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Quantum Fisher Information and Estimation Strategies

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Quantum Fisher Information and Estimation Strategies Quantum metrology in the context of quantum information: quantum Fisher Information and estimation strategies Mitul Dey Chowdhury1 1James C. Wyant College of Optical Sciences, University of Arizona (Dated: December 9, 2020) A central concern of quantum information processing – the use of quantum mechanical systems to encode, store, and transmit information – is the precision with which such information may be communicated and detected. Here, I review how quantum effects may be harnessed to measure a quantum system (quantum metrol- ogy) with increased precision beyond the reach of classical statistics. I present the concept of Fisher Information (FI) as a tool to understand how the Standard Quantum Limit (SQL) in parameter estimation could be surpassed and the Heisenberg Limit (HL) approached. I also outline specific probing and estimation strategies to beat the SQL such as entaglement, some of them already realized experimentally. Finally, I discuss how quantum en- hancement is vulnerable to noise, and the role of quantum Fisher Information (QFI) in understanding quantum decoherence and precision limits in noisy metrology. The extraction of information embedded in a system – coherent jαi input at port A (jαj2 = N is the average photon quantum or classical – is a three-step measurement process number or “power” input), and a vacuum input j0i in port B [1]: first, preparing a suitable probe; then, allowing the probe of the MZI shown in [Fig.2] [2]. to interact with the system; finally, the probe readout [Fig.1]. The second step affects some physical quantity (parameter) of the probe, which is measured in the readout stage to per- form a parameter estimation about the unknown system. For example, the phase difference between two arms of an opti- cal Mach-Zehnder interferometer (MZI) probed by a coher- ent field (described below) can be inferred by measuring the power difference between the two output arms. In addition to systematic errors which could be mitigated, the measure- ment process is subject to statistical errors—that depend on the very nature of the probe (such as quantum uncertainty principles), or readout schemes (averaging over many trials, Figure 2. MZI with 50:50 beamsplitters, homodyne detection. A and for example)—which set the ultimate bounds on measurement B are input ports, C and D are outputs. The difference in photocurrent between C and D is an estimator of φ. precision. Metrology is the study of these bounds and how to attain them. The task is to estimate the unknown phase φ introduced in one arm vis-a-vis the other (for example, due to the dis- placement of an inertial sensor). The output in each of the arms C and D is also a coherent state with average pow- 2 2 ers NC = N sin (φ/2) and ND = N cos (φ/2) respectively. Let Aˆ ≡ Nˆ D − NˆC denote the number difference opera- Figure 1. Concept of parameter estimation of a lossless quantum tor; the statistics of coherent states gives the mean hAi = channel. P is the probe, usually some quantum state ρ, that interacts ND − NC = N cos(φ), and variance in the power difference with the system U(φ) where φ is an unknown parameter. The readout 2 ˆ ˆ 2 ˆ 2 ∆ A = ∆ND + ∆NC = ND + NC = N. Evidently, the parameter (measurement M) of the probe gives an estimate of φ. The system φ may be estimated by a measurement of hAi. The imprecision may comprise multiple parameters fφig to be estimated. in the phase measurement is given by propagating the error in I. PARAMETER ESTIMATION: THE CLASSICAL estimating hAi to obtain the standard deviation, PERSPECTIVE d hAi 1 1 A classical measurement of a quantum system is one in ∆φ = ∆Aˆ= = p ∼ p : (1) dφ which the probes are not correlated or “entangled” in any way; Njsin φj N in addition, no entanglement is used at the measurement step. What is the best precision such a system can achieve? Additionally, it canp be shown that evenp a Fock state input at A gives the same 1= N scaling. This 1= N scaling of precision with the optical power—called the shot-noise limit (SNL)— A. Mach-Zehnder Interferometer is a consequence of the Poissonian statistics of classical light: the lack of cooperative behavior among the photons means The MZI is a simple paradigm that illustrates the statistical each photon probes the system stochastically. N independent limits of classical parameter estimation. Consider a classical, probes therefore decrease the variance by a factor N. 2 B. Translation to qubits: Ramsey Interferometer The MZI analysis may be extended beyond the context of optics. Of particular interest in quantum computing and infor- mation is the measurement of a qubit state. Consider Ram- sey interferometry which is topologically analogous to the MZI (π/2 pulses play the role of 50:50 beamsplitters). An j i j i atomic qubit is prepared in the state j i = 0 p+ 1 , and al- in 2 j i iφj i lowed to evolve to the state j i = 0 +pe 1 due to the dy- out 2 namics of the system. φ can be estimated by measuring the 2 probability that j outi = j ini, that is, p(φ) = jh inj outij = Figure 4. [1] Where to quantum(Q)-enhance over classical(C)? (a) cos2(φ/2) = (1 − cos φ)=2. The variance in p is ∆2 p(φ) = Uncorrelated probes, classical measurement; (b) entanglement at the h j (j i h j)2 j i − p2(φ) = p(φ) − p2(φ), giving ∆φ = measurement stage only; (c) entangle at input only; (d) use entangle- out in in out ∆p(φ)= dp(φ) = 1 [1]. Therefore, N trials of the experiment, ment at both input and measurement. CC and CQ have shot-noise dφ scaling, whereas QC and QQ attain Heisenberg scaling. or using N independent qubits,p will decrease the variance by N, giving the same ∆φ ∼ 1= N scaling. operators fEyg forming a POVM. n copies of ρx are measured to obtain a single final result y, from which we attempt to infer x. POVMs naturally lend themselves to the parlance of condi- tional probabilities; the conditional probability of outcome y ⊗n when the system was originally in ρx is pn(yjx) = Tr ρx Ey . After the y readout, we process the results to make a guess about x; this guess is the estimator z made with probability pn(zjy). Usually, z is an “unbiased” estimator, meaning we would like z to be as close as possible to x. The imprecision in the estimate of x is therefore dependent on the conditional P probability pn(zjx) = y pn(zjy)pn(yjx), and is given by the root-mean-square error (RMSE) [1] Figure 3. [4] Topological equivalence of MZI, Ramsey interferome- s X 2 ters δXn = (z − x) pn(zjx) (2) z C. Central limit theorem, SQL For an unbiased estimator, δXn is the standard deviation ∆Xn (this is the interpretation of “imprecision” or “error” I The SQL discussed above is a manifestation of the central have used in this paper). It can be shown that the error is limit theorem: the average of a large number N of independent lower-bounded by the famous Cramer-Rao Bound [1]: measurements of φ, each with a standard deviation ∆φ, will p ∆X ≥ 1= F (x); (3) converge to a Gaussian distributionp about the mean estimate n n with standard deviation ∆φ/ N. In order to beat this limit, where the (classical) Fisher Information F (x) or F(ρ⊗n) is [1] quantum correlations must be introduced in the measurement n x !2 process. X @p (yjx) F (x) ≡ n =p (yjx): (4) n @x n y II. TOWARDS QUANTUM-ENHANCED ESTIMATION: FI (Note that the summations in (2) and (4) should be replaced AND THE CRAMER-RAO BOUND with integrals for continuous outcomes y and estimates z.) Evidently, the task of minimizing ∆Xn is that of finding The statistical abstraction known as the Fisher Information, a system which (a) has a large FI, and (b) attains the CRB. together with the Cramer-Rao Bound (CRB), is essential in First, note that while the FI in (4) is dependent on the partic- understanding precision beyond the SQL. In this paper, I es- ular choice of POVM measurement scheme via pn(yjx), max- chew rigorous mathematical scrutiny in favor of presenting imizing FI over all conceivable POVMs yields a CRB that is important results and relevant properties of the FI. POVM-independent. This optimized bound at the measure- Before motivating the FI, it is germane to recall the con- ment stage of the estimation process is the quantum CRB ditional probabilites in the readout process. A system S has (QCRB) [1], some parameter x (for example, phase φ in the above exam- 1 1 ples), encoded in a set of probe states ρx. We perform a mea- ∆Xn ≥ p ≡ p : (5) surement on S, which we express as a set of Hilbert space maxPOVMs (Fn(x)) Jn(ρx) 3 Figure 5. [1] Ramsey parallel quantum enhancementp scheme: a. Classical estimation. As described in section Ib, the standard deviation in phase estimation scales as 1 for each trial, and 1= n (SQL) for n independent measurements. b. Quantum-enhanced: input state is the p p N-partite entangled state jai⊗n + jbi⊗n = 2, and the output is jai⊗n + eiNφ jbi⊗n = 2. Probability of the estimator (if output state is the same p as input) is (1 − cos Nφ)=2. The imprecision scales as ∆φ ≥ 1= nN ∼ 1=N (the HL) for n independent uses of the channel.
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