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 which 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 which 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 MPO (i.e. matrix product operator) reconstruction proposed by Baumgratz et al. in [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.
Realizing the Petz Recovery Map on an NMR Quantum Processor
Indian Institute of Science Education & Research Mohali（印度科学教育与研究所莫哈利分校）；Amrita Vishwa Vidyapeetham, Amritapuri（阿姆里塔大学阿姆里塔普里校区）；National Institute for Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）；Chennai Institute of Technology（金奈理工学院）；Universität Ulm（乌尔姆大学）；Universidade Federal de Pernambuco（伯南布哥联邦大学）
Experimental Tabletop Petz recovery of a photonic qubit
National Laboratory of Solid State Microstructures, College of Engineering and Applied Science, Nanjing University（南京大学工程与应用科学学院固体微结构物理国家重点实验室）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Physics, National University of Singapore（新加坡国立大学物理系）
An important issue in developing quantum technology is that quantum states are so sensitive to noise. We propose a protocol that introduces reverse dynamics, in order to precisely control quantum systems against noise described by the Lindblad master equation. The reverse dynamics can be obtained by constructing the Petz recovery map in continuous time. By providing the exact form of the Hamiltonian and jump operators for the reverse dynamics, we explore the potential of utilizing the near-optimal recovery of the Petz map in controlling noisy quantum dynamics. While time-dependent dissipation …
Quantum Darwinism and the quality of Petz recovery
University of Helsinki（赫尔辛基大学）；InstituteQ - the Finnish Quantum Institute（芬兰量子研究所）；Helsinki Institute of Physics（赫尔辛基物理研究所）；QTF Centre of Excellence, Department of Physics, University of Helsinki（赫尔辛基大学物理系量子技术卓越中心）
Simulation of Adjoints and Petz Recovery Maps for Unknown Quantum Channels
The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Wuhan University（武汉大学）；Hubei Center for Applied Mathematics（湖北应用数学中心）；Hubei Key Laboratory of Computational Science（湖北省计算科学重点实验室）；Wuhan Institute of Quantum Technology（武汉量子技术研究院）
We derive a new bound on the effectiveness of the Petz map as a universal recovery channel in approximate quantum error correction using the second sandwiched Rényi relative entropy $\tilde{D}_{2}$. For large Hilbert spaces, our bound implies that the Petz map performs quantum error correction with order-$ε$ accuracy whenever the data processing inequality for $\tilde{D}_{2}$ is saturated up to terms of order $ε^2$ times the inverse Hilbert space dimension. Conceptually, our result is obtained by extending arXiv:2011.03473, in which we studied exact saturation of the data processing inequality…
The Petz recovery channel plays an important role in quantum information science as an operation that approximately reverses the effect of a quantum channel. The pretty good measurement is a special case of the Petz recovery channel, and it allows for near-optimal state discrimination. A hurdle to the experimental realization of these vaunted theoretical tools is the lack of a systematic and efficient method to implement them. This paper sets out to rectify this lack: using the recently developed tools of quantum singular value transformation and oblivious amplitude amplification, we provide a…
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 channe…
At the heart of recent progress in AdS/CFT is the question of subregion duality, or entanglement wedge reconstruction: which part(s) of the boundary CFT are dual to a given subregion of the bulk? This question can be answered by appealing to the quantum error correcting properties of holography, and it was recently shown that robust bulk (entanglement wedge) reconstruction can be achieved using a universal recovery channel known as the twirled Petz map. In short, one can use the twirled Petz map to recover bulk data from a subset of the boundary. However, this map involves an averaging procedu…
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 \textit{reversal channel} that recovers the original states after the channel acts. Furthermore, this reversal channel can be constructed explicitly and is now called the \textit{Petz recovery map}. Recent developments h…
There are two parts to this work: First, we study the error correction properties of the real-space renormalization group (RG). The long-distance operators are the (approximately) correctable operators encoded in the physical algebra of short-distance operators. This is closely related to modeling the holographic map as a quantum error correction code. As opposed to holography, the real-space RG of a many-body quantum system does not have the complementary recovery property. We discuss the role of large $N$ and a large gap in the spectrum of operators in the emergence of complementary recovery…
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；Institute for Quantum Computing（量子计算研究所）；Leibniz Universität Hannover, Institut für Theoretische Physik（汉诺威莱布尼茨大学理论物理研究所）
We prove number of quantitative stability bounds for the cases of equality in Petz's monotonicity theorem for quasi-relative entropies defined in terms of an operator monotone decreasing functions. Included in our results is a bound in terms of the Petz recovery map, but we obtain more general results. The present treatment is entirely elementary and developed in the context of finite dimensional von Neumann algebras where the results are already non-trivial and of interest in quantum information theory.
We investigate the problem of reversing quantum dynamics, specifically via optimal Petz recovery maps. We focus on typical decoherence channels, such as dephasing, depolarizing and amplitude damping. We illustrate how well a physically implementable recovery map simulates an inverse evolution. We extend this idea to explore the use of recovery maps as an approximation of inverse maps, and apply it in the context of non-Markovian dynamics. We show how this strategy attenuates non-Markovian effects, such as the backflow of information.
The Data Processing Inequality (DPI) says that the Umegaki relative entropy $S(ρ||σ) := {\rm Tr}[ρ(\log ρ- \log σ)]$ is non-increasing under the action of completely positive trace preserving (CPTP) maps. Let ${\mathcal M}$ be a finite dimensional von Neumann algebra and ${\mathcal N}$ a von Neumann subalgebra if it. Let ${\mathcal E}_τ$ be the tracial conditional expectation from ${\mathcal M}$ onto ${\mathcal N}$. For density matrices $ρ$ and $σ$ in ${\mathcal N}$, let $ρ_{\mathcal N} := {\mathcal E}_τρ$ and $σ_{\mathcal N} := {\mathcal E}_τσ$. Since ${\mathcal E}_τ$ is CPTP, the DPI says th…
Holevo's just-as-good fidelity is a similarity measure for quantum states that has found several applications. One of its critical properties is that it obeys a data processing inequality: the measure does not decrease under the action of a quantum channel on the underlying states. In this paper, I prove a refinement of this data processing inequality that includes an additional term related to recoverability. That is, if the increase in the measure is small after the action of a partial trace, then one of the states can be nearly recovered by the Petz recovery channel, while the other state i…
We study the universal behavior of quantum information-theoretic quantities in thermalized isolated quantum many-body systems and evaporating black holes. In particular, we study a genuine mixed-state entanglement measure called the logarithmic negativity, other correlation measures including the Renyi negativities and the mutual information, and a signature of multipartite entanglement called the reflected entropy. We also probe the feasibility of recovering quantum information from subsystems of a thermalized quantum many-body system or from the radiation of an evaporating black hole, using …
We prove that for a large class of quantum Fisher information, a quantum channel is sufficient for a family of quantum states, i.e., the input states can be recovered from the output, if and only if the quantum Fisher information is preserved under the quantum channel. This class, for instance, includes Winger-Yanase-Dyson skew information. On the other hand, interestingly, the SLD quantum Fisher information, as the most popular example of quantum analog of Fisher information, does not satisfy this property. Our recoverability result is obtained by studying Riemannian monotone metrics on the q…
Quantum information is scrambled via chaotic time evolution in many-body systems. The recovery of initial information embedded locally in the system from the scrambled quantum state is a fundamental concern in many contexts. From a dynamical perspective, information recovery can measure dynamical instability in quantum chaos, fault-tolerant quantum computing, and the black hole information paradox. This article considers general aspects of quantum information recovery when the scrambling dynamics have conservation laws due to Lie group symmetries. Here, we establish fundamental limitations on …
In this work we study the nature of correlations among mixed states in the setup of two symmetric strips. We use various tools to determine how the bulk geometry could be reconstructed from the boundary mixed information. These tools would be the modular Hamiltonian and modular flow, OPE blocks, quantum recovery channels such as Petz map, Uhlmann holonomy and Wilson lines. We comment on the similarities and connections between these approaches in our symmetric setup of a mixed system. Specially, we use parameters such as dissipation which is being modeled by the mass of graviton, and also the …
Conditional distributions, as defined by the Markov category framework, are studied in the setting of matrix algebras (quantum systems). Their construction as linear unital maps are obtained via a categorical Bayesian inversion procedure. Simple criteria establishing when such linear maps are positive are obtained. Several examples are provided, including the standard EPR scenario, where the EPR correlations are reproduced in a purely compositional (categorical) manner. A comparison between the Bayes map, the Petz recovery map, and the Leifer-Spekkens acausal belief propagation is provided, il…
Bipartite quantum states admitting a causal explanation
Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Experimental Study Group, Massachusetts Institute of Technology（麻省理工学院实验研究组）
