Distilling Gaussian States with Gaussian Operations is Impossible J. Eisert, S. Scheel, and M. B. Plenio QOLS, Blackett Laboratory, Imperial College of Science, Technology and Medicine, London, SW7 2BW, United Kingdom (Received 15 April 2002; published 4 September 2002) We show that no distillation protocol for Gaussian quantum states exists that relies on (i) arbitrary local unitary operations that preserve the Gaussian character of the state and (ii) homodyne detection together with classical communication and postprocessing by means of local Gaussian unitary operations on two symmetric identically prepared copies. This is in contrast to the finite-dimensional case, where entanglement can be distilled in an iterative protocol using two copies at a time. The ramifications for the distribution of Gaussian states over large distances will be outlined. We also comment on the generality of the approach and sketch the most general form of a Gaussian local operation with classical communication in a bipartite setting. DOI: 10.1103/PhysRevLett.89.137903 PACS numbers: 03.67.–a, 03.65.Ud, 42.50.–p In most practical implementations of information proc- essing devices, sophisticated methods are necessary in order to preserve the coherence of the involved quantum states. Even the mere preparation of an entangled state of spatially distributed quantum systems requires such tech- niques: once prepared locally and then distributed, an entangled state will to some extent deteriorate from a highly entangled state to a less correlated state through the process of decoherence. This process can never be avoided entirely. However, one may prepare and distribute several identical entangled states and then apply appro- priate partly measuring local quantum operations and classical communication to obtain states that are similar to the highly entangled original state. This is possible only at the expense that one has fewer identically pre- pared systems or copies at hand, but this is a small price to pay. Appropriately indeed, this process has been given the name distillation [1], as fewer more highly entangled states are ‘‘distilled’’ from a supply of many less en- tangled states. It has been realized that, remarkably, for two-level systems such a distillation procedure may be performed on only two copies at a time, and it requires only two steps: (i) a local unitary operation and (ii) a local measurement, together with the classical commu- nication about the measurement outcome. Based on the measurement outcome, further local unitary operations are then implemented. Such distillation protocols may also be of crucial im- portance in the infinite-dimensional setting. Quantum information science over continuous variables has seen enormous progress recently, both in theory and experi- ment, mostly involving Gaussian states of field modes in a quantum optical setting [2 – 4]. Quite naturally, one should expect that a similar distillation procedure also works for Gaussian states in the infinite-dimensional case, also under the preservation of the Gaussian charac- ter of the state. If one transmits two pure two-mode squeezed Gaussian states through lossy optical systems such as fibers, the corresponding modes being from now on labeled A1, A2, B1, and B2, one obtains two identical copies of less entangled symmetric states [5]. A feasible distillation protocol preserving the Gaussian character may consist of the subsequent steps (see Fig. 1): (i) Application of any local Gaussian unitary opera- tion. That is, one may implement any unitary operations U A and U B on both A1 and A2 on one hand and B1 and B2 on the other hand, corresponding to symplectic transfor- mations [6] S A ;S B 2 Sp4; R [7]. This set includes all two-mode and one-mode squeezings, mixing at beam splitters, and phase shifts. To specify these operations, 20 real parameters are necessary. Note that we do not require both parties to realize the same transformation. (ii) A homodyne measurement on the modes A2 and B2. The parties communicate classically about the out- come of the measurement and may postprocess the states of modes A1 and B1 with unitary Gaussian operations. The main result of this Letter is that very much as a surprise, none of these protocols amounts to a distillation protocol. No matter how ingeniously the local unitary operation is chosen, the degree of entanglement cannot be increased. The optimal procedure is simply to do nothing at all [8]. The degree of entanglement will be measured in terms of the log-negativity, which is defined as E N  logk T A k for a state , where kk denotes the trace norm, U U B A B1 A1 B2 A2 FIG. 1 (color online). The class of considered feasible distil- lation protocols. VOLUME 89, NUMBER 13 PHYSICAL REVIEW LETTERS 23 SEPTEMBER 2002 137903-1 0031-9007= 02=89(13)=137903(4)$20.00  2002 The American Physical Society 137903-1