Quantum Metrology In The Context Of Quantum Information: Quantum Fisher .
Quantum metrology in the context of quantum information: quantum Fisher Information and estimation strategies Mitul Dey Chowdhury1 1 James 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 – quantum or classical – is a three-step measurement process [1]: first, preparing a suitable probe; then, allowing the probe 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, for example)—which set the ultimate bounds on measurement precision. Metrology is the study of these bounds and how to attain them. Figure 1. Concept of parameter estimation of a lossless quantum channel. P is the probe, usually some quantum state ρ, that interacts with the system U(φ) where φ is an unknown parameter. The readout (measurement M) of the probe gives an estimate of φ. The system may comprise multiple parameters {φi } to be estimated. I. PARAMETER ESTIMATION: THE CLASSICAL PERSPECTIVE A classical measurement of a quantum system is one in which the probes are not correlated or “entangled” in any way; in addition, no entanglement is used at the measurement step. What is the best precision such a system can achieve? A. Mach-Zehnder Interferometer The MZI is a simple paradigm that illustrates the statistical limits of classical parameter estimation. Consider a classical, coherent αi input at port A ( α 2 N is the average photon number or “power” input), and a vacuum input 0i in port B of the MZI shown in [Fig.2] [2]. Figure 2. MZI with 50:50 beamsplitters, homodyne detection. A and B are input ports, C and D are outputs. The difference in photocurrent between C and D is an estimator of φ. The task is to estimate the unknown phase φ introduced in one arm vis-a-vis the other (for example, due to the displacement of an inertial sensor). The output in each of the arms C and D is also a coherent state with average powers NC N sin2 (φ/2) and ND N cos2 (φ/2) respectively. Let  N̂D N̂C denote the number difference operator; the statistics of coherent states gives the mean hAi ND NC N cos(φ), and variance in the power difference 2  N̂D2 N̂C2 ND NC N. Evidently, the parameter φ may be estimated by a measurement of hAi. The imprecision in the phase measurement is given by propagating the error in estimating hAi to obtain the standard deviation, φ Â/ d hAi 1 1 . dφ N sin φ N (1) 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)— is a consequence of the Poissonian statistics of classical light: the lack of cooperative behavior among the photons means each photon probes the system stochastically. N independent 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 Ramsey interferometry which is topologically analogous to the MZI (π/2 pulses play the role of 50:50 beamsplitters). An , and alatomic qubit is prepared in the state ψin i 0i 1i 2 iφ 1i due to the dylowed to evolve to the state ψout i 0i e 2 namics of the system. φ can be estimated by measuring the probability that ψout i ψin i, that is, p(φ) hψin ψout i 2 cos2 (φ/2) (1 cos φ)/2. The variance in p is 2 p(φ) hψout ( ψin i hψin )2 ψout i p2 (φ) p(φ) p2 (φ), giving φ 1 [1]. Therefore, N trials of the experiment, p(φ)/ dp(φ) dφ or using N independent qubits, will decrease the variance by N, giving the same φ 1/ N scaling. Figure 3. [4] Topological equivalence of MZI, Ramsey interferometers Figure 4. [1] Where to quantum(Q)-enhance over classical(C)? (a) Uncorrelated probes, classical measurement; (b) entanglement at the measurement stage only; (c) entangle at input only; (d) use entanglement at both input and measurement. CC and CQ have shot-noise scaling, whereas QC and QQ attain Heisenberg scaling. operators {Ey } forming a POVM. n copies of ρ x 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 when the system was originally in ρ x is pn (y x) Tr ρ n E y . x 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] sX (z x)2 pn (z x) (2) δXn z C. Central limit theorem, SQL The SQL discussed above is a manifestation of the central limit theorem: the average of a large number N of independent measurements of φ, each with a standard deviation φ, will converge to a Gaussian distribution about the mean estimate with standard deviation φ/ N. In order to beat this limit, quantum correlations must be introduced in the measurement process. II. TOWARDS QUANTUM-ENHANCED ESTIMATION: FI AND THE CRAMER-RAO BOUND The statistical abstraction known as the Fisher Information, together with the Cramer-Rao Bound (CRB), is essential in understanding precision beyond the SQL. In this paper, I eschew rigorous mathematical scrutiny in favor of presenting important results and relevant properties of the FI. Before motivating the FI, it is germane to recall the conditional probabilites in the readout process. A system S has some parameter x (for example, phase φ in the above examples), encoded in a set of probe states ρ x . We perform a measurement on S, which we express as a set of Hilbert space For an unbiased estimator, δXn is the standard deviation Xn (this is the interpretation of “imprecision” or “error” I have used in this paper). It can be shown that the error is lower-bounded by the famous Cramer-Rao Bound [1]: p Xn 1/ Fn (x), (3) where the (classical) Fisher Information Fn (x) or F(ρ n x ) is [1] X pn (y x) !2 /pn (y x). (4) Fn (x) x y (Note that the summations in (2) and (4) should be replaced with integrals for continuous outcomes y and estimates z.) Evidently, the task of minimizing Xn is that of finding a system which (a) has a large FI, and (b) attains the CRB. First, note that while the FI in (4) is dependent on the particular choice of POVM measurement scheme via pn (y x), maximizing FI over all conceivable POVMs yields a CRB that is POVM-independent. This optimized bound at the measurement stage of the estimation process is the quantum CRB (QCRB) [1], Xn 1 maxPOVMs (Fn (x)) p 1 Jn (ρ x ) . (5)
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, na recipe exists to compute the quantum FI (QFI) o n 2 Jn (x) Tr ρ x L(ρ x ) for any ρ x from the so-called Symmetric Logarithmic Derivative (SLD) L(ρ x ) (expression found in [1, x 3, 4]), which is a function only of ρ x and ρ x , independent of measurement. Also important is the property that the FI is non-negative, and additive for uncorrelated events [4]; for a set of n uncorrelated measurements of ρ x , the total FI is Jn (x) nJ(ρ x ), 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, retrieving the SQL scaling. Finally, in the asymptotic limit of large n (the number of times the system is probed independently), the QCRB is always achievable using local and non-quantum-enhanced operations [1], implying that quantum entanglement is not necessary at the measurement or readout stage. Therefore, to beat the SQL, one must turn to non-classical strategies at the probe-preparation stage. Before embarking on a quest for SQL-beating FIs, let us take stock of the physical meaning behind FI, the statistical construct. A. FI: physical intuition The pn x(y x) term in equation (4) suggests that in essence, the FI is a measure of the change in the probability of the outcome y corresponding to a change in the actual parameter x. Indeed, 2 an alternate way of expressing FI is Fn (x) ln p xn (y x) , y clearly linking FI with the rate of change of conditional probability. A higher FI indicates that the probe outcome pn (y x) has amplified the changes in the system parameter x, viz, a smaller change in x can be resolved. A related concept central to POVM measurements that sheds light on FI is the fidelity of states. The fidelity, q 2 F (ρ1 , ρ2 ) Tr ρ1 ρ2 ρ1 , quantifies the distinctness of quantum states (and the parameters that depend on them). Consider a small change dX of the parameter X, that changes the state from ρ x to ρ0x . The resulting distinguishability between states is given by F (ρ x , ρ0x ) 1 14 Jn (ρ x )dX 2 [4], thereby linking it to the QFI. That is, if the states are not well distinguishable at the output, the FI suffers, as does the resolution of parameter estimation. Finally, note that although the Fisher information (4) features conditional probabilities that appear Bayesian, the Fisherian approach to parameter estimation is subtly different from the Bayesian one. In the latter, a prior probability distribution of the parameter X is known, whereas the FI estimates an entirely unknown X from posterior probabilities [4]. B. Quantum parameter estimation: optimized probes The QCRB stated in equation (5) was optimized at the measurement stage, assuming a fixed set of probe states ρ n x . Relaxing this condition allows us to optimize the QFI over different choices of input states. Denoting the input or probe quantum state as ρ0 (which is converted to ρ x at the measurement stage at the output of the quantum channel), the QCRB may be re-written as [1] 1 Xn p . maxρ0 Jn (ρ x ) (6) We allow ρ0 and ρ x to be entangled. The search for the optimum state can be narrowed by considering only pure states. The convexity of the QFI vindicates this: pure states have higher QFI than mixed states. If the quantum channel is unitary (represented by hamiltonian H) as is the case in a closed system without loss or decoherence, a pure state input ρ0 is changed to ρ x eiHX He iHX (assume HX to be dimensionless for simplicity). In the case of pure states, the QFI assumes an appealing form: Jn (ρ x ) 4 2 H [4], allowing us to reformulate the QCRB for n independent trials as [1] Xn 1 , 2 H n (7) a rather elegant form that is tantalizingly reminiscent of the energy-time uncertainty relation. Quantum enhancement to Xn is achieved by entangling the probe states. Maximally entangling states maximizes the QFI. EFor example, consider a 2-level system with a state n n ψ(N) / 2. This is a maximally entangled ai bi 0 state at the input of a generic Ramsey set-up [Fig.5]. The QFI for this state scales with the degree of entanglement as N,
4 giving the scaling 1 1 , Xn & N r nN (8) where n is the number of parallel or independent uses of the channel. r nN quantifies the concept of “resource scaling”: typically, the total amount of resources available to a measurement—to be divided between the degree of entanglement N and the number of independent trials n—is capped (for instance, in quantum-enhanced interferometry, this could be a limit on optical power). Figure 6. [20] (a)-(c) Visualizing decoherence models of 2-level atomic systems illustrated by their effect on the Bloch sphere: (a) depolarization; (b) dephasing; (c) spontaneous emission; parameter φ is the rotation about the z-axis, and “shrinkage” η quantifies the decoherence; (d) Lossy interferometer with power loss η in each arm. In all cases, the final precision bounds found via the upper bound on QFI is a function of η. D. C. The Heisenberg Limit The 1/N scaling in equation (8) is commonly referred to as the Heisenberg Limit, improving upon the SQL by a factor of N. A “Ramsey scheme” of phase estimation involving 2-level systems is described in Fig.5; this is a prototype that illustrates how the HL is attained by quantum enhancement of the input probe, deployment of the probes across parallel channels, and a local measurement at the output. In practice, several experimentally demonstrated architectures in atomic physics are capable of realizing the Ramsey scheme by generating entangled inputs. These include spinsqueezed states and Schrodinger cat states which are generated in diverse platforms including BECs, trapped ions, and NMR among others [1, 5, 6]. Other quantumly-correlated probe states beating the SQL include correlated Fock states in matter-wave interferometry for phase estimation [7]. Additional examples of systems that have experimentally achieved sub-shot-noise scaling are described in Pezze et al.[6], a comprehensive and recent review of quantum metrology using atomic-ensemble platforms. The translation of the Ramsey scheme to the optical domain is not experimentally straightforward [1, 8]. The optical analog of entangled qubit states that saturate the HL scaling are the celebrated NOON states, featuring entanglement between Fock states and the vacuum: ( Ni 0i 0i Ni) / 2. Notwithstanding 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 with the help of ancilla probes; these would need ( N)-partite entanglement to reach the HL (for instance, in [10]), suggesting that parallel and sequential schemes could be combined for the ideal measurement strategy. Heisenberg Limit: two approaches The HL is often defined as 1/N scaling, where N may be the average photon number (or degree of entanglement) in optical (or atomic) phase estimation problems. However, Boixo et al. proposed [11]that X may indeed scale by 1/N 2 , using sequential schemes. One proposal [12] considers probing a system with Kerr nonlinearity using a coherent state (1/N 3/2 ). Roy and Braunstein [13] showed that, in fact, with the full amount of entanglement in the available Hilbert space, X 1/2N . Pioneering experiments have also accomplished 1/N 1 scaling in parameter estimation, notably, Napolitano et al. achieved 1/N 3/2 using nonlinearities in atomic ensembles introduced by Faraday rotation pulses [14]. All of this begs the question, can the HL be surpassed? Not if the definition of the HL is extended beyond simply 1/N scaling. As pointed out by Zwierz et al. [15], the hamiltonians of nonlinear systems do not scale linearly with power. In the Kerr case, hHi N 2 . From the definition of the QFI and QCRB in equation (7) in terms of the variance of the Hamiltonian, one obtains the scaling X 1/N 2 , the true HL for this system. Re-interpreting the HL as a bound on the energy uncertainty reconciles the 1/N surpassing. As outlined in [15], philosophically, this reinforces the “resource-scaling” concept of precision measurement, where it is not enough to know how the precision scales with power (N), but how the resource “cost” (say, the number of queries/interactions/independent probes) does too. Figure 7. [20] Generic dependence of quantum-enhanced parameter estimation as a function of independent probes N in the presence of decoherence. For small N, the system retains HL scaling, but asymptotically veers to the classical SNL scaling for increased N, albeit with some “const” enhancement factor.
5 III. QFI AS A FRAMEWORK IN NOISY QUANTUM METROLOGY Lastly, the QFI plays a vital role in determining precision bounds in lossy or open quantum systems. While a rigorous mathematical analysis of this topic [16–19], is beyond this paper’s scope, I present a brief recipe. Instead of unitary operations, the quantum channel now features non-unitary transformations of the input probe, generalized by Kraus operators Πl (x, η), which contain informa- [1] V. Giovannetti, S. Lloyd, and L. Maccone, Nature photonics 5, 222 (2011). [2] V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004). [3] S. L. Braunstein and C. M. Caves, Physical Review Letters 72, 3439 (1994). [4] R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, in Progress in Optics, Vol. 60 (Elsevier, 2015) pp. 345–435. [5] J. Huang, S. Wu, H. Zhong, and C. Lee, in Annual Review of Cold Atoms and Molecules (World Scientific, 2014) pp. 365– 415. [6] L. Pezze, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Reviews of Modern Physics 90, 035005 (2018). [7] B. Lücke, M. Scherer, J. Kruse, L. Pezze, F. Deuretzbacher, P. Hyllus, J. Peise, W. Ertmer, J. Arlt, L. Santos, et al., Science 334, 773 (2011). [8] M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, Nature Photonics 4, 357 (2010). [9] S. Daryanoosh, S. Slussarenko, D. W. Berry, H. M. Wiseman, and G. J. Pryde, Nature communications 9, 1 (2018). [10] W. Wang, Y. Wu, Y. Ma, W. Cai, L. Hu, X. Mu, Y. Xu, Z.-J. Chen, H. Wang, Y. Song, et al., Nature communications 10, 1 tion about the parameter X and the specific loss (η) proP cess (Fig.6) [20]. The probe evolves to ρ x l Πl ρ0 Π†l . The QFI is then upper bounded by the computable quantity i h P Π† l C(ρ0 , Πl ) 4 hH1 (x)i hH2 (x)i2 , where H1 (x) l xl Π x , P Π†l and H2 (x) i l x Πl [16]. In the lossless case, this reduces to equation (7). These theoretical bounds have been calculated in depth [17], but Fig.7 [20] represents the general impact of noise on a precision measurement. All considered, the QFI continues to be an indispensable tool in deciphering the possibilities of quantum metrology. (2019). [11] S. Boixo, S. T. Flammia, C. M. Caves, and J. M. Geremia, Physical review letters 98, 090401 (2007). [12] J. Beltrán and A. Luis, Physical Review A 72, 045801 (2005). [13] S. Roy and S. L. Braunstein, Physical review letters 100, 220501 (2008). [14] M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. Sewell, and M. W. Mitchell, Nature 471, 486 (2011). [15] M. Zwierz, C. A. Pérez-Delgado, and P. Kok, Physical review letters 105, 180402 (2010). [16] B. Escher, R. de Matos Filho, and L. Davidovich, Nature Physics 7, 406 (2011). [17] J. F. Haase, A. Smirne, S. Huelga, J. Kołodynski, and R. Demkowicz-Dobrzanski, Quantum Measurements and Quantum Metrology 5, 13 (2016). [18] X. Feng and L. Wei, Scientific Reports 7, 1 (2017). [19] S. P. Nolan and S. A. Haine, Physical Review A 95, 043642 (2017). [20] R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, Nature communications 3, 1 (2012).
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,
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