Robust quantum metrological schemes based on protection of quantum Fisher information

Fragile quantum features such as entanglement are employed to improve the precision of parameter estimation and as a consequence the quantum gain becomes vulnerable to noise. As an established tool to subdue noise, quantum error correction is unfortunately overprotective because the quantum enhancement can still be achieved even if the states are irrecoverably affected, provided that the quantum Fisher information, which sets the ultimate limit to the precision of metrological schemes, is preserved and attained. Here, we develop a theory of robust metrological schemes that preserve the quantum Fisher information instead of the quantum states themselves against noise. After deriving a minimal set of testable conditions on this kind of robustness, we construct a family of $2t+1$ qubits metrological schemes being immune to $t$-qubit errors after the signal sensing. In comparison at least five qubits are required for correcting arbitrary 1-qubit errors in standard quantum error correction.

In quantum metrology, delicate and fragile quantum features are being used to enhance the sensitivity of experimental apparatus, e.g., non-classical probe states were used for the high sensitivity of optical interferometer and atomic spectroscopy [1][2][3][4][5][6][7][8]. However the quantum enhancement for the sensitivity may be compromised by the presence of ubiquitous and inevitable noises [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Therefore it is of utmost significance to investigate the robustness of the optimal strategies for the high sensitivity against noise. Most recently, quantum error correction (QEC) was employed in quantum metrology [27][28][29][30] to overcome noise problem, where by protecting the quantum states, on which a signal parameter is imprinted, the measurement precision for that parameter is protected. This is no wonder because the standard QEC was originally designed for protecting all the information encoded in quantum states, i.e., the logical states in the universal quantum computation, against the noise [31][32][33][34][35][36]. In quantum metrology, however, what matters essentially is the distinguishability about the signal parameter that is sensed by quantum systems and encoded in quantum states. According to quantum estimation theory [37][38][39] this distinguishability is measured by quantum Fisher information (QFI). Therefore, preserving the QFI of a given family of states against noises is sufficient for quantum metrological schemes to work under noisy environment. Since the QFI represents only partial information encoded in quantum states, the use of the QEC for quantum states is obviously overprotective, which leads to unnecessary waste of resources. Our main goal is to establish a variant theory of QEC designed for quantum metrology, namely, robust quantum metrological schemes, by taking the QFI instead of the fidelity of quantum states as the figure of merit.
A standard quantum metrological scheme to detect and estimate a signal parameter θ can be depicted by the following sensing transformation ρ → ρ θ = e −iθH ρe iθH with H being a known Hermitian operator and ρ the probe state. The value of the parameter is estimated through the classical data processing on the measurement outcomes obtained by repeating experiments in ρ θ . From estimation theory [37][38][39], the regularized rootmean-square error of the estimatorθ is limited by the Cramér-Rao bound where ν is the number of repetitions of the experiments, and is the (classical) Fisher information extracted by the measurement {M x } with p θ (x) := Tr(M x ρ θ ) being the probabilities of obtaining outcomes x. Here, M x are positive operators satisfying x M x = 1 with 1 being the identity operator. The maximal Fisher information over all possible measurements is given by the so-called QFI F (ρ θ ) := Tr(ρ θ L 2 θ ), where the symmetric logarithmic derivative (SLD) operator L θ is defined as the Hermitian operator satisfying dρ θ /dθ = 1 2 {L θ , ρ θ } with {·, ·} being the anti-commutator [37][38][39][40][41]. More importantly, the Cramér-Rao bound is asymptotically achieved [37,38], therefore, QFI can be considered as a measure on the distinguishability about the parameter in the parametric family of quantum states.
An optimal strategy of the quantum parameter estimation comprises the probe state maximizing the QFI and the measurement attaining the maximal Fisher information. Taking appropriate entangled states as the probe states, a quantum metrological scheme may achieve the Heisenberg scaling of precision 1/N , where N is the number of resources employed in the experiment, e.g., the number of probes [6][7][8]. This is a considerable improve-arXiv:1405.4052v2 [quant-ph] 16 Jun 2014 Abstract model for quantum parameter estimation. a, Set-ups for entanglement-enhanced metrology with noise being assumed after the sensing transformation U ⊗n θ , where n is the number of the qubits. b, Heisenberg-limited metrological scheme with the QEC protection. The entangling operation (labeled with "Entanglement" in the figure) acts only on the first qubit (thick line) in each block, and produce the Greenberger-Horne-Zeilinger state. The encoders (labeled with "Enc") encode the first qubit in each block into a phase-flip code. The sensing transformation U θ is a logical phase-shift operation on each phase-flip code space. Here, we note that the "Entanglement" operation plays a dual role: on one hand, it supplies the Heisenberg-scaling of the QFI, and on the other hand, it supplies a higher level of bit-flip code in addition to the phase-flip code. ment over the standard quantum limit 1/ √ N . Nevertheless, those entanglement-enhanced strategies that are optimal for the noiseless systems, easily lose the quantum gain for the noisy systems [9,11,12,[14][15][16][17][18][19][20][21][22][23][24][25][26].
We now turn to the question of the optimal strategy in noisy cases. We assume that the noise can be deferred until after the sensing transformation, i.e., states to be measured are N (ρ θ ) = j E j ρ θ E † j where N denotes a noisy channel with Kraus operators {E j }. We emphasize that this noise model is applicable to those noises that commutes with the generator of the sensing transformation, occurs during the transmission or storage in the interval between the sensing and the measurement, or induced by the measurement imperfection. This noise model can also be considered as an approximation of the noisy process of sensing a weak signal parameter, namely, θ 1. A general entanglement-enhanced metrology scenario of this type is depicted in Fig 1a. At first glance, the optimal strategy for such noisy cases might be established by seeking the optimal probe states maximizing the QFI of N (ρ θ ) and the corresponding optimal measurements. Technically, this straightforward optimization needs to diagonalize the parametric family of states N (ρ θ ), which is often formidable and even impossible without the details of the noises. Therefore, the optimal strategies obtained in this way are very restricted.
Based on the above considerations, protecting the involved parametric family of states with quantum errorcorrecting codes [31][32][33][34][35], which is applicable for the whole class of noisy channels with the operation elements being arbitrary linear combinations of the correctable error elements, is a good candidate of a robust strategy for quantum metrology [27][28][29][30]. However, QEC is overprotective because the measurement precision remains the same as long as the QFI is preserved and attained, even if the quantum states might be affected by some uncor-rectable errors. Here we shall develop a theory of QEC specialized for metrology: on the one hand our theory aims at preserving the QFI instead of all the information encoded in states, which ensures that our robust quantum metrological schemes are not overprotective; on the other hand our specialized theory also possesses the great advantage of the standard QEC that the errors can be digitalized for the preservation and attainment of QFI, which is our first main result: with L θ being the SLD operator for N (ρ θ ). If the QFI of ρ θ is preserved under a known noisy channel N , then it is preserved under all noisy channels whose Kraus operators are arbitrary linear combinations of {E j }, with the optimal measurement being the eigenstates of L θ .
We split the proof of Theorem 1 into three parts. First, we prove in the Methods that equation (1) is the necessary and sufficient condition on QFI-preserving under a known noisy channel. Second, for an arbitrary noisy channel K with Kraus operators {K j = i c ij E j } being unknown linear combinations of {E j }, we note that equation (1) still holds with E j being replaced by K j , therefore the QFI is preserved under the unknown noisy channel K. Third, it is known that the complete set of the eigenstates of the symmetric logarithmic derivative operator is the optimal measurement basis attaining the maximal Fisher information [39]. Through equation (1), the SLD operator L θ for N (ρ θ ) can be readily checked to be also the SLD operator for K(ρ θ ). Therefore, the measurement with respect to eigenstates of L θ is also optimal for K(ρ θ ). Remarkably, if the measurement basis is fixed, then a recovery operation is demanded to transform the basis of the optimal measurement to the fixed Geometrical picture for preserving QFI. On the manifold of pure states, there is a Riemannian metric known as the Fubini-Study metric [43,44]. Along the parametric states |ψ θ , a line element is given by ds(|ψ θ , |ψ θ+dθ ) = |D θ ψ θ dθ with |D θ ψ θ := (1 − |ψ θ ψ θ |)|dψ θ /dθ being the covariant derivative vector. This geometric metric is connected to parameter estimation theory in twofold: on one hand, the QFI is given by F (|ψ θ ψ θ |) = 4 |D θ ψ θ 2 ; on the other hand, the SLD operator L θ is a Hermitian representation of the covariant derivative as L θ |ψ θ = 2|D θ ψ θ , and the projective measurement with respect to eigenstates of L θ attains the maximal Fisher information. A set {Ej} of errors transforms parametric state vectors to Ej|ψ θ , while covariant derivative vectors to Ej|D θ ψ θ . The QFI is preserved under the errors if and only if there exists a Hermitian operator Q transforming all the erroneous state vectors to the corresponding erroneous covariant derivative vectors, i.e., QEj|ψ θ = 2Ej|D θ ψ θ for all j; this operator Q actually is the SLD operator for all the noisy states under the errors (See the Methods). The projective measurement with respect to eigenstates of Q attains the maximal Fisher information in noisy states.
one; otherwise, we only need to perform the optimal measurement for noisy states, and no recovery operation is demanded. In Fig. 2, we show a geometric picture to intuitively understand the conditions on the preservation of QFI.
Theorem 1 concerns the robustness of quantum parameter estimation with respect to noise. It suggests that there might be a probe state which does not maximize the QFI under a specific noisy channel but ensures the QFI to be preserved and attained under an entire class of noisy channels. The necessary and sufficient condition (1) on preserving the QFI against a set of errors needs the SLD operator L θ for the corresponding noisy state N (ρ θ ). Our second main result is the testable conditions on preserving QFI without referring to the SLD operators of the noisy state. These testable conditions are useful for finding good probe states for certain errors, or identifying those errors to which the QFI with certain probe state is immune.
for all j and k, and (ii) j α j E j |ψ θ = 0 for some α j ∈ C infers j α j E j L θ |ψ θ = 0. For mixed state ρ θ the QFI is preserved if and only if the above two conditions hold for all the states |ψ θ in the range of ρ θ .
The proof is sketched in the Methods (a full version is deferred to the Supplementary Information). For a unitarily parameterized family of pure states, |ψ θ = exp(−iθH)|ψ , by noting L θ |ψ θ = −2i∆H|ψ θ with ∆H := H − ψ|H|ψ , we simplify the two testable conditions into (i) for all j and k and (ii) j α j E j |ψ θ = 0 for some α j ∈ C infers j α j E j H|ψ θ = 0. The testable conditions describe the minimal requirements for the robustness of a parameter estimation scheme against noise, and are looser than that of QEC for the parametric family of states. Recall that a set {E j } of errors is correctable for a code space if and only if are satisfied for all j, k and all pairs of orthonormal state vectors |φ and |ϕ in the code space [35]. Let us choose |ψ and H to be in a standard quantum error-correcting code so that |φ ∝ |ψ θ and |ϕ ∝ L θ |ψ θ are two orthonormal states in the coding subspace. In such case, the first and second testable conditions are implied by equation (3a) and equation (3b), respectively. Henceforth, we simply say that ρ θ is a robust metrological scheme with respect to a set {E j } of errors if the QFI of ρ θ is preserved under {E j }. We show below that concrete robust metrological schemes can be easily constructed based on the stabilizer formalism [34]. A stabilizer code C(S) is the joint +1 eigenspace of the stabilizer group S, which is an Abelian subgroup of the nqubit Pauli group, i.e., S i |ψ = |ψ for all S i ∈ S and all |ψ ∈ C(S). A set {E j } of Pauli errors-E j are also elements of the n-qubit Pauli group-are correctable for this stabilizer code, if each E † j E k is either in the stabilizer group, or detectable, i.e., anticommutes with at least one element of the stabilizer group [34].
Theorem 3. In a metrological scheme |ψ θ = e −iHθ |ψ where the probe state |ψ is taken from the coding subspace of a stabilizer code C(S) capable of correcting errors {E j } and [H, S] = 0, the QFI is also immune to the errors {E jX }, whereX is a Pauli error that commutes with S while anticommutes with ∆H. If the coding subspace is two-dimensional then the optimal measurement is the joint measurement of S andX.
The proof is sketched in the Methods (see Supplementary Information for full proof). As examples, we construct at first a family of metrological scheme on 2t + 1 physical qubits capable of correcting arbitrary t-qubit errors while preserving the QFI, and then a Heisenberglimited version. Example 1. We consider a system composed of n = 2t + 1 qubits that are labeled with the index set I = {1, 2, · · · , n}. Let us denote X i , Y i and Z i the tensor products of the Pauli matrices X, Y , and Z on the ith qubit and identity operators on other qubits, respectively, and X α = i∈α X i with α ⊆ I. Let C be the 2-dimensional subspace stabilized by {X α | |α| = even}, which is exactly the coding subspace of a stabilizer code capable of correcting all t-qubit phase flip errors {Z α } with |α| ≤ t. For any state |ψ ∈ C such that ψ|Z I |ψ = 0, the metrological scheme |ψ θ = exp(−iθZ I )|ψ preserves QFI against all t-qubit phase flip errors plus errors of type {Z α X I }, which include essentially arbitrary errors on no more than t qubits. That is to say, in terms of QFI, the t-qubit phase flip codes can be used to protect a metrological scheme from arbitrary t-qubit errors. In comparison, at least five physical qubits are required in a standard quantum error-correcting code to correct arbitrary single qubit errors, while our scheme requires only three physical qubits. This is one of the advantages brought in by considering the preservation of QFI instead of the protection of quantum states. The maximal Fisher information is attained by the joint measurement of the stabilizers of C and X I , i.e., all the observables {X j | j ∈ I}, without any recovery operation.
Example 2. Beating the stand quantum limit by quantum entanglement is one of the most fascinating aspects of the quantum-enhanced metrology [6][7][8]. A canonical example is utilizing the m-qubit Greenberger-Horne-Zeilinger (GHZ) state as the probe states for the parallel samplings of a unitary sensing transformation, wherein QFI scales quadratically with m-the Heisenberg limit. Combing this entanglement-enhancement with QEC for preserving QFI can provide us a metrological scheme that achieves the Heisenberg limit and is immune to noisy processes to a certain extent. The basic idea is to replace all qubits in the Heisenberg-limited scheme by logical qubits of some phase-flip codes and the sensing transformation to a logical Z rotation on each logical qubit. Then, the parametric family of states read j are the logical Pauli Z operators on the i-th block, and |0 and |1 are the logical states. Let α | |α| is even} be the stabilizer group of the n-qubit phase-flip code for the i-th qubit in the original scheme. Further, assume that m and n = 2t + 1 are odd and m ≥ n. Considering the structure of the GHZ state, all parametric family of states are in a standard t-qubit-error correcting code with one higher-level logical qubit, generalizing Shor's construction. Obviously, this metrological scheme preserves the QFI against arbitrary t-qubit errors. Additionally the QFI is preserved also against a set of collective errors j which anti commutes with H. Moreover, the joint measurement of all the stabilizers X of the stabilizer code together withX attains the maximal Fisher information. In ref. [27], the authors considered the same type of sensing transformation generated by the many-body interaction, and employed QEC for metrology. In their scenario, only the protective capability from the error-correcting codes encoding the qubits in GHZ state was explored; we note that the GHZ state itself provides a higher-level bit-flip code with which more errors are harmless to QFI.
We note that the scenario of Fig. 1b is almost equivalent to another one that the sensing transformation is a Z rotation on the first qubit of each block, which is prepared in the GHZ state; after that, every qubit involved in the GHZ state is encoded into the phase-flip code. The only difference is that the former scenario can also preserve the QFI against the errors that do not occur temporally after the sensing process but can be deferred until after the sensing process, e.g., the errors that commute with the generator of the sensing transformation.
In summary, we have established a new theory of error correction designed for quantum metrology in the context of quantum estimation theory. The purpose of our specialized QEC is to preserve the QFI, which determines the best precision of estimating the value of a parameter, instead of the quantum states themselves. We gave testable conditions to identify the errors to which the QFI is immune, and constructed the optimal measurements in noisy states for the best estimation precision. While in the standard QEC any states, mixed or pure, in the coding subspace can be used in a metrological scheme, our conditions do not generally give rise to a subspace, instead only a special set of states that can serve our purpose. Our method can be readily applied for some parameter estimation problems, especially for those in the stabilizer formalism. Comparing with the standard stabilizer codes, our theory has the advantages of, firstly, being capable of preserving QFI against more errors using the same amount of resources and, secondly, sparing the recovery operations.

Methods
Let us start with a crucial observation on the loss of QFI after a known noisy channel. For a given channel N with Kraus operators {E j }, we denote V a unitary representation on the system plus an ancilla in the state ρ a such that where Tr a is the partial trace over the ancilla. For the sake of rigorousness, we assume that we always have bounded SLD operators henceforth, i.e., SLD operators L θ and L θ for states θ and N (ρ θ ), respectively, exist and are finite. The loss of QFI can be expressed as (see Supplementary Information) where is exactly the square of the measurement error used by Ozawa to derive his error-disturbance uncertainty relation [42]. From equation (4a), we see that the loss of QFI can be understood as the minimal measurement error of measuring a Hermitian operator Q after the given noisy channel compared with measuring L θ before the noisy channel. We note that due to equation (4a) and (4b), the following statements are equivalent: (a) ∆ F (ρ θ , N ) = 0.
(b) There exists a Hermitian operator Q such that QE j √ ρ θ = E j L θ √ ρ θ are satisfied for all j.
The necessary and sufficient condition (1) for the preservation of QFI under a known noisy channel follows from the equivalence between (a) and (c). Theorem 1 is a consequence of equation (1). The geometric picture illustrated in Fig. 2 is due to the equivalence between (a) and (b). Theorem 2 is implied by the equivalence between (a) and (b) together with the following lemma, for which we give a constructive proof in Supplementary Information.
for all j and k and (ii) for all α j such that j α j |s j = 0, j α j |d j = 0 must be satisfied. Theorem 3 follows from the satisfaction of the two testable conditions in Theorem 2 for the errors {E jX τ } with τ = 0, 1, whereX is a Pauli error that commutes with the stabilizer of the code S and anticommutes with ∆H. The full proof is presented in Supplementary Information.

Supplementary Information
Proof of equation (4). Let Q be an arbitrary Hermitian operator on the Hilbert space associated with the outputs of N , and L θ be an SLD operator for N (ρ θ ) and assumed bounded. Then, where we have used the following relations: As a result of equation (S1), we have ∆ F (ρ θ , N ) ≤ (ρ θ , N , Q) for every Hermitian operator Q, and the equality holds if (and only if) QN (ρ θ ) = L θ N (ρ θ ). Thus, we obtain equation (4a). Moreover, by using the Kraus operators E j for N , it can be readily checked that In the case of mixed states ρ θ , equation (1) is equivalent to L θ E j |ψ θ = E j L θ |ψ θ for all states |ψ θ in the range of ρ θ . Choosing a set of linearly independent states {|ψ l,θ } in the range of ρ θ , e.g., the eigenstates of ρ θ corresponding to nonzero eigenvalues, applying Lemma 1 to two indexed families of states {E j |ψ l,θ } and {E j L θ |ψ l,θ } with composite index (j, l), we obtain the testable conditions for mixed states.
Proof of Lemma 1. Necessity. If there exists a Hermitian operator Q such that Q|s j = |d j for all j, then condition (i) can be obtained by using the hermicity of Q as whilst, condition (ii) is obvious by noting that j α j |d j = Q j α j |s j , which vanishes if j α j |s j does. Sufficiency. Assume that conditions (i) and (ii) are satisfied, then we shall explicitly construct a desired Hermitian Q such that Q|s j = |d j for all j. Firstly, let us choose a maximal subset of linearly independent vectors |s j and denote the set of corresponding indices by J. Secondly, condition (ii) implies that restricting j in J is sufficient for constructing the Q. Note that every |s j with j / ∈ J can be expressed as with α j being complex numbers. If Q|s j = |d j for all j ∈ J, then from condition (ii) and equation (S4), we have Therefore, condition (ii) ensures that every Hermitian operator Q satisfying Q|s j = |d j for all j restricted in J must satisfy that for all j. Thirdly, we explicitly construct Q as follows. Condition (i) implies that the matrix g defined by g jk = s j |d k is Hermitian. Therefore, g can be diagonalized as where u is a unitary matrix, and c j are real numbers. Then, the Hermitian operator Q is explicitly constructed as where |s ⊥ j are vectors satisfying s ⊥ j |s k = δ jk for all j and k. Such a set of |s ⊥ j always exists as long as |s j are linearly independent [45]. It is easy to check that the Hermitian operator Q defined above satisfies Q|s j = |d j for all j. Therefore, are satisfied for all j and the proof is finished.
Proof of Theorem 3. We have only to check that two conditions in Theorem 2 are satisfied for the errors {E jX τ } with τ = 0, 1, whereX is a Pauli error that commutes with the stabilizers S of the code and anticommutes with ∆H. The first condition reads which holds true due to the follow facts: (i) When γ = τ , or γ = τ and E † k E j is detectable, then both sides of equation (S10) vanish. (ii) When γ = τ and E † k E j is a stabilizer, equation (S10) is ensured by the facts that L θ |ψ θ = −2i∆H|ψ θ andX anticommutes with ∆H.
To show that the second condition is also satisfied, we at first identify an independent set {E j } j∈J of errors such that E † k E j are detectable errors for arbitrary j, k ∈ J. We denote by J j the set of indices l such that E † j E l is a stabilizer. It is easy to check that {E j (1 ±X)|ψ θ } j∈J , as well as {E j (1 ±X)L θ |ψ θ } j∈J , is a set of mutually orthogonal states. As a result, both sets {E jX τ |ψ θ } j∈J and {E jX τ L θ |ψ θ } j∈J are linearly independent. If there are complex numbers α jτ such that j,τ α jτ E jX τ |ψ θ = 0, then we have j∈J,τ α jτ E jX τ |ψ θ = 0 with α j,τ = l∈Jj α l,τ , from which it follows that α jτ = 0 for all j ∈ J and τ = 0, 1 and j,τ α jτ E jX τ L θ |ψ θ = j∈J,τ α jτ E jX τ L θ |ψ θ = 0.
In order to investigate the property of the optimal measurement, we denote Due to the fact that the errors E j with the index j being restricted in J is independent, it is easy to check that X τ E † j QE jX τ |ψ θ = Q|ψ θ = L θ |ψ θ for arbitrary τ = 0, 1 and j. It follows that QE jX τ |ψ θ = E jX τ L θ |ψ θ , therefore this Hermitian operator Q is an SLD operator not only for |ψ θ but also for noisy states under the set {E j } of errors, therefore, the measurement with respect to the eigenstates of Q are optimal for the noisy states. Moreover, Q commutes with all the stabilizers andX, so they have common eigenstates. When the code space is 2-dimensional, all the stabilizer generator andX constitute a complete set of mutually commuting observables, therefore the joint measurement of them is equivalent to that with respect to eigenstates of Q.