Activating hidden metrological usefulness

We consider entangled states that cannot outperform separable states in any linear interferometers. Then, we show that these states can still be more useful metrologically than separable states if several copies of the state are provided or an ancilla is added to the quantum system. We present a general method to find the Hamiltonian for which a given quantum state provides the largest gain compared to separable states.

Entanglement lies at the heart of quantum mechanics and plays an important role in quantum information theory [1]. Recently, it has been realized that entanglement can be a useful resource in very general metrological tasks. By using entangled states it is possible to overcome the shot-noise limit, corresponding to classical interferometers, in the precision of parameter estimation [2-7]. On the other hand, separable states, i.e., states without entanglement cannot overcome the classical limit. It has even been shown that quantum states with a very weak form of entanglement, called bound entanglement [8,9], can also be metrologically useful in this sense [10,11]. However, there are highly entangled states that are not useful for metrology [12].
In what sense is metrological usefulness the property of the quantum state? It is clear that, starting from many entangled quantum states that are not useful for metrology, with local operations and classical communication (LOCC) it is possible to distill singlets, which are metrologically useful. This finding is almost trivial, as metrological "uselessness" is not conserved by LOCC operations. On the other hand, in most quantum metrology experiments LOCC operations are not possible. Thus, it would be interesting to consider simpler operations, such as adding an ancilla to the bipartite quantum state or consider several copies of the state, as is the case with the activation of bound entanglement [13] and nonlocality [14] (see Fig. 1).
In this paper, we show that some entangled quantum states that are not useful in linear interferometers become useful in the cases mentioned above. These findings are quite surprising: including uncorrelated ancilla qubits can make a state metrologically useful. We present a general method to find the local Hamiltonian for which a given quantum state provides the largest gain compared to separable states. Note that this task is different, and in a sense more complex, than maximizing the quantum Fisher information. The reason is that by changing the Hamiltonian, the sensitivity achievable by separable states can also change.
Quantum Fisher information.-Before discussing our main results, we review some of the fundamental rela- tions of quantum metrology. A basic metrological task in a linear interferometer is estimating the small angle θ for a unitary dynamics U θ = exp(−iHθ), where the Hamiltonian is the sum of local terms. In particular, for bipartite systems it is where H n are single-subsystem operators. The precision is limited by the Cramér-Rao bound as [15][16][17] ( where the quantum Fisher information, a central quantity in quantum metrology is defined by the formula [15] Here, λ k and |k are the eigenvalues and eigenvectors, respectively, of the density matrix , which is used as a probe state for estimating θ.
Metrological usefulness of a quantum state.-We call a quantum state metrologically useful, if it can outperform separable states in some metrological task, i.e., if It is an intriguing task to find the operator H, for which a given state outperforms the most separable states. For that we define the metrological gain compared to separable states by We are interested in g( ) = max localH g H ( ), where a local Hamiltonian is just the sum of single system Hamiltonians as in Eq. (1). The maximization task looks challenging since we have to maximize a fraction, where both the numerator and the denominator depend on the Hamiltonian. We also define the robustness of metrological usefulness, r( ), which is the amount of admixed white noise that is needed to make the quantum state metrologically not useful [11].
Maximally entangled state.-As we have mentioned, it is a difficult task to obtain g( ) and the optimal local Hamiltonian for any . As a first step, we consider the d × d maximally entangled state, which is defined as Due to the symmetry of the state, the optimal Hamiltonian can straightforwardly be obtained as where the diagonal matrix D is given as The details are given in the Supplementary Material [18]. For the 3 × 3-case, the robustness of the metrologically usefulness is [18] r(|Ψ (me) ) = 25 − √ 177 32 ≈ 0.3655.
Activation by an ancilla qubit.-Now we consider the previous state, after a pure ancilla qubit is added where (p) The setup is depicted in Fig. 1(a). Then, with the operator where an operator acting on the ancilla and A is Hence larger part of the noisy maximally entangled states are useful in the case with the ancilla.
In summary, there are states with the following properties. (i) They are not more useful than separable states considering any local Hamiltonian. (ii) With an ancilla, they are more useful than separable states for some local Hamiltonian.
Activation by adding extra copies.-We consider now two copies of the noisy 3 × 3 maximally entangled state The setup is shown in Fig. 1(b). Then, with the two-copy operator Hence larger part of the noisy maximally entangled states are useful in the two-copy case, than with a single copy. So far, we exploited the symmetries of quantum states to obtain the Hamiltonian leading to the largest metrological gain. We now present a general method to compute g( ) numerically.
Method for finding the optimal Hamiltonian.-We need to maximize F Q [ , H] over H for a given . However, since it is convex in H, maximizing it over H is a difficult task. Instead of the quantum Fisher information, let us consider first the variance of the parameter estimation assuming that the expectation value of some operator M is measured for the estimation. In this case, the variance of the estimation is given by the error propagation formula as We can use the fact that F Q [ , H] = max M 1/(∆θ) 2 M , which is due to Eq. (2).
Observation 1.-The error propagation formula given in Eq. (15) can be minimized over H for a given M and as follows.
Proof. Simple algebra yields where n = 1, 2 and c n > 0 is some constant. This way we make sure that σ min (H n ) ≥ −c n , and σ max (H n ) ≤ +c n , for n = 1, 2, where σ min (X) and σ max (X) denote the smallest and largest eigenvalues of X. The optimal H n is the one that maximizes Tr(A n H n ) under these constraints. It can straightforwardly be obtained as where the eigendecompisition of A is given as has the same eigenvectors as A n and has only eigenvalues +c n and −c n .
We already know how to optimize H for a given M. However, how do we find the optimal M ? This can be done with the well-known formula for the symmetric logarithmic derivative [17] Iterative method.-We can now construct the following procedure. First we choose a random M. Then, repeat the following two steps. (Step 1) Determine the optimal H for a given M using Observation 1. (Step 2) Determine the optimal M for a given H using Eq. (19). A see-saw procedure similar in spirit has been used to make the optimization of the metrological performance over density matrices in Refs. [11,22,23].
After several iterations of the two steps above, we obtain the maximal quantum Fisher information over a certain set of Hamiltonians. Based on that, we can calculate the quantity g c1,c2 ( ) = max H1,H2 where we assumed that H n are constrained with Eq. (17 which leads to F (sep) Q (c 1 , c 2 ) = 4(c 2 1 + c 2 2 ). Then, the gain can be expressed as where the optimization is only over c 2 , and, without the loss of generality, we set c 1 = 1. The optimial c 2 can be obtained from an analytical formula [18]. Hence we computed the maximum of the fraction, (5), for local Hamiltonians.
It is now clear that the eigenvalues of the optimal H n in Eq. (20) are ±c n . To show this, let us assume the contrary. Let us assume that for a state and for given c 1 , c 2 we know the optimal H 1 and H 2 , and H n fulfill Eq. (17), but not all eigenvalues are ±c n . Then, M defined in Eq. (19) is the optimal operator to be measured, and the quantum Fisher information equals the sensitivity (∆θ) 2 M defined in Eq. (15). However, one can show that we can always replace the eigenvalues of H n by ±c n such that Tr(i[ , M ]H) will not decrease, and 1/(∆θ) 2 M will not decrease either.
This fact can be the basis of a simple, alternative numerical search for the optimal H n , where we assume that H n is of the form (18). We setD n = c n diag(+1, +1, ..., +1, −1, −1, ..., −1) and then vary U n in order to get the maximal F Q ( , H 1 ⊗ 1 + 1 ⊗ H 2 ).
Using the numerical method above, we obtain a slightly larger value for the robustness of metrological usefulness for the state with an ancilla, (10). g( (anc) ) > 1 if p < 0.3941. The same is true for the case of the two copies of the maximally entangled state, (13). We obtain g( (tc) ) > 1 if p < 0.4169.
For states with a high symmetry, such as isotropic states [25,26], and Werner states [27], we obtained the optimal Hamiltonian analytically and determined the subset of these states that are metrologically useful [18]. We also used that to verify our numerical methods.
Activation of a bound entangled state by a separable state.-While bound entangled or non-distillable states [8,9] are considered weakly entangled, they can share many properties with highly entangled states. For example, there are bound entangled states that can reach the Heisenberg scaling in metrological applications [10]. It has also been shown that bipartite bound entangled states, which have a positive semidefinite partial transposition (PPT), can be useful for metrology [11]. Moreover, bipartite PPT entangled states can even have a high Schmidt-rank [28].
Let us now consider a PPT entangled state (PPT) AB that is not useful for quantum metrology. Then, we look for a [11] such that g1,1( AB) = 1.0000, that is, they are not useful metrologically.
separable state (sep) such that A B becomes useful. Hence, in this case we have to optimize not only over H, M, but also over the separable state. Simple convexity arguments show that the maxiumum is taken when we have a pure product state, , which corresponds to two ancillas at the two parties. In fact, even a single ancilla qubit is sufficient for activation.
Activation of a PPT entangled state by an ancilla qubit.-We now consider a PPT entangled state, that is not useful metrologically, and g( AB ) = 1. However, with an ancilla it becomes useful, g( (aA)(B) ) > 1. We show here examples for d × d dimensional PPT states found in Ref.
[11] for odd dimensions d up to d ≤ 11. See Table I for the numerical results.
Note that here we fixed c i = 1 for the coefficients of the local Hamiltonians H i , i = 1, 2. However, numerics suggests that optimization over c i does not help to increase g in the case of two ancillas (last column), and helps only marginally in the case of one ancilla (third column). For instance, in the case of d = 7, the g value raises from 1.0096 (corresponding to c 2 = 1) to 1.0098 (corresponding to c 2 1.034) if we optimize over c 2 .
How large part of quantum states are useful.-The scaling of the quantum Fisher information with the dimension has been considered for random states and for the best local Hamiltonian in Ref. [29]. We used our optimization algorithm to determine the distribution of the quantum Fisher information and obtain exactly how large part of pure or mixed quantum states are useful. The random pure states and mixed states have been generated according to Ref. [30]. For d = 3, the results are shown in Fig. 2. It suggests that almost no random mixed states are useful. Pure states are useful almost with a maximal usefulness. Usefulness of entangled bipartite pure states.-In general, we can always consider a bipartition of the particles for any multipartite state.
Observation 2.-All entangled bipartite pure states are metrologically useful, which is also demonstrated numerically in Fig. 2.
Proof.-Let us consider a pure state with a Schmidt decomposition where s is the Schmidt number, and the real non-negative σ k Schmidt coefficients are in a descending order. We define wheres is the largest even number for whichs ≤ s, and We define H B in a similar manner. We also define the collective Hamiltonian Then, we have H AB Ψ = 0. Direct calculation yields which is larger than the separable bound, F (sep) Q = 8, whenever the Schmidt rank is larger than 1. For even s, this can be seen noting that holds, where we used Eq. (27) to evaluate the left-hand side of Eq. (28), and we also took into account that σ 1 > σ 2 > 0, σ n ≥ 0 for n = 3, 4, 5, ..., and s n=1 σ 2 n = 1. For odd s, we need that holds, where we used that σ 1 σ 2 > σ 2 s . We can even consider several copies of a quantum state. In the Supplement, we prove that infinite number of copies of entangled pure quantum states are maximally useful [18].
Conclusions.-We showed that entangled quantum states that cannot outperform separable states in any linear interferometer can still be more useful than separable states, if several copies of them are considered or an ancilla is added to the system. This is surprising result which shows that the relationship between quantum metrology and the structure of quantum states requires further study. We presented a method to find the Hamiltonian for carrying out metrology in a linear interferometer with a given quantum state that provides the largest gain compared to the precision achievable by separable states.
We thank I. Apellaniz  The supplemental material contains some additional results. We present some details of the optimization over the c 2 parameter of the Hamiltonian. We calculate the optimal Hamiltonian analytically for isotropic states and Werner states. We consider metrology with mult-iparticle states, if some particles are united into a single party. We determine the maximum achievable precision. We also consider metrology with an infinite number of copies of arbitrary entangled pure states.

OUR OPTIMIZATION METHOD FINDS THE GLOBAL OPTIMUM
The maximization of the error propagation formula can be expressed using a variational formulation as [22] where M takes the role of αM. Then, the function is concave in M and linear in H, and the two-step see-saw algorithm we have described will find better and better Hamiltonians. However, the function in Eq. (S1) is not strictly concave in (M , H). Hence, our iterative numerical procedure will always lead to Hamiltonians with an increasing quantum Fisher information, however, it is not guaranteed to find a global optimum. Based on extensive numerical experience, in bipartite systems it converges very fast, and from 10 trials at least 2-3, typically more will lead to the global optimum. The 10 trials mentioned can take 2-3 minutes for the system sizes considered on a usual laptop computer. We can understand the expression better as follows. If we subtract a term 4 H 2 from the expression appearing on the right-hand side of Eq. (S1), then we will arrive at where the non-Hermitian matrix is defined as The optimal value is at Without the loss of generality, we set c 1 = 1, then c 2 can be obtained from Eq. (S7). One can add a third step to the two-step procedure of the paper, in which c 2 is updated according to the formula Eq. (S7). For a smoother convergence, one can change c 2 not abruptly, but only by a small value changing it in the direction of the value suggested by Eq. (S7).

METROLOGY WITH ISOTROPIC STATES
We will now consider quantum metrology with isotropic states, which are defined as [25] where P (+) d is a projector to the maximally entangled state |Ψ (me) defined in Eq. (6).
We consider a Hamiltonian of the form The subscript "coll" indicates that the Hamiltonian acts on both subsystems, in contrast to H 1 and H 2 that act only on one of the subsystems. The Hamiltonian is local, since it does not contain interactions terms. Isotropic states are invariant under transformations of the type where U is a single-qudit unitary and " * " denotes element-wise conjugation. Hence, isotropic states are invariant under the Hamiltonian where K is a Hermitian operator. Observation S1.-For short times, the action of the Hamiltonian H coll given in Eq. (S9) is the same as the action of where the single party Hamiltonian is defined as Proof. Let us define In the rest of the section, we omit the superscript "iso" in H . Then, simple algebra shows that (S15) Hence, for small t holds. The isotropic state is invariant under the action of H inv (∆), since the corresponding unitary is of the form given in Eq. (S10). Hence, the action of H coll is the same as the action of H (iso) coll (H) for small t. . Note that in the quantum metrology problems we consider we always estimate the parameter t around t = 0 assuming that it is small. Hence, the approximate equality in Eq. (S16) is sufficient.
Observation S2.-Replacing the evolution by H coll given in Eq. (S9) by the evolution by H (iso) coll given in Eq. (S12) does not decrease the metrological gain. Hence, when looking for the Hamiltonian with the largest metrological gain, it is sufficient to look for Hamiltonians of the form (S12).
Proof. When the evolution by H coll given in Eq. (S9) is replaced by the evolution by H . Knowing that f is matrix convex, we obtain that (S18) We will now use that for a pure state mixed with white noise it is possible to obtain a closed formula for the quantum Fisher information for any operator A as a function of p as [4] where we used that for the reduced state of |Ψ (me) we have ρ red1 = ρ red2 = 1 1/d. Next, we use the fact that holds. Hence, for the quantum Fisher information we obtain (S22) Based on Eq. (S22) and on Eq. (21), the metrological gain for a given Hamiltonian H (iso) coll is obtained as where r(H) is defined as and h k denote the eigenvalues of H.
Let us now consider the metrological gain for the isotropic state for various Hamiltonians.
Observation S3.-Isotropic states have the best metrological performance with respect to separable states with the Hamiltonian given by Based on Eq.
(3), the corresponding quantum Fisher information is where α is defined as α = 0 for even d, 1 for odd d.
No other Hamiltonian H corresponds to a better performance. Equation (S26) is maximal for p = 1 and has the value coll (H best )) = 2 which is 2 for even d and approaches 2 for large d for odd d.
Proof. Without the loss of generality, let us set h min = −1 and h max = +1. Then, the denominator of Eq. (S24) is 8. Let us consider now the numerator. The maximum of the numerator of Eq. (S24) will be clearly taken by a configuration for which h k = ±1. The first term is d 2 .
Next, we determine which isotropic states are useful metrologically.
Observation S4.-If holds then the isotropic state p is useful for metrology with the Hamiltonian (S25). Otherwise, the isotropic state is not useful with any other Hamiltonian. Proof. We look for the p for which the righ-hand side of Eq. (S28) is 1.
Let us now look for the Hamiltonian of the type (S12) with which the isotropic states have the worst metrological performance.
Observation S5.-Isotropic states have the worst metrological performance with respect to separable states with the Hamiltonian given by The corresponding quantum Fisher information is No other Hamiltonian H corresponds to a worst performance. Note that we considered collective Hamiltonians of the type (S12). Other collective Hamiltonians H coll can lead to a worse performace and can even have g( p , H coll ) = 0. In particular, this is the case for Hamiltonians given in Eq. (S11), where K can be any Hamiltonian.
The metrological gain given in Eq. (S31) is maximal for p = 1 and has the value If d ≥ 4, then the right-hand side of Eq. (S32) is not larger than one. Hence, with H worst , no isotropic state can be useful for d ≥ 4. For d = 3, on the other hand the right-hand side of Eq. (S32) is larger than one. Hence, for d = 3, the maximally entangled state |Ψ (me) is useful with the Hamiltonian H worst . We can also see that for d = 3 the maximally entangled state |Ψ (me) is useful with any Hamiltonian H (iso) coll . In Fig. S1, we plot the results of simple numerics for d = 3, 4 and 5. The random mixed states have been generated according to Ref. [30].

METROLOGY WITH WERNER STATES
We now examine whether another type of bipartite states with a rotational symmetry, i.e, Werner states defined as [27]  outperform separable states in metrology. Here −1 ≤ φ ≤ +1 and V is the flip operator.
We will consider a general evolution of the type Eq. (S9). Werner states are invariant under transformations of the type where U is a single-qudit unitary. Hence, Werner states are invariant under the Hamiltonian where J is a Hermitian operator. Observation S6.-For short times, the action of the Hamiltonian H coll given in Eq. (S9) is the same as the action of where the single party Hamiltonian H is defined as Proof. Let us define ∆ (W) as In the rest of the section, we omit the superscript "W" in H . Then, simple algebra shows that coll given in Eq. (S11) does not decrease the metrological gain. Hence, when looking for the Hamiltonian with the largest metrological gain, it is sufficient to look for Hamiltonians of the form (S11).
Proof. The proof is similar to the proof of Observation S2.
Werner states, given in Eq. (S33), can also be defined as where P s and P a are the projectors to the symmetric and antisymmetric subspace, respectively. We will be interested in the case φ ≤ 0. The quantum Fisher information for Werner states for a Hermitian operator A is where k ∈ S and l ∈ A denote the indices of symmetric and antisymmetric eigenstates, respectively. From Eq. (S41), the eigenvalues of the Werner states can be obtained, yielding If the operator A is of the form given in Eq. (S11), then for any symmetric states |Ψ s and antisymmetric states |Ψ a Ψ s |A|Ψ s = Ψ a |A|Ψ a = 0 (S44) hold. Hence, we can return to sums over all eigenvectors and write Then, we need that Hence, we obtain a general formula for the quantum Fisher information for Werner states as Based on Eq. (S47) and on Eq. (21), the metrological performance is given by where r(H) is defined in Eq. (S24). Let us now look for the Hamiltonian that provides the largest metrological gain for Werner states.
Observation S8.-Werner states have the best metrological performance with respect to separable states with the Hamiltonian H best given in Eq. (S25). The corresponding quantum Fisher information is (check!) where the optimization is carried out over collective Hamiltonians of the form (S11). No other such collective Hamiltonian corresponds to a better performance. Equation (S49) is maximal for φ = −1 and has the value which is close to 1 for large d.
Proof. The best H operator is the one for which r(H) defined in Eq. (S24) is the largest. In other words, we can look for the H for a constant (h max − h min ) 2 that maximizes [dTr(H 2 ) − Tr(H) 2 ]. The details of the proof are similar to the proof of Observation S3.
Next, we determine which Werner states are useful metrologically.
Observation S9.-If holds, then the Werner state is useful for metrology with the Hamiltonian (S25). Otherwise, the Werner state is not useful with any other Hamiltonian. Proof. We look for the φ for which the right-hand side of Eq. (S49) is 1.
Let us now look for the Hamiltonian of the type (S11) with which the Werner states have the worst metrological performance.
Observation S10.-Werner states have the worst metrological performance with respect to separable states with the Hamiltonian given in Eq. (S30). The corresponding quantum Fisher information is No other Hamiltonian corresponds to a worst performance.
Proof. This can be seen noting that Eq. (S48) is minimized for this case.
Note that we considered Hamiltonians H (W) coll (H) of the type (S11). Other collective Hamiltonians H coll can lead to a worse performace and can even reach to g( W , H coll ) = 0. In particular, this is the case for collective Hamiltonian of the form given in Eq. (S11). Equation (S52) is maximal for φ = −1 and has the value We can see that for d ≥ 3 the right-hand side of Eq. (S53) is not larger than one, hence the Werner state is not useful with the Hamiltonian H worst . We can also see that the metrological gain, (S53), is close to 0 for large d.
In Fig. S2, we plot the results of simple numerics for d = 3, 4 and 5. The random mixed states have been generated according to Ref. [30].

ESTIMATION OF THE QUANTUM FISHER INFORMATION FOR GENERAL QUANTUM STATES
Recently, there have been several methods presented to find lower bounds on the quantum Fisher information based on few operator expectation values [2,19]. Our results on isotropic states and Werner states can be used to construct lower bounds for the metrological gain g based on a single operator expectation value.
In order to proceed, we note that any d × d state can be depolarized into an isotropic state given in Eq. (S8) with the U ⊗ U * twirling operation as where M is a unitarily invariant probability measure. The state iso (F ) is just the isotropic state given in Eq. (S8), defined with a different parametrization as is the entanglement fraction of the state , which is alternatively called the singlet fraction [25,26]. Based on Eq. (S26), the maximum metrological performance of the isotropic state is given by where α is zero for even d, and one otherwise. Here, we remember that the metrological gain is defined as g( ) = max localH g H ( ), where g H ( ) is given in Eq. (5). Next, we show that g( ) cannot increase under twirling defined in Eq. (S54), i.e., g( ) ≥ g( iso (F )). (S58) We use a series of inequalities where H = (U † 0 ⊗U * † 0 )H(U 0 ⊗U * 0 ) and U 0 is some unitary. To arrive at the second line we used the property of the quantum Fisher information that it is convex in the state, Noting also that the eigenvalues of H are the same as that of H, and that F Based on Eq. (S58), the metrological gain of any quantum state can be bounded from below as g( ) ≥ g( iso (F )), where g( iso (F )) is defined in Eq. (S57) and F is just the entanglement fraction of . Based on Eq. (S56), F equals the expectation value of the projector to |Ψ (me) . Hence, our lower bound is based on a single operator expectation value. Similar calculations can be carried out for Werner states, using the fact that any quantum state can be depolarized into a Werner state using the U ⊗ U twirling Then, we can construct a lower bound where the Eq. (S49) gives the right-hand side of Eq. (S62) as a function of the parameter φ. The quantity φ is related to the expectation value of the flip operator V as

UNITING QUDITS
In most of the paper, we considered bipartite examples. In the multipartite case, the usefulness of a quantum state is always relative to the partitioning of the parties. From this point of view, it is worth to look at metrological usefulness of a multipartite state when we put the parties into two groups, and return to the bipartite problem. For instance, the four-qubit ring cluster state is not useful, F Q /F