrum of $\mathcal{I}_{A}\otimes\mathcal{R}_{k}(\rho)$ is in region $[0,k]$ or not. If we write the spectrum of $\mathcal{I}_{A}\otimes\mathcal{R}_{k}(\rho)$ in the form of Rodan measure, then the corresponding moments are $s_{n}=\text{Tr}[(\mathcal{I}_{A}\otimes\mathcal{R}_{k}(\rho))^{n}]$. We use the following theorem as the main criterion for Schmidt number certification:

Conversely,

Now the question is reduced to how to estimate the moments $\text{Tr}[(\mathcal{I}_{A}\otimes\mathcal{R}_{k}(\rho))^{n}]$. A standard approach is the randomized measurement [brydges2019probing, huang2020predicting, elben2023randomized]. There has been an extensive literature about the principle and applications of this method in quantum information science, so we do not elaborate on the details here. By estimating non-linear functions of $\rho$ and $\rho_{A}$ with classical shadow methods [huang2020predicting], we are able to construct the Hankel matrices $B_{N}$. Up to this step, we obtain a practical protocol for Schmidt number detection. The formal algorithm writes:

When we choose $N^{\ast}=3$, we obtain the simplest form of our protocol, and we prove the following theorem about its sample complexity:

In the remaining part of the work, we evaluate the performance of our protocol by testing what type of states can be certified by it. The question can be stated as follows: given a Schmidt number detection protocol and an ensemble of quantum states, when does the optimal certifiable Schmidt number lower bound equal the true Schmidt number? Or given several protocols, which one can give the best estimation for the true Schmidt number? We focus our attention on three types of state: pure states, pure states with depolarized noise and induced metric random states [aubrun2012phase]. We also analyze the correlation matrix method from this aspect. We conclude that the $k$-positive map criterion itself performs better than the correlation matrix criterion for states with high purity, while the applicable scope of the moment-based protocol is rather limited comparing to that of the correlation matrix method.

The main results are summarized in Table 1.