Coherent measurements in quantum metrology

It is well known that a quantum correlated probe can yield better precision in estimating an unknown parameter than classically possible. However, how such a quantum probe should be measured remains somewhat elusive. We examine the role of measurements in quantum metrology by considering two types of readout strategies: coherent, where all probes are measured simultaneously in an entangled basis; and adaptive, where probes are measured sequentially, with each measurement one way conditioned on the prior outcomes. Here we firstly show that for classically correlated probes the two readout strategies yield the same precision. Secondly, we construct an example of a noisy multipartite quantum system where coherent readout yields considerably better precision than adaptive readout. This highlights a fundamental difference between classical and quantum parameter estimation. From the practical point of view, our findings are relevant for the optimal design of precision-measurement quantum devices.

It is well known that a quantum correlated probe can yield better precision in estimating an unknown parameter than classically possible. However, how such a quantum probe should be measured remains somewhat elusive. We examine the role of measurements in quantum metrology by considering two types of readout strategies: coherent, where all probes are measured simultaneously in an entangled basis; and adaptive, where probes are measured sequentially, with each measurement conditioned on the prior outcomes. Here we firstly show that for classically correlated probes the two readout strategies yield the same precision. Secondly, we construct an example of a noisy multipartite quantum system where coherent readout yields considerably better precision than adaptive readout. This highlights a fundamental difference between classical and quantum parameter estimation. From the practical point of view, our findings are relevant for the optimal design of precision-measurement quantum devices.
Introduction.-Quantum mechanical systems can be used to outperform classical ones in information processing [1]. Quantum correlations can be employed to beat the shot-noise (standard) limit in metrology protocols. Such parameter estimation methods are crucial for both theoretical advances and the development of technologies. However, almost all quantum technologies operate with some level of noise and how quantum enhancement fares in the presence of noise is still unclear. Another intrinsic quantum feature is the measurement process, which, in general, disturbs the system being measured and the outcome depends on a basis choice. Here, we explore this distinction to uncover a difference between coherent and adaptive readout strategies in a quantum metrology protocol in the presence of noise.
A general framework to estimate a parameter involves a suitable probe and an interaction that physically manifests the parameter. The probe, initially in state , acquires some information about the parameter, φ, yielding the encoded state φ , which is then read out by some convenient strategy. The estimation process depends on how much information about the parameter is encoded in the probe. The precision of the estimation protocol, which can be saturated in a large number of trials, is limited by the Cramér-Rao relation [2][3][4], in which the root mean square error, ∆φ, associated with an unbiased estimator, is bounded by the Fisher information [5], F, * kaonan.bueno@ufabc.edu.br † dar55@cam.ac.uk ‡ felix.pollock@gmail.com § lucas@chibebe.org ¶ serra@ufabc.edu.br * * kavan.modi@monash.edu as ∆φ ≥ 1/ √ F. Fisher information is a key concept in metrology and gives us knowledge about the effectiveness of a parameter estimation protocol. Measuring the encoded probe state affords the probability distribution p φ (x), from which F can be computed directly from its definition, . The quantum Fisher information (QFI) is defined as the maximum of F over all possible measurements, and to attain it we need to employ a readout procedure that yields an appropriate distribution p φ (x).
A valuable ingredient in parameter estimation is the use of correlated probes, which can be employed to improve the estimation. For instance, in the quantum case nonclassical correlations offer considerable advantages in quantum metrology [6][7][8] even in noisy scenarios [9][10][11]. Such studies are tractable because QFI can be computed using only the initial probe state and the generator of the encoding [12][13][14][15][16]. In this Letter we address the need for correlations in the readout procedure, that is, the need for entangling measurements to produce a suitable p φ (x). Specifically, we show that for some correlated probe states a coherent quantum readout procedure yields a better estimate for the parameter of interest than an adaptive classical readout procedure. We begin by defining coherent and adaptive measurements below. However, for simplicity's sake we only consider a bipartite probe for now and address the general multipartite case at the end of the Letter.
Coherent and adaptive measurements.-Let us consider the following game between three characters: Alice, Bob, and Charlie. They receive the same bipartite encoded probe φ . However, Charlie has access to the whole state, while Alice and Bob have access only to the local partitions A and B, respectively. We suppose that Charlie has an apparatus able to perform joint measure-ments on the whole state yielding a bipartite probability distribution p φ (a, b). The precision that Charlie can attain about φ is bounded by Fisher information with a and b being the outcomes associated with partitions A and B, respectively. Alice and Bob are allowed to communicate and perform any operations on their own partitions. To measure φ with optimal precision Alice and Bob can employ an adaptive strategy. First, Bob performs a suitable measurement on his partition and observes outcome b with probability q φ (b). He then communicates his result to Alice, and, based on that Alice performs a measurement on her partition observing outcome a with probability q φ (a|b). Putting their outcomes together, the precision with which Alice and Bob can attain φ is bounded by the Fisher information of the joint probability distribu- (1).) Alternatively, the precision with which Bob can attain φ is bounded by Bob's Fisher information Similarly, Alice's precision is bounded by the conditional Fisher information for the conditional distribution with F(A|B = b) ≡ a q φ (a|b) [∂ φ ln q φ (a|b)] 2 being the Fisher information for Alice conditioned on Bob's outcome b. Together their precision of φ is bounded by We are now ready to formally state our problem: we consider a challenge by Charlie to Alice and Bob to attain the same precision for φ as he does using the coherent strategy. When Alice and Bob are able to meet Charlie's challenge, no quantum resources are necessary for the readout. We will show below that this is indeed the case in classical metrology and even N 00N or the equivalent N −qubit Greenberger-Horne-Zeilinger (NGHZ) state (both to be defined below) metrology in the absence of noise. We then show that with the introduction of noise there is a gap in the achievable precision between the two readout strategies. We have graphically depicted the two readout strategies in Fig. 1.
Classical metrology.-In classical metrology, the state of the probe is simply a probability distribution p 0 (a, b) and the encoded state is also a probability distribution p φ (a, b). Classical probabilities satisfy FIG. 1. Two readout strategies for parameter estimation. In (i) the coherent readout process consists of applying an entangling operation U followed by measuring both partitions. In (ii) Bob measures his partition and observes outcome b, which he communicates to Alice. Next, Alice measures her partition, conditioned on Bob's outcome, and observes outcome a|b. For classical metrology, (i) yields the left-hand side of Eq. (5) and (ii) yields the right-hand side of Eq. (5). The key ingredient for coherent readout is the ability to perform an entangling unitary transformation U .
The left-and right-hand sides of Eq. (4) correspond to Charlie's coherent readout, and Alice and Bob's adaptive readout, respectively. This means that for classical metrology we have the following well-known additivity relation for Fisher information [4] F(A, B) = F(B) + F(A|B).
The implication is that Alice and Bob are able to attain the same precision as Charlie for φ and thus meet Charlie's challenge. We have given a full derivation of Eq. (5), including the multipartite case, in [17]. Quantum metrology.-In the quantum setting, the discrete probability distribution p φ (x) is now obtained from a positive operator-valued measure (POVM) Π with elements {Π x } acting on the encoded quantum state φ : Remarkably, there is a simple formula that yields the optimal Fisher information over all POVMs [12]: where λ i and |ψ i are the eigenvalues and eigenvectors of respectively, while the Hamiltonian H is the Hermitian generator of the unitary encoding U = e −iHφ , where φ = U U † . This formula allows us to compute the QFI without having to worry about the measurements in the readout phase. The implication here is that there exists a POVM that will yield the optimal Fisher information, but we do not know much about it -we shall return to this point later.
Let us now go back to the game between Charlie versus Alice and Bob. Again the rule is that Charlie is able to make operations with entangling power, while Alice and Bob do not possess any entangling power. We will shortly show that Charlie's coherent measurement allows better precision in estimating φ than Alice and Bob's adaptive measurements.
Types of measurements.-To attain the optimal Fisher information requires, in general, an entangling operation involving all parts of the probe. We will refer to such a measurement as a coherent measurement. We want to compare the coherent measurement strategy to a measurement strategy where we have no entangling power. Such a strategy then simply involves probabilistic-projective measurements on qubits. We will refer to such a measurement as an adaptive measurement. It is easy to see that F co ≥ F ad since the set of coherent measurements contain the set of adaptive measurements. In this Letter we give examples where F co > F ad , which proves that in general entangling power in the readout phase of quantum metrology is necessary for optimal estimation. Also see [18] for a hierarchy of measurements for QFI.
Parameter dependence of measurements.-QFI in Eq. (6) does not tell us much about the optimal POVM. The derivation of Eq. (6) does identify the necessary and sufficient condition for the optimal measurement, namely that for constant k x ∈ R and L φ is the symmetric logarithmic derivative defined by ∂ φ φ = 1 2 ( φ L φ + L φ φ ) [12]. Although there is no known general solution for the POVM elements Π x , a sufficient condition is that they are projectors onto eigenspaces of L φ . This construction yields measurements that are in general coherent and also local in parameter space (parameter dependent).
We do not know whether optimality can be attained with an adaptive measurement nor do we know whether there exists a globally optimal (parameter independent) measurement. The first issue is inconvenient as we consequently have to search for optimal adaptive measurements on a case-by-case basis, but the second is of purely theoretical interest and of little importance in practice for the following reason. The parameter φ is estimated from a large number of measurements, M , so we can employ an adaptive strategy in which M (M M ) suboptimal measurements are performed to estimate φ, after which we can fix the measurement to be optimal at the approximate parameter valueφ. It is a well known result of classical statistics that a maximum likelihood estimator (MLE) for φ saturates the Cramér-Rao bound as M → ∞, but we can only construct an approximate MLE since we have used a measurement that is optimal atφ as opposed to φ. Fortuitously, it is nonetheless found to share the same asymptotic properties as the true MLE, provided that M is sufficiently large, meaning that globally optimal measurements confer no estimation advantage in the limit of large M [19,20].
In fact, we can reach a useful conclusion regarding the possibility of global optimality: when φ is full-rank (and thus strictly positive) for all φ, we show in [17] that no globally optimal measurement exists. Moreover, noise perturbs any zero eigenvalues consequently making the result applicable to any practically realizable state.
Classically correlated quantum states.-Let us first prove that classically correlated probes do not require any entanglement in the readout. Suppose we use as probe a bipartite (or multipartite) quantum state that is classically correlated: = ab q(a, b) |ab ab| which is which is the definition of a classically correlated state [21,22]. However, there exists a measurement Π that yields the outcome probabilities p a φ b φ = tr[Π φ ] leading to optimal QFI. Using Eq. (8) and the cyclic invariance of the trace we get which is a measurement that can be applied in the adaptive form.
NGHZ and N00N states.-Before we give an example exhibiting a gap between the coherent and adaptive Fisher information, let us discuss the important case of N 00N states. A N 00N state is an optical state of N photons in superposition with the vacuum in two arms of an interferometer: (|N 0 + |0N )/ √ 2. For linear encoding, N 00N states saturate the ultimate bound for Fisher information, the Heisenberg limit N 2 [6].
Here we will work with N −qubit GHZ (NGHZ) states instead of N 00N states, but in principal the two are equivalent [23][24][25][26]. An NGHZ state before the encoding is where all qubits are in the state |0 in superposition with all qubits in state |1 . It is written as Here, |0 and |1 are the eigenstates of the Pauli matrix σ z .
The coherent readout of the latter state requires a series of C-not gates between the first qubit and the remaining N − 1 qubits, resulting in the state (1/ √ 2)(|0 + e iN φ |1 ) ⊗ |0 ⊗N −1 . The first qubit can now be measured in the |± = (|0 ± |1 ) / √ 2 basis and the desired Fisher information is achieved. However, the same result is achieved adaptively by measuring each qubit in the |± basis. After N − 1 qubits are measured the state of the remaining qubit is (|0 + (−1) k e iN φ |1 )/ √ 2, where k is the number of times the outcome |− was observed. If k is odd we apply a σ z operation, but no correction is required otherwise.
Strangely, this example shows that coherent processing is unnecessary for readout in metrology with NGHZ states. It seems that for pure probes the adaptive strategy could be equivalent to the coherent strategy, as hinted at in [6]. We were unable to prove this, nor able to find a counterexample. However, in a real experiment one never has a pure state, so we will now introduce the addition of noise to the problem.
Werner state.-Let us now choose the probe as the Werner state W = (1 − η) 1 The parameter η ∈ [0, 1] is the strength of the signal, while 1−η indicates the amount of white noise present in the probe state. Using Eq. (6) we can compute the optimal Fisher information that Charlie can attain.
Now, we need to compare this result to the adaptive strategy. Bob's local Fisher information vanishes because his local state is maximally mixed: He gets completely random bits for any POVM he implements. Furthermore, we can fine-grain the POVM to a projector onto the state |β The conditional state of Alice is as if Alice had prepared a state W A|β = (1 − η) 1 1 2 + η |β β| and sent it through an interferometer with the phase generated by the Hamiltonian 2 |1 1|. Therefore we can simply compute the QFI using Eq. (6).
We know that the QFI will be maximized for b 0 = b 1 = 1/ √ 2, therefore we can conclude that Bob will make measurements |± . A σ z operation is applied to Alice's state when Bob's outcome is minus to make the two conditional states the same. The adaptive Fisher information is Note that there is a finite difference between the coherent readout and adaptive readout. The difference vanishes only when η = 0 or 1. Therefore, in this example, Charlie's ability to perform coherent interactions plays a non-trivial role in phase estimation in the presence of noise. This is our central result, but we will discuss its consequence after we generalize to the case of N qubits.
Multipartite states.-We can represent a multipartite probe with white noise as W N = (1 − η) 1 The phase shift is now introduced by the unitary U ⊗N with U = e iφ|1 1| . Charlie's Fisher information is computed using Eq. (6) (see [17] for the details): For the adaptive strategy, once again, the local Fisher information for any party is null since the local states are maximally mixed. The sequence of adaptive measurements on the space spanned by the N -parties is equivalent to Alice applying N times the phase on her qubit, resulting in the conditional Fisher information: which is greater than zero for a noisy probe (η = 0, 1). The derivation of Eq. (13) can be found in [17]. Analysis and conclusions.-Above, the coherent strategy offers a quadratic enhancement over the adaptive strategy in η. The ratio of the two Fisher information amounts to the number of times the adaptive strategy has to be repeated to match the precision attained by the coherent strategy. Even for a handful of qubits we have 1 − η 2 N −1 and For highly mixed states with a very small value of η, the adaptive readout performs extremely poorly compared to the coherent readout. This has huge implications for magnetic field sensing in the nuclear magnetic resonance setup [27,28], where η ≈ 10 −5 and implies more than three-hundred times better precision in φ due to coherent operations. Interestingly, the gap in the Fisher information is reminiscent of the gap in the Holevo quantity between coherent decoding versus adaptive decoding, which is shown to be quantum discord in [29]. There are several characteristic traits of quantum mechanics that distinguish it from the classical theory: besides non-classical correlations like entanglement and discord, the possibility of performing coherent interactions between different partitions of the probe does not have a classical analogue [30,31].
In this Letter we showed that, in general, coherent readout leads to better precision over adaptive readout in quantum parameter estimation. We also showed that the two readouts are equivalent for classical probe states. Finally, the noise in some quantum correlated probes can be quadratically suppressed with the use of quantum coherent operations, leading to better precision for parameter estimation. In this manner we have highlighted the importance of coherent measurements in quantum metrology in the presence of quantum correlations.
Acknowledgments.-LC, KMi, and RS are grateful for the financial support from CNPq, CAPES, and INCT-IQ. KMi and RS are grateful for the financial support from FAPESP. DR gratefully acknowledges the hospitality of the University of Oxford, and the Institute of Physics and Nuffield Foundation for financial support. FAP thanks the Leverhulme Trust for financial support. The John Templeton Foundation, the National Research Foundation, and the Ministry of Education of Singapore supported KMo during the completion of this work. KMo thanks UFABC for their hospitality.
FIG. 2. Two readout strategies for parameter estimation for the multipartite case with probe state W N . In (i) the coherent readout process consists of applying a series of C-not gates, which are entangling operations, followed by measuring all partitions. In (ii) each party measures their partition in the |± basis and communicates the outcome to the first party. The first party applies a σz if the number of |− outcomes was odd. Finally, she makes a measurement on her qubit. The key ingredient for coherent readout is the ability to perform an entangling unitary transformation, which is the C-not gate here.

Appendix C: Coherent Fisher information for W N
Here we wish to compute the quantum Fisher information (QFI) with coherent readout for the following state where To do this we need the eigenvectors and eigenvalues of the state above. Eigenvector |G N comes with eigenvalue η + (1 − η)/2 N and all other 2 N − 1 eigenvectors come with eigenvalue (1 − η)/2 N . Next note that the Hamiltonian that encodes the parameter to be estimated is Now we simply utilize Eq. (6) in main text and compute QFI. We begin by noting that the eigenstates with the same eigenvalues do not contribute and the action of the Hamiltonian on |G N is (1/ √ 2 |1 ⊗N ). Therefore the only other eigenvector that matters is |Ḡ N = (|0 This is because all other eigenstates of N are orthonormal to the |1 ⊗N term. The QFI is then: Let us imagine that the last party has a measurement outcome along some direction m| = m 0 0| + m 1 1|, with |m 0 | 2 + |m 1 | 2 = 1. The corresponding conditional state of the remaining qubits is We want to find values of m 0 and m 1 that maximize the Fisher information of the last state. We note that the last state is exactly as if we had prepared the state of N − 1 and sent it through a process generated by the Hamiltonian N N −1 i |1 1|. The coherent QFI of the last state can be computed to be which is maximum for m 0 = m 1 = 1/ √ 2. Therefore we can conclude that the first measurement is in the |± basis. The resultant conditional state of the N − 1 qubits is But this state looks exactly like the N qubit state we started with in Eq. (D1). Therefore by carrying out the exact same analysis we find that the next measurement also has to be in the |± basis. After all but the last party has measured the final qubit has the state Let us denote the number of parties that observe |− with k. If k is odd then a σ z is applied to change the minus sign in the superposition. This state is exactly as if we had prepared the state and sent it through a process generated by Hamiltonian N |1 1|. The Fisher information for this Hamiltonian and the last state is the Fisher information for adaptive readout: