Simultaneous Detection of High-Dimensional Entanglement for Two Unknown Quantum States

Abstract

The state overlap, quantified via $\tr[ρσ]$, is a metric widely used to assess the closeness between two quantum states $ρ$ and $σ$. Although global state overlap alone does not directly capture entanglement properties, we uncover that incorporating local state overlaps provide profound insights into the entanglement characteristics of quantum states. To be precise, the ratio of global to local state overlaps provides a lower bound on the Schmidt number, which is usually used for quantifying high-dimensional entanglement. Unlike conventional methods for detecting entanglement, the approach here can simultaneously reveal entanglement information for two unknown quantum states. Moreover, state overlap can be efficiently determined through local randomized measurement methods, which ensures the experimental feasibility of our approach. In a special case, our criterion reduces to an entanglement criterion that is more powerful than the two criteria used most in experiment--the purity criterion and the fidelity-based criterion and also outperform the $p_3$-PPT method in specific instances. Our findings highlight a promising direction for advancements in entanglement detection experiments.

Figures (4)

• Figure 1: This figures shows the points $(x,y)$ where the Schmidt number of $\rho_{\mathrm{Iso}}(x)$ and $\rho_{\mathrm{Iso}}(y)$ can be detected via $r$-IPC ($r=1,2,3,4,5$ for $d=10$). That is, the corresponding $\rho_{\mathrm{Iso}}(x)$ and $\rho_{\mathrm{Iso}}(y)$ violate the inequality \ref{['eq:Inner_ineq_SCHM']}.
• Figure 2: This figure shows how to use local randomized measurements to detect the Schmidt number. In view of our current context, we propose a slight modification to the existing methods for estimating state overlap through local randomized measurements (the full details of the method can be found in Ref. IPC_Elben20). Consider the simplest case where $\mathcal{H}_A=\otimes_{i=1}^m \mathcal{H}_{A_i}$ and $\mathcal{H}_B=\otimes_{j=1}^n \mathcal{H}_{B_j}$ with each $\mathcal{H}_{A_i}$ and $\mathcal{H}_{B_j}$ having a dimension of $\ell \geq 2$. We apply the same unitary $U_{AB}=U_A\otimes U_B$, where $U_A=\otimes_{i=1}^m u_{A_i}$ and $U_B=\otimes_{j=1}^n u_{B_j}$, to both $\rho$ and $\sigma$. Here the $u_{A_i}$ and $u_{B_j}$ are sampled from a unitary 2-design with dimension $\ell$. Next, we perform projective measurements in the computational basis $|\mathbf{s}_{AB}\rangle =|\mathbf{s}_A\rangle\otimes |\mathbf{s}_B\rangle =(\otimes_{i=1}^m|s_{A_i}\rangle) \otimes (\otimes_{j=1}^n|s_{B_j}\rangle )$, recording the outcomes $\mathbf{s}_{AB}$ and $\mathbf{t}_{AB}$ for the two states respectively. Repeating this measurement for the same $U_{AB}$ allows us to estimate: $\mathcal{P}(U_{X}, \mathbf{s}_{X}) = \text{Tr}[U_{X} \rho_X U_{X}^\dagger |\mathbf{s}_{X}\rangle \langle \mathbf{s}_{X}| ], \mathit{Q}(U_{X}, \mathbf{t}_{X}) = \text{Tr}[U_{X} \sigma_X U_{X}^\dagger |\mathbf{t}_{X}\rangle \langle \mathbf{t}_{X} | ],$ where $X\in \{A,B,AB\}$ and $\rho_{AB}=\rho,\sigma_{AB}=\sigma$. By repeating this procedure with varying $U_{AB}$, we can estimate the state overlaps $\text{Tr}[\rho_X \sigma_X]$ using second-order cross-correlations: $\text{Tr}[\rho_X \sigma_X]= d_X \sum_{\mathbf{s}_X,\mathbf{t}_X} (-\ell)^{-\mathcal{D}(\mathbf{s}_X,\mathbf{t}_X)} \mathbb{E}_{U_X} [\mathcal{P}(U_{X}, \mathbf{s}_{X}) \mathit{Q}(U_{X}, \mathbf{t}_{X})],$ where $d_X$ denotes the dimension of system $X$, $\mathcal{D}(\mathbf{s}_X, \mathbf{t}_X)$ is the Hamming distance between $\mathbf{s}_X$ and $\mathbf{t}_X$, and $\mathbb{E}_{U_X}$ indicates the ensemble average over random unitaries of the form $U_X$. As $\langle \rho_X, \sigma_X\rangle= \text{Tr}[\rho_X\sigma_X]$, we can estimate $\mathcal{S}(\rho,\sigma) = \max\{\frac{\langle \rho, \sigma\rangle}{\langle \rho_A, \sigma_A\rangle},\frac{\langle \rho, \sigma\rangle}{\langle \rho_B, \sigma_B\rangle}\}$. Then we get the estimation of $\mathrm{SN}(\rho),\mathrm{SN}(\sigma) \geq \lceil\mathcal{S}(\rho,\sigma) \rceil$ (see also in Eq. \ref{['eq:Schmidt_estimation']}).
• Figure 3: The figure presents two complementary results for the state defined in Eq. \ref{['eq:example2']}, which is represented by a point $(d,x)$ in the parameter space. Note that $\rho(x)$ is entangled for all $x>0$, and that larger values of $x$ correspond to a higher degree of entanglement in $\rho(x)$. (a) The $(r+1)$-UFS-IP states are indicated by the five colored regions ($r=1,2,3,4,5$). For each region, the lower boundary is determined by the saturation condition $\mathcal{S}(\rho(x_*),\sigma(y_{x_*})) = r$, above which $\mathcal{S}(\rho(x),\sigma(y_x)) > r$ holds and the states can therefore be detected by the $r$-IPC. The upper boundary of the corresponding region is given by the equality $\lambda(\rho(x)) = r/d$, below which the states necessarily belong to the $(r+1)$-UFS by Proposition 1. (b) The IPC exhibits superior detection performance compared to three experimentally accessible entanglement criteria, namely the $p_3$-PPT criterion, the FBC, and the purity criterion (PC). This advantage is illustrated by the four blue curves: states lying above a given curve are detected as entangled by the corresponding method, and a lower curve therefore indicates a stronger entanglement-detection capability.
• Figure 4: This figure shows that the boundary obtained in Proposition 1 is almost tight. Specifically, we plot the two boundaries corresponding to Eqs. \ref{['eq:r-FBC_I']} and \ref{['eq:r-FBC_R']} for $r=1,2,3,4$, ordered from bottom to top. The blue curves correspond to the bound given by Eq. \ref{['eq:r-FBC_R']}, namely $x+\frac{1-x}{d(d-1)}=\frac{r}{d},$ above which the states can be detected by the $r$-FBC via the particular witness $\mathcal{W}_{r\text{-}\mathrm{FBC}}(\Psi)$. The red curves correspond to the bound specified by Eq. \ref{['eq:r-FBC_I']}, i.e., $\Delta=\frac{r}{d},$ below which the states cannot be detected by the $r$-FBC.

Theorems & Definitions (5)

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