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arXiv:2003.10559 · PRX Quantum

Asymptotic Theory of Quantum Channel Estimation

Sisi Zhou, Liang Jiang

Published Mar 16, 2021• 67 citations
arXivHTMLPDF

Abstract

The quantum Fisher information (QFI), as a function of quantum states, measures the amount of information that a quantum state carries about an unknown parameter.The (entanglement-assisted) QFI of a quantum channel is defined to be the maximum QFI of the output state assuming an entangled input state over a single probe and an ancilla.In quantum metrology, people are interested in calculating the QFI of N identical copies of a quantum channel when N → ∞, which is called the asymptotic QFI.Over the years, researchers found various types of upper bounds of the asymptotic QFI, but they were proven achievable only in several specific situations.It was known that the asymptotic QFI of an arbitrary quantum channel grows either linearly or quadratically with N .Here we show that a simple criterion can determine whether the scaling is linear or quadratic.In both cases, the asymptotic QFI and a quantum error correction protocol to achieve it are computable via a semidefinite program.When the scaling is quadratic, the Heisenberg limit, a feature of noiseless quantum channels, is recovered.When the scaling is linear, we show that the asymptotic QFI is still in general larger than N times the single-channel QFI and, furthermore, that sequential estimation strategies provide no advantage over parallel ones.

DOI
10.1103/prxquantum.2.010343

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The quantum Fisher information (QFI), as a function of quantum states, measures the amount of information that a quantum state carries about an unknown parameter.The (entanglement-assisted) QFI of a quantum channel is defined to be the maximum QFI of the output state assuming an entangled input state over a single probe and an ancilla.In quantum metrology, people are interested in calculating the QFI of N identical copies of a quantum channel when N → ∞, which is called the asymptotic QFI.Over the years, researchers found various types of upper bounds of the asymptotic QFI, but they were proven achievable only in several specific situations.It was known that the asymptotic QFI of an arbitrary quantum channel grows either linearly or quadratically with N .Here we show that a simple criterion can determine whether the scaling is linear or quadratic.In both cases, the asymptotic QFI and a quantum error correction protocol to achieve it are computable via a semidefinite program.When the scaling is quadratic, the Heisenberg limit, a feature of noiseless quantum channels, is recovered.When the scaling is linear, we show that the asymptotic QFI is still in general larger than N times the single-channel QFI and, furthermore, that sequential estimation strategies provide no advantage over parallel ones.
The quantum Fisher information (QFI) measures the amount of information that a quantum state carries about an unknown parameter. The (entanglement-assisted) QFI of a quantum channel is defined to be the maximum QFI of the output state assuming an entangled input state over a single probe and an ancilla. Both the channel QFI and the optimal input state could be solved via a semidefinite program (SDP). In quantum metrology, people are interested in calculating the QFI of NNN identical copies of a quantum channel when N→∞N \rightarrow \inftyN→∞, which we call the asymptotic QFI. It was known that the asymptotic QFI grows either linearly or quadratically with NNN. Here we obtain a simple criterion that determines whether the scaling is linear or quadratic. In both cases, we found a quantum error correction protocol achieving the asymptotic QFI and an SDP to solve the optimal code. When the asymptotic QFI is quadratic, the Heisenberg limit, a feature once thought unique to unitary quantum channels, is recovered. When the asymptotic QFI is linear, we show it is still in general larger than NNN times the channel QFI, showing the non-additivity of the channel QFI of general quantum channels.
The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of NNN resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with NNN better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources NNN is strictly constrained, I show that the final measurement uncertainty is πππ times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.
We derive ultimate precision bounds for estimating parameters encoded in \emph{time-dependent} Hamiltonians in the presence of general Markovian noise, allowing for arbitrary adaptive protocols with fast controls and noiseless ancillas. Extending the minimization-over-purifications framework to time-varying continuous channels, we obtain a differential upper bound on the achievable quantum Fisher information (QFI) that can be evaluated at all times via semidefinite programming. For parameter-independent noise, we prove a universal long-time scaling law: if the coherent (noiseless) dynamics yields Qcoh(T)∼T2kQ_{\mathrm{coh}}(T)\sim T^{2k}Qcoh​(T)∼T2k, then under Markovian noise the QFI scales at most as Q(T)∼T2kQ(T)\sim T^{2k}Q(T)∼T2k in the DHNLS regime, whereas in the DHLS regime it is fundamentally limited to Q(T)∼T2k−1Q(T)\sim T^{2k-1}Q(T)∼T2k−1. We illustrate these behaviors on paradigmatic driven-qubit sensors, exhibiting T4T^{4}T4 and T3T^{3}T3 scalings under dephasing and spontaneous emission, respectively. Finally, we provide explicit continuous exact and approximate quantum error correction constructions -- supplemented by spin-squeezed probes -- that asymptotically saturate the bounds, establishing their tightness.
The aim of this thesis is to develop a theoretical framework to study parameter estimation of quantum channels. We begin by describing the classical task of parameter estimation that we build upon. In its most basic form, parameter estimation is the task of obtaining an estimate of an unknown parameter from some experimental data. This experimental data can be seen as a number of samples of a parameterized probability distribution. In general, the goal of such a task is to obtain an estimate of the unknown parameter while minimizing its error. We study the task of estimating unknown parameters which are encoded in a quantum channel. A quantum channel is a map that describes the evolution of the state of a quantum system. We study this task in the sequential setting. This means that the channel in question is used multiple times, and each channel use happens subsequent to the previous one. A sequential strategy is the most general way to use, or process, a channel multiple times. Our goal is to establish lower bounds on the estimation error in such a task. These bounds are called Cramer--Rao bounds. Quantum channels encompass all possible dynamics allowed by quantum mechanics, and sequential estimation strategies capture the most general way to process multiple uses of a channel. Therefore, the bounds we develop are universally applicable. We consider the use of catalysts to enhance the power of a channel estimation strategy. This is termed amortization. The reason we do so is to investigate if an n-round sequential estimation strategy does better than a simpler parallel strategy. Quantitatively, the power of a channel for a particular estimation task is determined by the channel's Fisher information. Thus, we study how much a catalyst quantum state can enhance the Fisher information of a quantum channel by defining the amortized Fisher information. In the quantum setting, there are many Fisher information quantities that can be defined. We limit our study to two particular ones: the symmetric logarithmic derivative (SLD) Fisher information and the right logarithmic derivative (RLD) Fisher information. We establish our Cramer--Rao bounds by proving that for certain Fisher information quantities, catalyst states do not improve the performance of a sequential estimation protocol. The technical term for this is an amortization collapse. We show how such a collapse leads directly to a corresponding Cramer--Rao bound. We establish bounds both when estimating a single parameter and when
Quantum metrology with indefinite causal order (ICO) has attracted intense interest due to its potential to surpass the limitations of conventional fixed-order strategies. A key open question is whether ICO can fundamentally enhance asymptotic precision scaling. In this work, we bridge this gap for the estimation of a single parameter encoded in NNN identical uses of a finite-dimensional quantum channel. We first establish a universal Heisenberg-scaling upper bound for the full general ICO process-matrix class and show that for unitary channels its optimal quantum Fisher information (QFI) coincides exactly with that of parallel strategies. For noisy channels, a structurally refined bound shows that channels restricted to the standard quantum limit (SQL) under parallel strategies remain SQL-limited under general ICO strategies. Most significantly, an asymptotically tight (AT) bound is derived to close the remaining possibility of an asymptotic ICO advantage by showing that general ICO and optimal parallel strategies have exactly the same leading QFI coefficient in both the SQL and Heisenberg regimes. For the operationally motivated class of quantum circuits with quantum control of causal order, we further obtain an iterative constraint on finite-query precision whose asymptotic limit agrees with that of the AT bound. Our results clarify the ultimate role of indefinite causality as a metrological resource for quantum channel estimation.
Extensive research has been dedicated to the asymptotic theory of quantum metrology, where the goal is to determine the ultimate precision limit of quantum channel estimation when many accesses to the channel are allowed. The ultimate limit has been well established, but in general, a noiseless and controllable ancilla is required for attaining it. Little is known about the metrological performance without noiseless ancillae, which is more relevant in practical circumstances. In this Letter, we present a novel theoretical framework to address this problem, bridging quantum metrology and the asymptotic theory of quantum channels. Leveraging this framework, we prove sufficient conditions for achieving the Heisenberg limit with repeated applications of the channel to be estimated both with and without applying interleaved unitary control operations. For the latter case, we design an algorithm to identify the control operation explicitly.