Optimal Continuous- to Discrete-Variable Bipartite Entanglement Conversion

Abstract

Discrete-variable (DV) entanglement is crucial for numerous quantum applications, yet its deterministic generation in many bosonic systems remains experimentally challenging. In contrast, continuous-variable (CV) entanglement can be produced efficiently. We propose two optimal schemes for converting CV bipartite entanglement into DV entanglement using only local operations and classical communication. The first scheme extracts maximally entangled qubit pairs at the theoretically maximal rate, while the second probabilistically produces a maximally entangled qudit pair with the highest average entanglement. In both schemes, we quantify the optimal performance and identify the measurement operators required for implementation. Notably, using only a sequence of binary measurements, our approach can succeed in a finite number of measurement rounds on average, even though the CV resource is infinite-dimensional. Our schemes improve the feasibility of implementing DV-based quantum technologies on bosonic platforms.

Figures (8)

• Figure 1: Illustration of CV-to-DV entanglement conversion by transforming a TMSV state into a maximally entangled qudit pair via LOCC. (a) The first scenario considers transforming a TMSV state into a maximally entangled qubit pair with the highest possible rate. (b) In the second scenario, a TMSV is transformed into a maximally entangled qudit pair with random dimension $d$. $\mathcal{P}_d$ is the probability of getting a qudit pair with dimension $d$. The scheme is optimal when the set of probabilities $\{\mathcal{P}_d\}$ gives the average entanglement is the highest.
• Figure 2: (a) The illustration of the probabilistic generation of a maximally entangled single-photon time-bin qubit pair from the SPDC photon source. (b) The conversion rates of our optimal scheme and several existing schemes. The vertical gray-dashed line indicates the threshold squeezing strength. The dashed box indicates the weak-squeezing regime. (c) The zoom-in of the weak-squeezing regime.
• Figure 3: The number of POVMs yielded by different entanglement transformation schemes against the boson number truncation $N$ of a TMSV. The presented data corresponds to $\lambda = 0.5$, but the behavior is typical for other values of $\lambda$. The Nielsen's scheme, Birkhoff-von Neumann's algorithm and the method of areas require $2^{N-1}$, $(N-1)^2/2+2$ and $N$ POVMs, respectively.
• Figure 4: Illustration of Hardy's method of areas. (a) The chart represents the initial TMSV state. Every column is painted in a different colour, and the area covered by each column is given by the squared Schmidt coefficients $\{\alpha_n\}$. (b) The transformed chart in the deterministic regime. The two columns have the same height $\beta_0=\beta_1=1/2$, so the chart coincides that represents the target state. The boundaries of adjacent colours are labeled by $\{y_l\}$. The probability of obtaining $\ket{\text{Bell}_{nm}}$, $P_{nm}$, is determined by the area covered by the segment of columns between two adjacent boundaries, and the corresponding Fock states involved, $\{n,m\}$, are determined by the colours of the segment. (c) The transformed chart in the probabilistic regime. The heights of the first and second columns are respectively $\alpha_0$ and $\sum_{n=1}^\infty\alpha_n$. Since $\alpha_0 > 1/2$, there is a segment consists of only the first but not the second column. This segment, whose height and covered area are both $P_{\text{fail}}$, corresponds to the failed conversion.
• Figure 5: Illustration of the method of areas for the optimal random qudit scheme. The segment boundaries (dashed) are located at the heights of the columns. The probability of obtaining a qudit pair with dimension $d$ is given by the total area of columns between boundaries at $\alpha_{d-1}$ and $\alpha_d$, and the corresponding Fock states involved are determined by the columns included in that segment.