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arXiv:2206.11842 · Physical Review A

Constraints on Gaussian error channels and measurements for quantum communication

Alex Kwiatkowski, Ezad Shojaee, Sristy Agrawal, Akira Kyle, Curtis L. Rau, Scott Glancy, Emanuel H. Knill

Published Apr 10, 2023• 5 citations
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Abstract

Joint Gaussian measurements of two quantum systems are important for quantum communication between remote parties and are often used in continuous-variable teleportation or entanglement-swapping protocols. Many of the errors in real-world implementations can be modeled by independent Gaussian error channels acting prior to measurement. In this work we study independent single-mode Gaussian error channels on two modes $\mathsf{A}$ and $\mathsf{B}$ that take place prior to a joint Gaussian measurement. We determine the set of pairs of such channels that render all Gaussian measurements separable and therefore unsuitable for entanglement swapping or teleportation of arbitrary input states. For example, if the error channels are loss with parameters ${l}_{\mathsf{A}}$ and ${l}_{\mathsf{B}}$ followed by added noise with parameters ${n}_{\mathsf{A}}$ and ${n}_{\mathsf{B}}$, then all Gaussian measurements are separable if and only if ${l}_{\mathsf{A}}+{l}_{\mathsf{B}}+{n}_{\mathsf{A}}+{n}_{\mathsf{B}}\ensuremath{\ge}1$.

DOI
10.1103/physreva.107.042604

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We characterise Gaussian quantum channels that are Gaussian incompatibility breaking, that is, transform every set of Gaussian measurements into a set obtainable from a joint Gaussian observable via Gaussian postprocessing. Such channels represent local noise which renders measurements useless for Gaussian EPR-steering, providing the appropriate generalisation of entanglement breaking channels for this scenario. Understanding the structure of Gaussian incompatibility breaking channels contributes to the resource theory of noisy continuous variable quantum information protocols.
We consider an arbitrary continuous-variable three-party Gaussian quantum state which is used to perform quantum teleportation of a pure Gaussian state between two of the parties (Alice and Bob). In turn, the third party (Charlie) can condition the process by means of local operations and classical communication. We find the best measurement that Charlie can implement on his own mode that preserves the Gaussian character of the three-mode state and optimizes the teleportation fidelity between Alice and Bob.
Gaussian quantum information processing with continuous-variable (CV) quantum information carriers holds significant promise for applications in quantum communication and quantum internet. However, applying Gaussian state distillation and quantum error correction (QEC) faces limitations imposed by no-go results concerning local Gaussian unitary operations and classical communications. This paper introduces a Gaussian QEC protocol that relies solely on local Gaussian resources. A pivotal component of our approach is CV gate teleportation using entangled Gaussian states, which facilitates the implementation of the partial transpose operation on a quantum channel. Consequently, we can efficiently construct a two-mode noise-polarized channel from two noisy Gaussian channels. Furthermore, this QEC protocol naturally extends to a nonlocal Gaussian state distillation protocol.
Loss and noise limit the use of light to distribute entanglement, transmit quantum information, and establish secret keys. Gaussian states have a compact description in terms of means and covariance matrices, but their entanglement still requires global optimization. We develop a framework for computing Gaussian-restricted versions of the relative entropy of entanglement and Rains entanglement. An integral representation reveals a convex geometry under matrix geometric means and reduces Gaussian Rains entanglement to relative entropy minimized over Gaussian states with positive partial transpose. We obtain unique optimal reference states and additivity for independent Gaussian input states. Our convex-majorization algorithm has an explicit geometric convergence rate, uniform in the faithful regularization parameter and valid for pure input states. Controlled numerical integration and approximate convex subproblem solutions give a bit-complexity bound that is polynomial in accuracy bits for each fixed rational input covariance matrix, with explicit dependence on covariance size. These calculations bound extractable entanglement and, through the relative entropy of entanglement, extractable secret key. Teleportation simulation, energy allocation, and a convex optimization of passive-basis loss cores give bounds on quantum and private communication under an input-energy constraint, allowing arbitrary two-way classically assisted protocols. Two- and three-mode examples compare the bounds with achievable rates, including matching pure-loss benchmarks. Finally, we separate the Gaussian-restricted quantities from their unrestricted counterparts and give a photon-number truncation method with controlled error for computing the latter.
We present a full experimental characterization of continuous-variable quantum-communication channels established by shared entanglement together with local operations and classical communication. The resulting teleportation channel was fully characterized by measuring all elements of the covariance matrix of the shared two-mode squeezed Gaussian state. From the experimental data we determined the lower bound to the quantum channel capacity, the teleportation fidelity of coherent states, and the logarithmic negativity and purity of the shared state. Additionally, a positive secret key rate was obtained for two of the established channels.
Suppose we want to benchmark a quantum device held by a remote party, e.g., by testing its ability to carry out challenging quantum measurements outside of a free set of measurements M\mathcal{M}M. A very simple way to do so is to set up a binary-state discrimination task that cannot be solved efficiently by means of free measurements. If we can find pairs of orthogonal states that become arbitrarily indistinguishable under measurements in M\mathcal{M}M, in the sense that the error probability in discrimination approaches that of a random guess, we say that there are data hiding against M\mathcal{M}M. Here we investigate data hiding in the context of continuous-variable quantum systems. First, we look at the case where M\mathcal{M}M denotes the set of measurements implementable with local operations and classical communication. While previous studies have placed upper bounds on the maximum efficiency of data hiding in terms of the local dimension and are thus not applicable to continuous-variable systems, we establish more general bounds that rely solely on the local mean photon number of the states employed. Along the way, we perform a rigorous quantitative analysis of the error introduced by the nonideal Braunstein-Kimble quantum teleportation protocol, determining how much squeezing and local detection efficiency are needed in order to teleport an arbitrary multimode local state of known mean energy with a prescribed accuracy. Finally, following a seminal proposal by Sabapathy and Winter, we look at data hiding against Gaussian operations assisted by the feedforward of measurement outcomes, providing an example of a relatively simple scheme that works with a single mode only.