Robust entanglement detection in arbitrary two-mode Gaussian state: a Stokes-like operator-based approach

Abstract

Detection of entanglement in quantum states is one of the most important problems in quantum information processing. However, it is one of the most challenging tasks to find a universal scheme which is also desired to be optimal to detect entanglement for all states of a specific class--as always preferred by experimentalists. Although, the topic is well studied at least in case of lower dimensional compound systems, e.g., two-qubit systems, but in the case of continuous variable systems, this remains as an open problem. Even in the case of two-mode Gaussian states, the problem is not fully solved. In our work, we have tried to address this issue. At first, a limited number of Hermitian operators is given to test the necessary and sufficient criterion on the covariance matrix of separable two-mode Gaussian states. Thereafter, we present an interferometric scheme to test the same separability criterion in which the measurements are being done via Stokes-like operators. In such case, we consider only single-copy measurements on a two-mode Gaussian state at a time and the scheme amounts to the full state tomography. We further analyze the robustness of the proposed detection method against experimentally relevant imperfections and demonstrate that the separability test remains reliable under moderate levels of detection inefficiency. Although this latter approach is a linear optics-based one, nevertheless it is not an economic scheme. Resource-wise a more economical scheme than the full state tomography is obtained if we consider measurements on two copies of the state at a time. However, optimality of the scheme is not yet known.

Figures (3)

• Figure 1: a) A schematic diagram of the measurement setup to estimate all elements of a covariance matrix associated to a single mode of a Gaussian state. k-th mode of an unknown Gaussian state and a reference state which are represented by annihilation operators $\hat{a}_k,$ and $\hat{a}_r,$ respectively, interfere at a 50-50 beam-splitter ($\text{B.S}).$ A phase shifter is introduced at the reference mode. The outputs of the beam-splitter are represented by annihilation operators $\hat{\tilde{a}}_1,$ and $\hat{\tilde{a}}_2.$ The detectors generate photocurrents $\tilde{I}_1$ and $\tilde{I}_2$ which are proportional to the intensities at the output modes of B.S. b) A schematic diagram of the measurement setup to obtain all elements of a covariance matrix corresponding to a two-mode Gaussian state. Two modes of $\rho_{12}$ represented by annihilation operators $\hat{a}_1,$ and $\hat{a}_2,$ interfere at a 50-50 beam-splitter 1 ($\text{B.S}_1).$ The outputs of $\text{B.S}_1$ are made to interfere at two separate 50-50 beam-slitters, e.g., beam-splitter 2 ($\text{B.S}_2$) and beam-splitter 3 ($\text{B.S}_3$) along with the two reference states represented by annihilation operators $\hat{a}_c,$ and $\hat{a}_d.$ Phase shifters are introduced at the reference modes. The outputs of the measurement setup are represented by annihilation operators $\hat{a}_3,$$\hat{a}_4,$$\hat{a}_5,$ and $\hat{a}_6.$ Two pairs of photocurrents $(I_3, I_4),$ and $(I_5, I_6)$ are proportional to the intensities of the output modes of $\text{B.S}_2$ and $\text{B.S}_3,$ respectively.
• Figure 2:
• Figure 3: A schematic diagram of the measurement setup to estimate all elements of the matrix $C$ associated to a two-mode Gaussian state. Two copies of the state are shared between Alice and Bob. $(\hat{a}^{(1)}_{1}, \hat{a}^{(2)}_{1})$ and $(\hat{a}^{(1)}_{2}, \hat{a}^{(2)}_{2})$ are two pairs of annihilation operators associated to the two pairs of input modes of the OPAs on Alice's and Bob's side, respectively. Note that, the correlated modes are $\hat{a}^{(j)}_{1},$ and $\hat{a}^{(j)}_{2}.$$\text{Pump}_1,$ and $\text{Pump}_2$ are sources of pump beams incident on the OPA on Alice's and Bob's side, respectively. The output modes of the OPA on Alice's and Bob's side are represented by annihilation operators $\hat{A}_k,$ and $\hat{B}_k,$ respectively. Here $j\in\{1, 2\},$ and $k\in\{3, 4\}.$