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arXiv:1811.04956 · Journal of Mathematical Physics

Recovery map for fermionic Gaussian channels

Brian G. Swingle, Yixu Wang

Published Jul 1, 2019• 11 citations
arXivHTMLPDF

Abstract

A recovery map effectively cancels the action of a quantum operation to a partial or full extent. We study the Petz recovery map in the case where the quantum channel and input states are fermionic and Gaussian. Gaussian states are convenient because they are totally determined by their covariance matrix and because they form a closed set under so-called Gaussian channels. Using a Grassmann representation of fermionic Gaussian maps, we show that the Petz recovery map is also Gaussian and determine it explicitly in terms of the covariance matrix of the reference state and the data of the channel. As a by-product, we obtain a formula for the fidelity between two fermionic Gaussian states. We also discuss subtleties arising from the singularities of the involved matrices.

DOI
10.1063/1.5093326

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A recovery map effectively cancels the action of a quantum operation to a partial or full extent. We study the Petz recovery map in the case where the quantum channel and input states are fermionic and Gaussian. Gaussian states are convenient because they are totally determined by their covariance matrix and because they form a closed set under so-called Gaussian channels. Using a Grassmann representation of fermionic Gaussian maps, we show that the Petz recovery map is also Gaussian and determine it explicitly in terms of the covariance matrix of the reference state and the data of the channel. As a by-product, we obtain a formula for the fidelity between two fermionic Gaussian states. We also discuss subtleties arising from the singularities of the involved matrices.
Recently, there has been focus on determining the conditions under which the data processing inequality for quantum relative entropy is satisfied with approximate equality. The solution of the exact equality case is due to Petz, who showed that the quantum relative entropy between two quantum states stays the same after the action of a quantum channel if and only if there is a reversal channel that recovers the original states after the channel acts. Furthermore, this reversal channel can be constructed explicitly and is now called the Petz recovery map. Recent developments have shown that a variation of the Petz recovery map works well for recovery in the case of approximate equality of the data processing inequality. Our main contribution here is a proof that bosonic Gaussian states and channels possess a particular closure property, namely, that the Petz recovery map associated to a bosonic Gaussian state \sigma and a bosonic Gaussian channel N is itself a bosonic Gaussian channel. We furthermore give an explicit construction of the Petz recovery map in this case, in terms of the mean vector and covariance matrix of the state \sigma and the Gaussian specification of the channel N.
Optical systems are a main platform for quantum information processing. A main challenge is information loss due to scattering in unmonitored modes. These losses are modeled as state-independent beam-splitter interactions, with a thermal state (for all practical purposes, the vacuum) in the second input port. The perfect correction of these Gaussian lossy channels with Gaussian operations alone is known to be impossible. In this work, we investigate the Petz recovery map as an approximate recovery. For single mode losses and Gaussian reference states, the Petz map is found to use either a beam-splitter or a state-independent amplifier, depending on the parameters. Then we study the recovery performance on several examples, showing that it is near-optimal among the considered class of protocols. We also obtain more specific comparisons: Petz is always better than just re-preparing the reference state; but it is worse than doing nothing if the reference state is far from the true state. Finally, we extend our study to losses on two modes, and compare the global Petz map to the local implementation on each mode separately.
Abstract Recently, there has been focus on determining the conditions under which the data processing inequality for quantum relative entropy is satisfied with approximate equality. The solution of the exact equality case is due to Petz, who showed that the quantum relative entropy between two quantum states stays the same after the action of a quantum channel if and only if there is a reversal channel that recovers the original states after the channel acts. Furthermore, this reversal channel can be constructed explicitly and is now called the Petz recovery map . Recent developments have shown that a variation of the Petz recovery map works well for recovery in the case of approximate equality of the data processing inequality. Our main contribution here is a proof that bosonic Gaussian states and channels possess a particular closure property, namely, that the Petz recovery map associated to a bosonic Gaussian state σ and a bosonic Gaussian channel N is itself a bosonic Gaussian channel. We furthermore give an explicit construction of the Petz recovery map in this case, in terms of the mean vector and covariance matrix of the state σ and the Gaussian specification of the channel N .
The reconstruction of the state of a multipartite quantum mechanical system represents a fundamental task in quantum information science. At its most basic, it concerns a state of a bipartite quantum system whose subsystems are subjected to local operations. We compare two different methods for obtaining the original state from the state resulting from the action of these operations. The first method involves quantum operations called Petz recovery maps, acting locally on the two subsystems. The second method is called matrix (or state) reconstruction and involves local, linear maps that are not necessarily completely positive. Moreover, we compare the quantities on which the maps employed in the two methods depend. We show that any state that admits Petz recovery also admits state reconstruction. However, the latter is successful for a strictly larger set of states. We also compare these methods in the context of a finite spin chain. Here, the state of a finite spin chain is reconstructed from the reduced states of a few neighbouring spins. In this setting, state reconstruction is the same as the matrix product operator reconstruction proposed by Baumgratz et al. [Phys. Rev. Lett. 111, 020401 (2013)]. Finally, we generalize both these methods so that they employ long-range measurements instead of relying solely on short-range correlations embodied in such local reduced states. Long-range measurements enable the reconstruction of states which cannot be reconstructed from measurements of local few-body observables alone and hereby we improve existing methods for quantum state tomography of quantum many-body systems.
This study delves into the efficacy of the Petz recovery map within the context of two paradigmatic quantum channels: dephasing and amplitude-damping. While prior investigations have predominantly focused on qubits, our research extends this inquiry to higher-dimensional systems. We introduce a novel, state-independent framework based on the Choi-Jamiołkowski isomorphism to evaluate the performance of the Petz map. By analyzing different channels and the (non-)unital nature of these processes, we emphasize the pivotal role of the reference state selection in determining the map's effectiveness. Furthermore, our analysis underscores the considerable impact of suboptimal choices on performance, prompting a broader consideration of factors such as system dimensionality.
In quantum error correction, the Petz map serves as a perfect recovery map when the Knill-Laflamme conditions are satisfied. Notably, while perfect recovery is generally infeasible for most quantum channels of finite dimension, the Petz map remains a versatile tool with near-optimal performance in recovering quantum states. This work introduces and proves, for the first time, the necessary and sufficient conditions for the optimality of the Petz map in terms of entanglement fidelity. In some special cases, the violation of this condition can be easily characterized by a simple commutator that can be efficiently computed. We provide multiple examples that substantiate our new findings.