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arXiv:quant-ph/0204069 · Physical Review Letters

Gaussian Transformations and Distillation of Entangled Gaussian States

Jaromı́r Fiurášek

Published Sep 4, 2002• 411 citations
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Abstract

We prove that it is impossible to distill more entanglement from a single copy of a two-mode bipartite entangled Gaussian state via local Gaussian operations and classical communication. More generally, we show that any hypothetical distillation protocol for Gaussian states involving only Gaussian operations would be a deterministic protocol. Finally, we argue that the protocol considered by Eisert et al. [preceding Letter, Phys. Rev. Lett. 89, 137903 ()]] is the optimum Gaussian distillation protocol for two copies of entangled Gaussian states.

DOI
10.1103/physrevlett.89.137904

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We prove that it is impossible to distill more entanglement from a single copy of a two-mode bipartite entangled Gaussian state via local Gaussian operations and classical communication. More generally, we show that any hypothetical distillation protocol for Gaussian states involving only Gaussian operations would be a deterministic protocol. Finally, we argue that the protocol considered by Eisert et al. [preceding Letter, Phys. Rev. Lett. 89, 137903 ()]] is the optimum Gaussian distillation protocol for two copies of entangled Gaussian states.
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.
We will 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 analysis shows that unlike the finite-dimensional case, where entanglement can be distilled in an iterative protocol using two copies at a time, there is no such procedure in the case of continuous variables for Gaussian initial states and the above Gaussian operations. The ramifications for the distribution of Gaussian states over large distances will be outlined. We will also comment on the generality of the approach and sketch the most general form of a Gaussian local operation with classical communication in a bi-partite setting.
We propose an entanglement distillation and purification scheme for symmetric two-mode entangled Gaussian states that allows to asymptotically extract a pure entangled Gaussian state from any input entangled symmetric Gaussian state. The proposed scheme is a modified and extended version of the entanglement distillation protocol originally developed by Browne et al. [Phys. Rev. A 67, 062320 (2003)]. A key feature of the present protocol is that it utilizes a two-copy degaussification procedure that involves a Mach-Zehnder interferometer with single-mode non-Gaussian filters inserted in its two arms. The required non-Gaussian filtering operations can be implemented by coherently combining two sequences of single-photon addition and subtraction operations.
A mixed quantum state $\ensuremath{\rho}$ shared between two parties is said to be distillable if, by means of a protocol involving only local quantum operations and classical communication, the two parties can transform some number of copies of $\ensuremath{\rho}$ into a single shared pair of qubits having high fidelity with the maximally entangled state $|{\ensuremath{\varphi}}^{+}\ensuremath{\rangle}=(|00\ensuremath{\rangle}+|11\ensuremath{\rangle})/\sqrt{2}$. In this Letter it is proved that there exist states that are distillable, but for which an arbitrarily large number of copies is required before any distillation procedure can produce a shared pair of qubits with even a small amount of entanglement. Specifically, for every positive integer nnn there exists a state $\ensuremath{\rho}$ that is distillable, but, given nnn or fewer copies of $\ensuremath{\rho}$, every distillation procedure outputting a single shared pair of qubits outputs those qubits in a separable (i.e., unentangled) state.
We recently proposed an entanglement distillation protocol based on local operations and classical communication that probabilistically and universally distills a maximally entangled Bell pair from multiple copies of an unknown two-qubit state [Phys. Rev. Lett. 135, 260202 (2025)]. Here, we investigate the robustness of this protocol against imperfections in its implementation. We consider several incoherent and coherent gate-error models, as well as measurement readout errors, and determine the regimes in which distillation enhances the fidelity with the target Bell state. We find that the protocol remains effective over substantial error ranges, although its robustness depends strongly on the type of error. We further analyze asymmetric errors between the two parties and identify conditions under which complementary postselection outcomes yield identical distillation dynamics, allowing the success probability to be enhanced.