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 metrology) 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 enhancement 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 |αi input at port A (|α|2 = N is the average photon quantum or classical – is a three-step measurement process number or “power” input), and a vacuum input |0i 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 perform a parameter estimation about the unknown system. For example, the phase difference between two arms of an optical Mach-Zehnder interferometer (MZI) probed by a coherent 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 measurement 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 {φi} 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/ = √ ∼ √ . (1) dφ which the probes are not correlated or “entangled” in any way; N|sin φ| N in addition, no entanglement is used at the measurement step. What is the best precision such a system can achieve? Additionally, it can√ be shown that even√ 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 information 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 | i | i atomic qubit is prepared in the state |ψ i = 0 √+ 1 , and al- in 2 | i iφ| i lowed to evolve to the state |ψ i = 0 +√e 1 due to the dy- out 2 namics of the system. φ can be estimated by measuring the 2 probability that |ψouti = |ψini, that is, p(φ) = |hψin|ψouti| = 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 hψ | (|ψ i hψ |)2 |ψ 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,√ will decrease the variance by N, giving the same ∆φ ∼ 1/ N scaling.

operators {Ey} 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 conditional probabilities; the conditional probability of outcome y ⊗n when the system was originally in ρx is pn(y|x) = 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(z|y). 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(z|x) = y pn(z|y)pn(y|x), 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(z|x) (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 distribution√ 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 (y|x) F (x) ≡ n /p (y|x). (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(y|x), 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 . (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 enhancement√ 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 √ √ N-partite entangled state |ai⊗n + |bi⊗n / 2, and the output is |ai⊗n + eiNφ |bi⊗n / 2. Probability of the estimator (if output state is the same √ 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. Crucially, a recipe exists to compute the quantum FI (QFI) Finally, note that although the Fisher information (4) fea- n ⊗n 2o Jn(x) = Tr ρx L(ρx) for any ρx from the so-called Symmetric tures conditional probabilities that appear Bayesian, the Fish- Logarithmic Derivative (SLD) L(ρx) (expression found in [1, erian approach to parameter estimation is subtly different from ∂ρx the Bayesian one. In the latter, a prior probability distribution 3,4]), which is a function only of ρx and ∂x , independent of measurement. of the parameter X is known, whereas the FI estimates an en- Also important is the property that the FI is non-negative, tirely unknown X from posterior probabilities [4]. and additive for uncorrelated events [4]; for a set of n uncorre- B. Quantum parameter estimation: optimized probes lated measurements√ √ of ρx, the total FI is Jn(x) = nJ(ρx), giving ∆Xn ≥ 1/ n J(x). For example, the FI of a coherent state probe with average photon number N scales as JN (φ) ∼ N, The QCRB stated in equation (5) was optimized at the mea- ⊗n retrieving the SQL scaling. surement stage, assuming a fixed set of probe states ρx . Re- Finally, in the asymptotic limit of large n (the number of laxing this condition allows us to optimize the QFI over dif- times the system is probed independently), the QCRB is al- ferent choices of input states. Denoting the input or probe ways achievable using local and non-quantum-enhanced op- quantum state as ρ0 (which is converted to ρx at the measure- erations [1], implying that quantum entanglement is not nec- ment stage at the output of the quantum channel), the QCRB essary at the measurement or readout stage. may be re-written as [1] Therefore, to beat the SQL, one must turn to non-classical 1 strategies at the probe-preparation stage. Before embarking ∆X ≥ . (6) on a quest for SQL-beating FIs, let us take stock of the physi- n p maxρ0 Jn(ρx) cal meaning behind FI, the statistical construct. We allow ρ0 and ρx to be entangled. A. FI: physical intuition The search for the optimum state can be narrowed by con- sidering only pure states. The convexity of the QFI vindicates ∂p (y|x) this: pure states have higher QFI than mixed states. If the The n term in equation (4) suggests that in essence, the ∂x quantum channel is unitary (represented by hamiltonian H) as FI is a measure of the change in the probability of the outcome is the case in a closed system without loss or decoherence, a y corresponding to a change in the actual parameter x. Indeed, iHX −iHX 2 pure state input ρ0 is changed to ρx = e He (assume ∂ ln pn(y|x) an alternate way of expressing FI is Fn(x) = ∂x , HX to be dimensionless for simplicity). In the case of pure y 2 clearly linking FI with the rate of change of conditional prob- states, the QFI assumes an appealing form: Jn(ρx) = 4∆ H [4], allowing us to reformulate the QCRB for n independent ability. A higher FI indicates that the probe outcome pn(y|x) has amplified the changes in the system parameter x, viz, a trials as [1] smaller change in x can be resolved. 1 A related concept central to POVM measurements that ∆Xn ≥ √ , (7) sheds light on FI is the fidelity of states. The fidelity, 2∆H n q √ √ 2 F (ρ1, ρ2) = Tr ρ1ρ2 ρ1 , quantifies the distinctness a rather elegant form that is tantalizingly reminiscent of the of quantum states (and the parameters that depend on them). energy-time uncertainty relation. Consider a small change dX of the parameter X, that changes Quantum enhancement to ∆Xn is achieved by entangling 0 the probe states. Maximally entangling states maximizes the the state from ρx to ρx. The resulting distinguishability be- 0 1 2 QFI. For example, consider a 2-level system with a state tween states is given by F (ρx, ρx) = 1 − Jn(ρx)dX [4], √ 4 (N)E ⊗n ⊗n thereby linking it to the QFI. That is, if the states are not well ψ0 ≡ |ai + |bi / 2. This is a maximally entangled distinguishable at the output, the FI suffers, as does the reso- state at the input of a generic Ramsey set-up [Fig.5]. The QFI lution of parameter estimation. for this state scales with the degree of entanglement as ∼ N, 4 giving the scaling

1 1 ∆Xn & √ = √ , (8) nN N r where n is the number of parallel or independent uses of the Figure 6. [20] (a)-(c) Visualizing decoherence models of 2-level channel. r ≡ nN quantifies the concept of “resource scal- atomic systems illustrated by their effect on the Bloch sphere: (a) ing”: typically, the total amount of resources available to a depolarization; (b) dephasing; (c) spontaneous emission; parameter measurement—to be divided between the degree of entangle- φ is the rotation about the z-axis, and “shrinkage” η quantifies the de- ment N and the number of independent trials n—is capped coherence; (d) Lossy interferometer with power loss η in each arm. (for instance, in quantum-enhanced interferometry, this could In all cases, the final precision bounds found via the upper bound on be a limit on optical power). QFI is a function of η.

D. Heisenberg Limit: two approaches

C. The Heisenberg Limit The HL is often defined as 1/N scaling, where N may be the average photon number (or degree of entanglement) in op- The 1/N scaling in equation (8) is commonly referred to tical (or atomic) phase estimation problems. However, Boixo 2 as the√ Heisenberg Limit, improving upon the SQL by a factor et al. proposed [11]that ∆X may indeed scale by 1/N , us- of N. A “Ramsey scheme” of phase estimation involving ing sequential schemes. One proposal [12] considers prob- 2-level systems is described in Fig.5; this is a prototype that ing a system with Kerr nonlinearity using a coherent state illustrates how the HL is attained by quantum enhancement (1/N3/2). Roy and Braunstein [13] showed that, in fact, with of the input probe, deployment of the probes across parallel the full amount of entanglement in the available Hilbert space, channels, and a local measurement at the output. ∆X ∼ 1/2N . Pioneering experiments have also accomplished >1 In practice, several experimentally demonstrated architec- 1/N scaling in parameter estimation, notably, Napolitano et 3/2 tures in atomic physics are capable of realizing the Ramsey al. achieved 1/N using nonlinearities in atomic ensembles scheme by generating entangled inputs. These include spin- introduced by Faraday rotation pulses [14]. All of this begs squeezed states and Schrodinger cat states which are gener- the question, can the HL be surpassed? ated in diverse platforms including BECs, trapped ions, and Not if the definition of the HL is extended beyond simply NMR among others [1,5,6]. Other quantumly-correlated 1/N scaling. As pointed out by Zwierz et al. [15], the hamil- probe states beating the SQL include correlated Fock states tonians of nonlinear systems do not scale linearly with power. in matter-wave interferometry for phase estimation [7]. Addi- In the Kerr case, hHi ∼ N2. From the definition of the QFI and tional examples of systems that have experimentally achieved QCRB in equation (7) in terms of the variance of the Hamilto- 2 sub-shot-noise scaling are described in Pezze et al.[6], a com- nian, one obtains the scaling ∆X ∼ 1/N , the true HL for this prehensive and recent review of quantum metrology using system. Re-interpreting the HL as a bound on the energy un- atomic-ensemble platforms. certainty reconciles the 1/N surpassing. As outlined in [15], The translation of the Ramsey scheme to the optical domain philosophically, this reinforces the “resource-scaling” concept is not experimentally straightforward [1,8]. The optical ana- of precision measurement, where it is not enough to know how log of entangled qubit states that saturate the HL scaling are the precision scales with power (N), but how the resource the celebrated NOON states, featuring entanglement between “cost” (say, the number of queries/interactions/independent √ probes) does too. Fock states and the vacuum: (|Ni |0i ± |0i |Ni) / 2. Notwith- standing the difficulty of generating NOON states for arbitrary N (N = 2 states may be produced by the Hang-Ou-Mandel effect), the primary obstacle is the fragility in the presence of incoherent losses that readily turn NOON states into statistical mixtures. Nevertheless, other entanglement strategies have managed to outperform the shot-noise limit [9]. I have focused on parallel estimation schemes since they readily illustrate both classical scaling by repeated measurement, as well as quantum enhancement. In these schemes, each of n probes samples the system once, requiring multi(N)- partite entanglement of the probe. Other schemes exist in which one probe sequentially measures n copies of the system Figure 7. [20] Generic dependence of quantum-enhanced parameter with the help of ancilla probes; these would need (< N)-partite estimation as a function of independent probes N in the presence entanglement to reach the HL (for instance, in [10]), suggest- of decoherence. For small N, the system retains HL scaling, but ing that parallel and sequential schemes could be combined asymptotically veers to the classical SNL scaling for increased N, for the ideal measurement strategy. albeit with some “const” enhancement factor. 5

III. QFI AS A FRAMEWORK IN NOISY QUANTUM tion about the parameter X and the speciﬁc loss (η) pro- METROLOGY P † cess (Fig.6) [20]. The probe evolves to ρx = l Πlρ0Πl . The QFI is then upper bounded by the computable quantity † h 2i P ∂Πl ∂Πl Lastly, the QFI plays a vital role in determining precision C(ρ0, Πl) = 4 hH1(x)i − hH2(x)i , where H1(x) = l ∂x ∂x , bounds in lossy or open quantum systems. While a rigorous ∂Π† and H (x) = i P l Π [16]. In the lossless case, this reduces mathematical analysis of this topic [16–19], is beyond this pa- 2 l ∂x l to equation (7). These theoretical bounds have been calculated per’s scope, I present a brief recipe. in depth [17], but Fig.7 [20] represents the general impact of Instead of unitary operations, the quantum channel now fea- noise on a precision measurement. tures non-unitary transformations of the input probe, gener- All considered, the QFI continues to be an indispensable alized by Kraus operators Πl(x, η), which contain informa- tool in deciphering the possibilities of quantum metrology.

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