Quantum metrology with one auxiliary particle in a correlated bath and its quantum simulation Wan-Ting He,1,∗ Huan-Yu Guang,1,∗ Zi-Yun Li,1,∗ Ru-Qiong Deng,2,1 Na-Na Zhang,1 Jie-Xing Zhao,1† Fu-Guo Deng,1 and Qing Ai,1,‡ Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China 2School of Physics and Electronics, Hunan Normal University, Hunan 410006, China (Dated: May 24, 2021) In realistic metrology, entangled probes are more sensitive to noise, especially for a correlated environment. The precision of parameter estimation with entangled probes is even lower than that of the unentangled ones in a correlated environment. In this paper, we propose a measurement scheme with only one auxiliary qubit, which can selectively offset the impact of environmental noise under this situation. We analyse the estimation precision of our scheme and find out that it approaches the Heisenberg limit when prepared in a proper auxiliary state. We further discuss employing auxiliary states to improve the precision of measurement in other environment models such as a partially-correlated environment. In order to verify our scheme, we apply a recently- developed quantum algorithm to simulate the quantum dynamics of our proposal and show that it outperform the other proposals with less resources. I. INTRODUCTION all, the quantum dynamics of open quantum systems are usually classified into Markovian and non-Markovian [17–22]. When the system-bath couplings are relatively Quantum metrology employs quantum entanglement large, or the number of degrees of freedom in the envi- and coherence to achieve an ultra-high precision for the ronment is not sufficiently large, e.g. natural photosyn- estimation of an unknown parameter [1, 2]. It has be- thetic complexes and NV centers in diamond, the open come an indispensable element of satellite navigation, quantum systems are subject to non-Markovian quan- aerospace measurement and control, mobile phone and tum dynamics [4, 23]. Using entangled probes in a non- computer chip processing. This opens a broad range of Markovian quantum dynamics allows for a higher mea- valuable applications of quantum mechanics, in addition surement precision than that in a Markovian quantum to quantum information processing [3], photosynthetic dynamics, which scales as n−3/4 [24]. On the other hand, exciton energy transfer [4, 5], avian magnetoreception when particles interact with a correlated environment, [4, 6–8] and quantum metamaterial [9–12]. As a result e.g. nuclear spins in a molecule, the decoherence rate per of the central-limit theorem, the estimated precision of particle will increase linearly with the number of parti- an unknown quantity critically depends on the number cles, i.e., superdecoherence [25]. In this regime, utilizing of resources available for the measurement. One of the entangled probes will no longer outperform unentangled primary goals of quantum metrology is to enhance the ones no matter the open quantum dynamics is Markovian precision of resolution with limited resources. An en- or non-Markovian. hanced resolution can be achieved if quantum entangle- ment is used to correlate the probes before making them In order to overcome decoherence in an uncorre- interact with the system to be measured [13, 14]. Taking lated bath, logical states are introduced to establish N advantage of n entangled quantum probes, one can at- decoherence-free subspace [26–28]. However, since log- tain the Heisenberg limit (HL) which scales as n−1, being ical qubits require n =2N physical qubits, it doubles the number of valuable resources used per experiment. In or- arXiv:2105.10260v1 [quant-ph] 21 May 2021 the ultimate limit in precision set by quantum mechan- ics. And this result has been demonstrated experimen- der to effectively reduce the usage of resources, here we tally [15, 16]. On the contrary, when using unentangled propose a measurement scheme with only one auxiliary probes, one can only reach the standard quantum limit qubit in a correlated bath. Using such a well-designed (SQL) which scales as n−1/2. Obviously, the use of en- auxiliary particle in quantum metrology can selectively tanglement can significantly enhance the precision when offset the impact of noise in a correlated environment. n is large. Comparing with previous proposals, we show that al- though we use less resources, we can still approach the In realistic scenario for experiments, quantum probes HL when preparing proper auxiliary states. are inevitably affected by noise. The achievable preci- sion decreases due to the decoherence. As known to On the other hand, although the theoretical scheme utilizing entanglement in a non-Markovian environment is appealing, it might be difficult to experimentally ver- ify it since it critically requires the homogeneity of the ∗These authors contributed equally to this work. qubits. For example, for NV centers in diamond, differ- †[email protected] ent spins manifest different Zeeman energies and decoher- ‡[email protected] ence rates in the same environment due to the inhomoge- 2 neous gyromagnetic ratios. Recently, based on the bath- When the entangled probes are in uncorrelated environ- engineering technique [29, 30] and the gradient ascent t ment, we can obtain Γn(t)= n 0 dτγ(τ)= nΓ(t), where pulse engineering (GRAPE) algorithm [31, 32], we theo- γ(t) and Γ(t) are respectively the´ dephasing rate and de- retically proposed and experimentally demonstrated that coherence factor for a single qubit. Thus, the uncertainty the open quantum dynamics with an arbitrary Hamilto- is explicitly written as [24, 37] nian and spectral density can be exactly and efficiently simulated [33, 34]. Thus, based on this algorithm, we 1 cos2 (φt) e−2nΓ(t) δφ2 = − . (4) show how to verify that our scheme can make the esti- nTt sin2 (φt) e−2nΓ(t) mation precision achieve the HL in a quantum simulation experiment [35, 36]. In order to attain the best precision, it is necessary This paper is organized as follows: Our measurement to optimize this expression of uncertainty against the scheme and its quantum dynamics for quantum metrol- duration of each single measurement t. The best in- ogy in a correlated bath are introduced in the next sec- terrogation time satisfies φte = kπ/2 with odd k and tion. Specifically, we offer an example of magnetic-field dΓ(t) 2nt t=te =1 [24], and thus yields sensing utilizing our measurement scheme. Then, in dt | Sec. III, we apply a recently-developed quantum algo- 2 1 2nΓ(te) rithm to simulate our measurement scheme with an aux- δφ e= 2 e , (5) | n Tte iliary qubit and show that it can approach the HL. In Appendix, we give a brief description for the open quan- where the subscript e indicates that the entangled probes tum dynamics of n qubits in a correlated bath and an are used. uncorrelated bath, respectively. When the entangled probes are in a correlated envi- ronment, the superdecoherence of the probes will modify the probability as II. MODEL AND DYNAMICS 1 2 P = 1+cos(nφt) e−n Γ(t) . (6) A. Quantum Metrology under Superdecoherence 2 h i The uncertainty of parameter φ reads In quantum metrology, using entangled probes can 2 −2n2Γ(t) obtain an increase in precision when assuming a fully- 2 1 cos (nφt) e δφ e = − . (7) coherent evolution, while in realistic scenario, there | nTt sin2 (nφt) e−2n2Γ(t) is always decoherence caused by environmental noise. 2 The optimal precision with entangled probes under By minimizing δφ e, i.e., requiring that φte = kπ/2 with decoherence was first analyzed in Markovian environ- 2 dΓ(t|) odd k and 2n t e =1, we have ment [37], and further discussed in non-Markovian en- dt |t=t vironment [24]. The best resolution in the estimation 2 2 1 2n Γ(te) was obtained with a given number of particles n and a δφ e = e . (8) | n2Tt total duration of experiment T , i.e., nT/t being the ac- e tual number of experiment. The entangled probes are When utilizing unentangled probes, the uncertainty of prepared in an n-qubit GHZ state ( 0 ⊗N + 1 ⊗N )/√2, parameter is δφ2 = 1 exp[2Γ(t )], and the best in- | i | i |u nT tu e and then evolve freely for a duration t. In the ideal terrogation time is given by φtu = kπ/2 with odd k and case, we have ( 0 ⊗N + exp( inφt) 1 ⊗N )/√2, where φ dΓ(t) 2t u =1 [24]. Following Ref. [24], we define is the parameter| i to be estimated.− However,| i in the pres- dt |t=t ence of dephasing noise, the final state becomes 0 ⊗N + δφ u ⊗N t{| i r = , (9) exp[ inφt Γn(t)] 1 /√2, with Γn (t)= dτγn(τ), | − − | i } 0 δφ e and γn(t) being the dephasing rate of n qubits.´ After a | π/2 pulse, the probability of finding these probes in the where δφ e and δφ u are the standard deviation of the initial state reads estimated| | parameter| | φ when using entangled and un- 1 entangled probes, respectively. Here, r characterizes P = 1 + cos(nφt)e−Γn(t) . (1) 2 the improved precision of measurement for using entan- h i gled probes instead of unentangled ones. For entan- The uncertainty of the measurement can be calculated as gled probes in an uncorrelated environment, we obtain 2 r = n(te/tu) exp[2Γ(tu) 2nΓ(te)], while for entan- 2 1 δφ = , (2) gled probes in a correlated− environment, we have r2 = (nT/t) F (φ) n(t /t ) exp[2Γ(t ) 2n2Γ(t )]. Obviously, r changes e u u − e where the Fisher information [38, 39] is along with the dependence of function Γ(t) on time [24], i.e., the dynamics of decoherence.