Separability-entanglement classifier via machine learning

The problem of determining whether a given quantum state is entangled lies at the heart of quantum information processing. Despite the many methods—such as the positive partial transpose criterion and the $k$-symmetric extendibility criterion—to tackle this problem, none of them enables a general, practical solution due to the problem's NP-hard complexity. Explicitly, separable states form a high-dimensional convex set of vastly complicated structures. In this work, we build a different separability-entanglement classifier underpinned by machine learning techniques. We use standard tools from machine learning to learn the entanglement feature of arbitrary given quantum states. We perform substantial numerical tests on two-qubit and two-qutrit systems, and the results indicate that our method can outperform the existing methods in generic cases in terms of both speed and accuracy. This opens up avenues to explore quantum entanglement via the machine learning approach.

Figure 1

(a) In the high-dimensional space, the set of all states is convex, while separable states form a convex subset. Many criteria, such as the linear (green straight line) or nonlinear (green curve) entanglement witnesses [7, 15] and PPT tests, are based on this geometric structure and detect a limited set of entangled states. (b) Classifier built from supervised learning has a decision boundary of highly complex shape.

Figure 2

(a) Illustration of the iterative algorithm for detecting the separability. Initially, we build a CHA $\mathcal{C}$. For a state $\rho$ with feature vector $\mathit{p}$, we find the maximum $\alpha$ such that $\alpha \mathit{p}$ is still in $\mathcal{C}$. If $\alpha \ge 1$, then $\rho$ is surely separable. Otherwise, suppose $\alpha \mathit{p}$ lies on a hyperplane $\mathcal{P}$, such that $\mathcal{P}\cap \mathcal{C}$ is the boundary of $\mathcal{C}$. Let ${\mathit{c}}_{i_1},...,{\mathit{c}}_{i_D}$ be extreme points of $\mathcal{C}$ that are in $\mathcal{P}\cap \mathcal{C}$. We enlarge $\mathcal{C}$ by sampling separable pure states that are near ${\mathit{c}}_{i_1},...,{\mathit{c}}_{i_D}$, then repeat the above procedure for many times, until $\alpha \ge 1$ or $\alpha$ converges. (b) Illustration of the learning algorithm: ensemble methods are models composed of multiple weaker models that are independently trained and whose predictions are combined in some way to make the overall prediction. For example, each time we draw a subset of training data (marked as yellow dots in the figure), we train a model based on this subset, which is a weak model on the whole training set. We repeat the process many times and obtain a batch of weak models, and combine them as a committee.

Figure 3

Results of the CHA approach and BCHA classifier. More details of the BCHA classifier can be found in Appendix pp5. (a) Test results of BCHA when $m={10}^3$ for the two-qubit case. The random density matrices in the test dataset are projected on a plane by projection $\pi :(\mathit{x},\alpha )\to (x_1,{\alpha }^{-1})$. Here $H_1=|0\rangle \langle 0|\otimes {\sigma }_z/\sqrt{2}$. The red points are states predicted as separable, while the blue points are states predicted as entangled. Some states with $\alpha <1$ are predicted as separable, which is different from CHA. (b) Test results of BCHA when $m=2\times {10}^4$ for the two-qutrit case. Here $P_1=\left(\right|00\rangle \langle 00|-|01\rangle \langle 01\left|\right)\sqrt{3}/2$. All the states in the test dataset are PPT states. The red points are states predicted as separable, while the blue points are states which are predicted as bound entangled. (c) Comparison between CHA and BCHA for two-qubit states. For the same $m$, BCHA clearly suppresses the error rate significantly. And to achieve the same error rate, BCHA requires much less running time (which mainly depends on the value of $m$). For instance, to decrease the error rate to less than $3\%$, CHA requires a convex hull with $m\approx 7\times {10}^3$, while BCHA only requires a convex hull with $m\approx {10}^3$, which considerably reduces the computational cost. (d) Comparison between CHA and BCHA for two-qutrit PPT states. From the data, similar to the two-qubit case, BCHA also outperforms CHA in terms of speed and accuracy.

Figure 4

Projections of the boundaries of the separable states and the $k$-symmetric extendible states ($k=2,...,12$) on the plane spanned by the operators $H_1=|0\rangle \langle 0|\otimes {\sigma }_z/\sqrt{2}$ and $H_2=({\sigma }_y\otimes {\sigma }_x-{\sigma }_x\otimes {\sigma }_y)/2$. Here ${\sigma }_x,{\sigma }_y,{\sigma }_z$ are the three Pauli operators. As we can see, there is still a large gap between ${\mathrm{\Theta }}_{12}$ and separable set.