PHYSICAL REVIEW A 99, 042304 (2019) Entanglement properties of a measurement-based entanglement distillation experiment Hao Jeng,1 Spyros Tserkis,2 Jing Yan Haw,1 Helen M. Chrzanowski,1,3 Jiri Janousek,1 Timothy C. Ralph,2 Ping Koy Lam,1 and Syed M. Assad1,* 1Centre for Quantum Computation and Communication Technology, Research School of Physics and Engineering, The Australian National University, Canberra, Australian Capital Territory 2601, Australia 2Centre for Quantum Computation and Communication Technology, School of Mathematics and Physics, University of Queensland, St. Lucia, Queensland 4072, Australia 3Institute of Physics, Humboldt-Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany (Received 15 November 2018; published 1 April 2019) Measures of entanglement can be employed for the analysis of numerous quantum information protocols. Due to computational convenience, logarithmic negativity is often the choice in the case of continuous-variable sys- tems. In this work, we analyze a continuous-variable measurement-based entanglement distillation experiment using a collection of entanglement measures. This includes logarithmic negativity, entanglement of formation, distillable entanglement, relative entropy of entanglement, and squashed entanglement. By considering the distilled entanglement as a function of the success probability of the distillation protocol, we show that the logarithmic negativity surpasses the bound on deterministic entanglement distribution at a relatively large probability of success. This is in contrast to the other measures which would only be able to do so at much lower probabilities, hence demonstrating that logarithmic negativity alone is inadequate for assessing the performance of the distillation protocol. In addition to this result, we also observed an increase in the distillable entanglement by making use of upper and lower bounds to estimate this quantity. We thus demonstrate the utility of these theoretical tools in an experimental setting. DOI: 10.1103/PhysRevA.99.042304 I. INTRODUCTION notably, the logarithmic negativity [14] as one can calculate it quite straightforwardly. On the one hand, quantum entanglement is a useful non- In this work, we analyze a continuous-variable classical resource. It can be used for the construction of measurement-based entanglement distillation experiment quantum gates [1] or for the distribution of cryptographic keys [5] using a collection of measures. We present two main in a secure manner [2]. On the other hand, implementations results. First, we show that the logarithmic negativity is of these tasks are usually limited in performance due to ex- distinct from the other measures; it crosses the “deterministic perimental imperfections. Utilizing a variety of methods such bound” before the other measures do, at a probability as photon subtraction [3,4] and noiseless linear amplification of success (of the distillation protocol) that is orders of [5,6], entanglement distillation protocols seek a potential res- magnitude greater. The deterministic bound is the maximum olution to this problem by concentrating weakly entangled entanglement that can be deterministically distributed across a states into subsets that are more entangled. given quantum channel (usually imperfect), assuming that one Here we address the problem of quantifying entanglement had an Einstein-Podolsky-Rosen (EPR) resource state with distillation; this will, in general, depend on the kind of system infinite squeezing. For instance, when the entanglement of that one is working with. In the case of discrete variables, the formation crosses this bound, we can take it to indicate a form fidelity with respect to some maximally entangled state [7] of error correction [15], thus giving an example of how the and nonlocality based on the Bell inequalities [8] are both use- logarithmic negativity can fail to capture important properties ful measures for observing quantum entanglement. However, of distillation protocols. Our results can be regarded as an these methods are not particularly suitable in the case of con- experimental demonstration of such an example. tinuous variables. Maximally entangled continuous-variable Our second result is the certification of an increase in states are unphysical, and a theorem from Bell precludes the distillable entanglement. Currently, there is no known the demonstration of nonlocality using Gaussian states and way to evaluate the distillable entanglement directly, which Gaussian measurements (the standard tools for continuous means that one is only able to look at it through the use variable experiments) [9], unless one introduces additional of upper and lower bounds [16,17]. The upper bound puts a assumptions on one’s system [10,11]. Thus far, the analy- limit on how much distillable entanglement we had prior to ses of continuous-variable entanglement distillation have in- distillation, while the lower bound guarantees at least how stead centered around inseparability criteria [12,13] and, most much we have after performing distillation. By observing a sufficient increase in the lower bound, we could certify that the distillable entanglement has indeed increased. We *[email protected] remark that the minimization of optical loss and the choice 2469-9926/2019/99(4)/042304(11) 042304-1 ©2019 American Physical Society HAO JENG et al. PHYSICAL REVIEW A 99, 042304 (2019) of a sharp upper bound turned out to be important factors in Any given covariance matrix σ can be put into the follow- order to observe the increase in the distillable entanglement. ing standard form: In particular, there are many possible choices for the upper ⎡ ⎤ bound, but the relative entropy of entanglement was found to m 0 c1 0 ⎢ 0 m 0 c ⎥ be the only one that was sufficiently stringent for this task. σ = ⎣ 2⎦, (2) We have organized this paper as follows. In Sec. II,we c1 0 n 0 provide some background in Gaussian quantum optics and 0 c2 0 n establish the notations and conventions that will be used and this can be done using only local Gaussian unitaries [13], throughout this paper. In Sec. III, definitions of the various which does not change the entanglement of the state. entanglement measures are provided, discussing their basic One can always diagonalize the covariance matrix using properties with an emphasis on operational interpretations. In only symplectic matrices, and the corresponding eigenval- Sec. IV, the final section, we briefly describe the experiment ues are called symplectic eigenvalues [20]. Of particular setup and present a discussion of the distillation results based importance are the symplectic eigenvalues of the partially on the measures. transposed state. For two-mode Gaussian states, the partial transpose flips the sign of the phase quadrature, equivalent to II. PRELIMINARIES flipping the sign of the c2 entry in the standard form of the covariance matrix [Eq. (2)]. The symplectic eigenvaluesν ˜± of Gaussian states can be characterized by the first and second the partially transposed state are given by moments of the quadrature operators [18]—also known re- spectively as the mean fields and the covariance matrix. Since ˜ ± ˜ 2 − σ 2 4 det measures of entanglement depend only on the covariance ν˜± = , (3) matrix, we will assume vanishing mean fields without the 2 loss of generality. For two-mode Gaussian states, the covari- where ˜ = det M + det N − 2 det C. ance matrix can be written in block form (using the xpxp For convenience, covariance matrices of the form notation [19]): ⎡ ⎤ m 0 c 0 σ = MC, ⎢0 m 0 −c⎥ T σ = ⎣ ⎦ (4) C N c 0 m 0 0 −c 0 m where M, N, and C are real 2 × 2 matrices. In general, the entries of the covariance matrix depend on the value of will be called symmetric, while those of the form h¯; in this paper, we will normalize to the variance of the ⎡ ⎤ vacuum field, which amounts to settingh ¯ = 2. In order for m 0 c 0 the covariance matrix to represent a physical state, it is also ⎢0 m 0 −c⎥ σ = ⎣ ⎦ (5) required to satisfy the Heisenberg uncertainty principle. c 0 n 0 The density matrix of closed quantum systems evolve 0 −c 0 n under unitaries:ρ ˆ → Uˆ ρˆUˆ †. In general, representations of these unitaries in the Fock basis require infinite-dimensional will be called quadrature symmetric. These states form strict matrices; if we restrict ourselves to Gaussian operations, subsets of general two-mode Gaussian states [Eq. (2)] by then this can be simplified to the evolution of covariance imposing different kinds of symmetries. matrices: σ → Sσ ST . For two-mode states, each S is a 4 × 4 square matrix and is symplectic with respect to the following III. ENTANGLEMENT MEASURES symplectic form: ⎡ ⎤ We present some background on the theory of entangle- 0100 ment measures, including definitions, properties, and formu- ⎢− ⎥ = ⎣ 10 0 0⎦. las that will be useful. We note that the problem of computing 0001 entanglement measures is NP-complete for many measures 00−10 [21], with analytical expressions available only in restricted cases that will generally possess some degree of symmetry. It satisfies the equation SST = . Every Gaussian unitary In the special case of the entanglement of formation, such a Uˆ is associated with a symplectic matrix. We will find the restriction will give rise to a simple operational interpretation following unitary useful: in terms of quantum squeezing and thus a connection to the logarithmic negativity which we briefly discuss. r(ˆabˆ−aˆ†bˆ† )/2 Sˆ2(r) = e , (1) which is known as the two-mode-squeezing operator. As A. Entanglement entropy ˆ usual,a ˆ and b denote the annihilation operators of the The entanglement entropy EV uniquely determines entan- two optical modes. The two-mode-squeezing operator is glement on the set of pure states. Given the von Neumann parametrized by the squeezing parameter r, with r = 0 cor- entropy S, with responding to no squeezing and r →∞to the limit of infinite squeezing. S(ˆρ) =−tr(ρ ˆ lnρ ˆ), (6) 042304-2 ENTANGLEMENT PROPERTIES OF A … PHYSICAL REVIEW A 99, 042304 (2019) wardly.