Maximally entangled states in discrete and Gaussian regimes

Abstract

We study a relation between discrete-variable quantum states and continuous-variable (especially, restricted on Gaussian) ones. In the previous work, we have investigated an information-theoretic correspondence between the Gaussian maximally mixed states and their purifications as Gaussian maximally entangled states in Jeong and Lim (Phys Lett A 380:3607, 2016). We here compare the purified continuous-variable maximally entangled state with a two-mode squeezed vacuum state, which is a conventional entangled state in Gaussian regime, by the explicit calculation of quantum fidelities between those states and an $N\times N$-dimensional maximally entangled state in the finite Hilbert space. Consequently, we naturally conclude that the purified maximally entangled state is more suitable to the Gaussian maximally entangled state than the two-mode squeezed vacuum state, in a sense that it might be useful for continuous-variable quantum information tasks in which entangled states are needed.

Appendix A: Correspondence framework: DV versus CV systems

Since a CV quantum state has the infinite degrees of freedom in general, it is not an overestimation that a generic CV quantum state possesses potentially higher resources than a DV state for performing a quantum information processing. In this sense, a DV state with arbitrary dimension can be possibly embedded on a CV state. Specifically, one of the methods is, in advance, finding a mapping between DV states with different dimensions, and taking the limit as the dimension goes to infinity for one of them. For example, we have generators for SU(n) algebra from the transition operators for an n-dimensional quantum state [38]. Then, we consider another N-dimensional quantum state with $N\ge n$ and assume that N can be divided by n for simplicity. By using an appropriate coarse-graining method, we have N / n of SU(n) algebras and sum those up to finally getting the generators of SU(n) algebra fully spanned in the N-dimensional space [14]. Now a problem, however, arises when we take the limit $N \rightarrow \infty$ for getting an indeed CV state. In this limit, roughly speaking, a non-MES can be mapped onto the MES of finite dimensions. In other words, there can be several ways of mapping a quantum state in finite dimensions to the CV state.

It is worth noting that there is another correspondence between CV and DV states, especially MES, under the nonlinear quantum optical settings [39,40,41]. In fact, a coherent state can be written as a superposition of pseudo-number states, which are d-modulo photon number states, so can be given by $\left| \right. \alpha \left. \right\rangle ={1 \over \sqrt{d}} \sum _{k=0}^{d-1}\left| \right. k_d\left. \right\rangle$, where $\left| \right. k_d\left. \right\rangle$ is a state having ‘$k \mod d$’ number of photons [41]. After a cross-Kerr interaction represented by $e^{{{2\pi i}\over {d}}\hat{n}_1\hat{n}_2}$ on the two-mode initial state $\left| \right. \alpha \left. \right\rangle _1\left| \right. \alpha \left. \right\rangle _2$, an MES can be produced, if $|\alpha | \gg d/2\pi$ holds, as follows:

$$\begin{aligned} \left| \right. \alpha \left. \right\rangle _1\otimes \left| \right. \alpha \left. \right\rangle _2\overset{\underset{{\text {Cross-Kerr}}}{}}{\longmapsto }{1\over \sqrt{d}}\sum _{k=0}^{d-1}\left| \right. k_d\left. \right\rangle _1\otimes |\tilde{k}_d\rangle _2, \end{aligned}$$

(14)

where $|\tilde{k}_d\rangle$ is a pseudo-phase state which is equivalent to a ${2\pi k}\over {d}$-phase-shifted coherent state $|e^{{2\pi k i}\over {d}} \alpha \rangle$. The simplest example is, when $d=2$, the entangled coherent state $\left| \right. \text {ECS}^{\pm }\left. \right\rangle =\left| \right. \alpha \left. \right\rangle \left| \right. \alpha \left. \right\rangle \pm \left| \right. -\alpha \left. \right\rangle \left| \right. -\alpha \left. \right\rangle$ [42, 43]. Since $|\left<{\alpha }|{-\alpha }\right>|=0$ when $\alpha$ is large, we obtain approximately a $2 \times 2$-MES. In fact, these are not the usual Bell states even $\alpha$ is large enough, because when $d=2$, we can calculate entanglement entropy using an orthogonal basis $\left| \right. \pm \left. \right\rangle =\left| \right. \alpha \left. \right\rangle \pm \left| \right. -\alpha \left. \right\rangle$. The result is that the entanglement entropy has dependence on $\alpha$ for $\left| \right. \text {ECS}^+\left. \right\rangle$, but no dependence on $\alpha$ in case of $\left| \right. \text {ECS}^-\left. \right\rangle$. So we call these quasi-Bell states [44]. This can be extended to any $d \times d$-MES, but the above scheme is valid only when $|\alpha |$ is much larger than $d/2\pi$ so thus coherent states with large amplitude are needed as d increases.

From the previous instances, we figure out the fact that there has not yet been any faithful or unambiguous correspondence between CV and DV quantum states. Now we move our focus onto the Gaussian states. A Gaussian state is defined by a quantum state having a Gaussian distribution of the Wigner function or characteristic function. Because we can always displace the average of a Gaussian distribution by a local operation, the only valuable information of the Gaussian states is lying on the second moments of canonical variables, which are encoded in the covariance matrix, positive and real symmetric $2n\times 2n$ matrix for an n-mode Gaussian state. For the dynamics, we consider the group of n-mode Gaussian unitaries, which is a projective representation of the inhomogeneous symplectic group ISp(2n,$\mathbb {R}$ [45]. Consequently, in the Gaussian quantum system, even though governing Hilbert space is still infinite dimensional, it is enough to handle matrices with finite degrees of freedom.

However, there still exists a subtle problem of dimension-mode matching, despite concerning finite number of parameters. For the easiest example, let us consider a DV quantum state of dimension 2 (i.e., a qubit) and a single-mode Gaussian state. In this case, the number of free parameters for both cases is 3 under the consideration of the normalization. Next, non-trivial example is a quantum state of dimension 4 (a two-qubit state) and a two-mode Gaussian state. Here, the number of free parameters of the two-qubit state is 15, but that of the two-mode Gaussian state is only 10 [46]. Accordingly, we cannot make a simple correspondence between a d-dimensional DV state and an n-mode Gaussian state in general although there have been many analogies between the n-qubit system and the n-mode Gaussian state.