Coherent measurements in quantum 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 ${{p}_{\phi }}(a,b)$. Classical probabilities satisfy

$$\begin{eqnarray}&&{{p}_{\phi }}(a,b)={{p}_{\phi }}(b){{p}_{\phi }}(a\mid b).\end{eqnarray} \tag{ 4 }$$

The left- and right-hand sides of equation (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]

$$\begin{eqnarray}&&\mathcal{F}(A,B)=\mathcal{F}(B)+\mathcal{F}(A\mid B).\end{eqnarray} \tag{ 5 }$$

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 equation (5), including the multipartite case, in appendix A.

In the quantum setting, the discrete probability distribution ${{p}_{\phi }}(x)$ is now obtained from a positive operator-valued measure (POVM) ${\boldsymbol{ \Pi }}$ with elements $\{{{\Pi }_{x}}\}$ acting on the encoded quantum state ${{\varrho }_{\phi }}$: ${{p}_{\phi }}(x)={\rm tr}[{{\varrho }_{\phi }}{{\Pi }_{x}}]$ with ${{\sum }_{x}}{{\Pi }_{x}}=\mathbb{I}$. Remarkably, there is a simple formula that yields the optimal Fisher information over all POVMs [12]:

$$\begin{eqnarray}&&\mathop{{\rm max} }\limits_{{\boldsymbol{ \Pi }} }\mathcal{F}({{\varrho }_{\phi }})=2\mathop{\sum }\limits_{ij}\frac{{{({{\lambda }_{i}}-{{\lambda }_{j}})}^{2}}}{{{\lambda }_{i}}+{{\lambda }_{j}}}\mid {{\langle {{\psi }_{i}}\mid H\mid {{\psi }_{j}}\rangle }\mid ^{2}},\end{eqnarray} \tag{ 6 }$$

where ${{\lambda }_{i}}$ and $\mid {{\psi }_{i}}\rangle$ are the eigenvalues and eigenvectors of $\varrho$ respectively, while the Hamiltonian H is the Hermitian generator of the unitary encoding $U={{{\rm e}}^{-{\rm i}H\phi }}$, where ${{\varrho }_{\phi }}=U\varrho {{U}^{\dagger }}$. Although the POVM does not explicitly depend on ϕ, in general the optimal POVM, in equation (6), may differ for different values of ϕ. We will discuss further this issue below.

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.

In general equation (6) does not tell us much about the optimal POVM. The derivation of equation (6) does identify the necessary and sufficient condition for the optimal measurement, namely that

$$\begin{eqnarray}&&\sqrt{{{\Pi }_{x}}}{{L}_{\phi }}\sqrt{{{\varrho }_{\phi }}}={{k}_{x}}\sqrt{{{\Pi }_{x}}}\sqrt{{{\varrho }_{\phi }}}\end{eqnarray} \tag{ 7 }$$

for constant ${{k}_{x}}\in \mathbb{R}$ and ${{L}_{\phi }}$ is the symmetric logarithmic derivative defined by ${{\partial }_{\phi }}{{\varrho }_{\phi }}=\frac{1}{2}\left( {{\varrho }_{\phi }}{{L}_{\phi }}+{{L}_{\phi }}{{\varrho }_{\phi }} \right)$ [12]. Although there is no known general solution for the POVM elements ${{\Pi }_{x}}$, a sufficient condition is that they are projectors onto eigenspaces of ${{L}_{\phi }}$. This construction yields measurements that are in general coherent and also local in parameter space (optimal for only certain values of ϕ).

We do not know whether optimality can be attained with an adaptive measurement nor do we know whether there exists a measurement that is globally optimal, i.e., for any value of ϕ. 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 $M^{\prime}$ ($M^{\prime} \ll M$) suboptimal measurements are performed to estimate ϕ, after which we can fix the measurement to be optimal at the approximate parameter value $\tilde{\phi }$. It is a well-known result of classical statistics that a maximum likelihood estimator (MLE) for ϕ saturates the Cramér–Rao bound as $M\to \infty$, but we can only construct an approximate MLE since we have used a measurement that is optimal at $\tilde{\phi }$ as opposed to ϕ. Fortuitously, it is nonetheless found to share the same asymptotic properties as the true MLE, provided that $M^{\prime}$ is sufficiently large, meaning that globally optimal measurements confer no estimation advantage in the limit of large M [21, 22].

In fact, we can reach a useful conclusion regarding the possibility of global optimality: when ${{\varrho }_{\phi }}$ is full-rank (and thus strictly positive) for all ϕ, we show in appendix D that no globally optimal measurement exists. Moreover, noise perturbs any zero eigenvalues consequently making the result applicable to any practically realisable state. In fact, such a strategy was recently implemented in an experiment reported in [23] and it could also be eventually used in a proof-of-principle experiment of the ideas introduced here.

Let us first prove that classically correlated probes do not require any entanglement in the readout. Suppose we use as a probe a bipartite (or multipartite) quantum state that is classically correlated: $\varrho ={{\sum }_{ab}}q(a,b)\mid ab\rangle \langle ab\mid$ which is mapped to ${{\varrho }_{\phi }}={{\sum }_{{{a}_{\phi }}{{b}_{\phi }}}}q(a,b)\mid {{a}_{\phi }}{{b}_{\phi }}\rangle \langle {{a}_{\phi }}{{b}_{\phi }}\mid$ due to the encoding, where $\mid {{j}_{\phi }}\rangle =U\mid j\rangle$ for $j=a,b$. Now, note that this state is invariant under local projective measurements

$$\begin{eqnarray}&&{{\varrho }_{\phi }}=\mathop{\sum }\limits_{{{a}_{\phi }}{{b}_{\phi }}}\mid {{a}_{\phi }}{{b}_{\phi }}\rangle \langle {{a}_{\phi }}{{b}_{\phi }}\mid {{\varrho }_{\phi }}\mid {{a}_{\phi }}{{b}_{\phi }}\rangle \langle {{a}_{\phi }}{{b}_{\phi }}\mid ,\end{eqnarray} \tag{ 8 }$$

which is the definition of a classically correlated state [24, 25]. However, there exists a measurement $\{{{\Pi }_{x}}\}$ that yields the outcome probabilities ${{p}_{\phi }}(x)={\rm tr}[{{\Pi }_{x}}{{\varrho }_{\phi }}]$ leading to optimal QFI. Using equation (8) and the cyclic invariance of the trace we get

$$\begin{eqnarray}&&{{p}_{\phi }}(x)=\mathop{\sum }\limits_{{{a}_{\phi }}{{b}_{\phi }}}\langle {{a}_{\phi }}{{b}_{\phi }}\mid {{\Pi }_{x}}\mid {{a}_{\phi }}{{b}_{\phi }}\rangle {\rm tr}\left[ \mid {{a}_{\phi }}{{b}_{\phi }}\rangle \langle {{a}_{\phi }}{{b}_{\phi }}\mid {{\varrho }_{\phi }} \right].\end{eqnarray} \tag{ 9 }$$

It is then straightforward to write the POVM elements as

$$\begin{eqnarray}&&{{\Pi }_{x}}=\mathop{\sum }\limits_{{{a}_{\phi }}{{b}_{\phi }}}\langle {{a}_{\phi }}{{b}_{\phi }}\mid {{\Pi }_{x}}\mid {{a}_{\phi }}{{b}_{\phi }}\rangle \mid {{a}_{\phi }}{{b}_{\phi }}\rangle \langle {{a}_{\phi }}{{b}_{\phi }}\mid ,\end{eqnarray} \tag{ 10 }$$

which is diagonal in a product basis.

Next, we assume that Alice and Bob have full knowledge of the elements ${{\Pi }_{x}}$. In order to implement ${{\Pi }_{x}}$ they would need to implement entangling operations. However, instead they may start with an estimate $\tilde{\phi }$ of the phase and apply the following POVM

$$\begin{eqnarray}&&{{\tilde{\Pi }}_{x}}=\mathop{\sum }\limits_{{{a}_{\tilde{\phi }}}{{b}_{\tilde{\phi }}}}\langle {{a}_{\tilde{\phi }}}{{b}_{\tilde{\phi }}}\mid {{\Pi }_{x}}\mid {{a}_{\tilde{\phi }}}{{b}_{\tilde{\phi }}}\rangle \mid {{a}_{\tilde{\phi }}}{{b}_{\tilde{\phi }}}\rangle \langle {{a}_{\tilde{\phi }}}{{b}_{\tilde{\phi }}}\mid .\end{eqnarray} \tag{ 11 }$$

Since they know the POVM element they can easily compute the probability distribution $\langle {{a}_{\tilde{\phi }}}{{b}_{\tilde{\phi }}}\mid {{\Pi }_{x}}\mid {{a}_{\tilde{\phi }}}{{b}_{\tilde{\phi }}}\rangle$. Then they simply make measurements in basis $\mid {{a}_{\tilde{\phi }}}{{b}_{\tilde{\phi }}}\rangle$, which is local. After each round they can update the value of $\tilde{\phi }$ so that it is closer to the actual value of ϕ. In this way they can adaptively approach the POVM in equation (10) using local measurements and classical communication.

This means that the such a measurement can be implemented without the aid of an entangling operation. Our main point here is that if ${{\varrho }_{\phi }}$ is not classical then it may not be possible to reproduce the optimal distribution ${{p}_{\phi }}(x)$ via local measurements on A and B. On the other hand, for classical probes states the POVM elements, as given in equation (10), can be implemented locally. A word of caution is necessary here. In general implementing a separable POVM may require entanglement [26]. However, in the construction above, the basis vectors of the POVM are locally distinguishable, i.e., $\langle a\mid a^{\prime} \rangle ={{\delta }_{a{{a}^{\prime }}}}$ and $\langle b\mid b^{\prime} \rangle ={{\delta }_{b{{b}^{\prime }}}}$. Therefore it is possible to implement this POVM via LOCC.

Finally, note that this result is fully independent of the formula for QFI in equation (6). Moreover, it might seem that in order for Alice and Bob to implement the POVM given above, they need to know the value of ϕ. It is important to keep in mind that the value of ϕ is not totally unknown, and it is the precision in the value of ϕ that we desire. Therefore it is reasonable to assume that Alice and Bob know the neighbourhood in which ϕ lies and implement $\Pi _{a,b,x}^{\phi }$ with increasing accuracy.

Before we give an example exhibiting a gap between the coherent and adaptive Fisher information, let us discuss the important case of $N00N$ states. A $N00N$ state is an optical state of N photons in superposition with the vacuum in two arms of an interferometer: $(\mid N0\rangle +\mid 0N\rangle )/\sqrt{2}$. For linear encoding, $N00N$ states saturate the ultimate bound for Fisher information, the Heisenberg limit N² [6].

Here we will work with NGHZ states instead of $N00N$ states, but in principal the two are equivalent [27–30]. An NGHZ state before the encoding is where all qubits are in the state $\mid 0\rangle$ in superposition with all qubits in state $\mid 1\rangle$. It is written as $\mid {{G}^{N}}\rangle ={{(\mid 0\rangle }^{\otimes N}}+\mid 1{{\rangle }^{\otimes N}})/\sqrt{2}$ which is transformed to $\mid G_{\phi }^{N}\rangle ={{(\mid 0\rangle }^{\otimes N}}+{{{\rm e}}^{{\rm i}N\phi }}\mid 1{{\rangle }^{\otimes N}})/\sqrt{2}$ after the encoding. That is, the phase is encoded by ${{U}^{\otimes N}}$ with $U={{{\rm e}}^{{\rm i}\phi \mid 1\rangle \langle 1\mid }}$. Here, $\mid 0\rangle$ and $\mid 1\rangle$ are the eigenstates of the Pauli matrix ${{\sigma }_{z}}$.

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 $(1/\sqrt{2})(\mid 0\rangle +{{{\rm e}}^{{\rm i}N\phi }}\mid 1\rangle )\otimes \mid 0{{\rangle }^{\otimes N-1}}$. The first qubit can now be measured in the $\mid \pm \rangle =\left( \mid 0\rangle \pm \mid 1\rangle \right)/\sqrt{2}$ basis and the desired Fisher information is achieved. However, the same result is achieved adaptively by measuring each qubit in the $\mid \pm \rangle$ basis. After $N-1$ qubits are measured the state of the remaining qubit is $(\mid 0\rangle +{{(-1)}^{k}}{{{\rm e}}^{{\rm i}N\phi }}\mid 1\rangle )/\sqrt{2}$, where k is the number of times the outcome $\mid -\rangle$ was observed. If k is odd we apply a ${{\sigma }_{z}}$ 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.

Let us now choose the probe as the Werner state $\mathcal{W}=(1-\eta )\frac{\mathbb{I}}{4}+\eta \mid {{\mathcal{B}}_{00}}\rangle \langle {{\mathcal{B}}_{00}}\mid$, where $\mid {{\mathcal{B}}_{00}}\rangle =\left( \mid 00\rangle +\mid 11\rangle \right)/\sqrt{2}$. The parameter $\eta \in [0,1]$ is the strength of the signal, while $1-\eta$ indicates the amount of white noise present in the probe state. Using equation (6) we can compute the optimal Fisher information that Charlie can attain

$$\begin{eqnarray}&&{{\mathcal{F}}_{{\rm co}}}(\mathcal{W})=\frac{8{{\eta }^{2}}}{1+\eta }.\end{eqnarray} \tag{ 12 }$$

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 $\mid {{\beta }^{*}}\rangle =b_{0}^{*}\mid 0\rangle +b_{1}^{*}\mid 1\rangle .$ The corresponding conditional states of Alice are $\mathcal{W}_{\phi }^{A\mid \beta }=(1-\eta )\frac{\mathbb{I}}{2}+\eta \mid {{\beta }_{\phi }}\rangle \langle {{\beta }_{\phi }}\mid$, where $\mid {{\beta }_{\phi }}\rangle ={{b}_{0}}\mid 0\rangle +{{{\rm e}}^{{\rm i}2\phi }}{{b}_{1}}\mid 1\rangle .$

The conditional state of Alice is as if Alice had prepared a state ${{\mathcal{W}}^{A\mid \beta }}=(1-\eta )\frac{\mathbb{I}}{2}+\eta \mid \beta \rangle \langle \beta \mid$ and sent it through an interferometer with the phase generated by the Hamiltonian $2\mid 1\rangle \langle 1\mid$. Therefore we can simply compute the QFI using equation (6).

We know that the QFI will be maximized for ${{b}_{0}}={{b}_{1}}=1/\sqrt{2}$, therefore we can conclude that Bob will make measurements $\mid \pm \rangle$. A ${{\sigma }_{z}}$ 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

$$\begin{eqnarray}&&{{\mathcal{F}}_{{\rm ad}}}(\mathcal{W})=4{{\eta }^{2}}.\end{eqnarray} \tag{ 13 }$$

Note that there is a finite difference between the coherent readout and adaptive readout. The difference vanishes only when $\eta =0\;{\rm or}\;1$. 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.

We can represent a multipartite probe with white noise as ${{\mathcal{W}}^{N}}=(1-\eta )\frac{\mathbb{I}}{{{2}^{N}}}+\eta \mid {{G}^{N}}\rangle \langle {{G}^{N}}\mid$ with $\mid {{G}^{N}}\rangle ={{(\mid 0\rangle }^{\otimes N}}+\mid 1{{\rangle }^{\otimes N}})/\sqrt{2}$. The phase shift is now introduced by the unitary ${{U}^{\otimes N}}$ with $U={{{\rm e}}^{{\rm i}\phi \mid 1\rangle \langle 1\mid }}$. Charlie's Fisher information is computed using equation (6) (see appendix B for the details):

$$\begin{eqnarray}&&{{\mathcal{F}}_{{\rm co}}}({{\mathcal{W}}^{N}})=\frac{{{2}^{N}}}{{{2}^{N}}\eta +2(1-\eta )}{{N}^{2}}{{\eta }^{2}}.\end{eqnarray} \tag{ 14 }$$

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:

$$\begin{eqnarray}&&{{\mathcal{F}}_{{\rm ad}}}({{\mathcal{W}}^{N}})={{N}^{2}}{{\eta }^{2}}\end{eqnarray} \tag{ 15 }$$

which is greater than zero for a noisy probe ($\eta \ne 0,1$). The derivation of equations (14) and (15) via a direct calculation of the QFI can be found in appendix B. On the other hand in appendix C, we provide an example where the optimal POVM for both coherent and adaptive strategies is obtained for a given value of ϕ considering a phase estimation protocol. The POVM approach presented in appendix C is shown to yield the same result as the derivation based on QFI.

Now, let us consider the three partition case ($a,b$ and c) with joint probability distribution ${{p}_{\phi }}(a,b,c)$. In this case two successive applications of the definition of conditional probability result in ${{p}_{\phi }}(a,b,c)={{p}_{\phi }}(a\mid bc){{p}_{\phi }}(b\mid c){{p}_{\phi }}(c)$. In this fashion, we can write

$$\begin{eqnarray*}\begin{array}{rcl} \mathcal{F}(A,B,C) & = & \mathop{\sum }\limits_{abc}{{p}_{\phi }}(a,b,c){{\left[ \frac{{{\partial }_{\phi }}{{p}_{\phi }}(a,b,c)}{{{p}_{\phi }}(a,b,c)} \right]}^{2}} \\ {} & = & \mathop{\sum }\limits_{abc}{{p}_{\phi }}(a\mid bc){{p}_{\phi }}(b\mid c){{p}_{\phi }}(c){{\left[ \frac{{{\partial }_{\phi }}[{{p}_{\phi }}(a\mid bc){{p}_{\phi }}(b\mid c){{p}_{\phi }}(c)]}{{{p}_{\phi }}(a\mid bc){{p}_{\phi }}(b\mid c){{p}_{\phi }}(c)} \right]}^{2}}, \\ \end{array}\end{eqnarray*}$$

which after some algebra turns out to be

$$\begin{eqnarray*}\begin{array}{rcl} \mathcal{F}(A,B,C) & = & \mathop{\sum }\limits_{c}\frac{{{\left[ {{\partial }_{\phi }}{{p}_{\phi }}(c) \right]}^{2}}}{{{p}_{\phi }}(c)}+\mathop{\sum }\limits_{c}{{p}_{\phi }}(c)\mathop{\sum }\limits_{b}\frac{{{\left[ {{\partial }_{\phi }}{{p}_{\phi }}(b\mid c) \right]}^{2}}}{{{p}_{\phi }}(b\mid c)} \\ {} & {} & +\mathop{\sum }\limits_{bc}{{p}_{\phi }}(bc)\mathop{\sum }\limits_{a}\frac{{{\left( {{\partial }_{\phi }}{{p}_{\phi }}(a\mid bc) \right)}^{2}}}{{{p}_{\phi }}(a\mid bc)} \\ {} & = & \mathcal{F}(C)+\mathcal{F}(B\mid C)+\mathcal{F}(A\mid BC). \\ \end{array}\end{eqnarray*}$$

By successive applications of the definition of conditional probability, it is easy to show that for a multipartite probe (with N partitions $\{{{X}_{1}},...,{{X}_{N}}\}$), we can write a fancy generalization as

$$\begin{eqnarray}&&\mathcal{F}({{X}_{1}}...{{X}_{N}})=\mathcal{F}({{X}_{N}})+\mathop{\sum }\limits_{k=1}^{N-1}\mathcal{F}({{X}_{k}}\mid {{X}_{k+1}}...{{X}_{N}}).\end{eqnarray} \tag{ A.3 }$$

Here we wish to compute the QFI with coherent readout for the following state

$$\begin{eqnarray}&&{{\mathcal{W}}^{N}}=(1-\eta )\frac{\mathbb{I}}{{{2}^{N}}}+\eta \mid {{G}^{N}}\rangle \langle {{G}^{N}}\mid ,\end{eqnarray} \tag{ B.1 }$$

where $\mid {{G}^{N}}\rangle ={{(\mid 0\rangle }^{\otimes N}}+\mid 1{{\rangle }^{\otimes N}})/\sqrt{2}.$ To do this we need the eigenvectors and eigenvalues of the state above. Eigenvector $\mid {{G}^{N}}\rangle$ comes with eigenvalue $\eta +(1-\eta )/{{2}^{N}}$ and all other ${{2}^{N}}-1$ eigenvectors come with eigenvalue $(1-\eta )/{{2}^{N}}$. Next note that the Hamiltonian that encodes the parameter to be estimated is ${{H}_{N}}=\oplus _{i}^{{}}{{H}_{i}}$ with ${{H}_{i}}=\mid 1{{\rangle }^{{}}}\langle 1\mid$.

Now we simply utilise equation (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 $\mid {{G}^{N}}\rangle$ is ${{(1/\sqrt{2}\mid 1\rangle }^{\otimes N}})$. Therefore the only other eigenvector that matters is $\mid {{\bar{G}}^{N}}\rangle ={{(\mid 0\rangle }^{\otimes N}}-\mid 1{{\rangle }^{\otimes N}})/\sqrt{2}$. This is because all other eigenstates of ${{\varrho }_{N}}$ are orthonormal to the $\mid 1{{\rangle }^{\otimes N}}$ term. The QFI is then:

$$\begin{eqnarray}&&{{\mathcal{F}}_{{\rm co}}}({{\mathcal{W}}^{N}})=4\frac{{{\left( \eta +(1-\eta )/{{2}^{N}}-(1-\eta )/{{2}^{N}} \right)}^{2}}}{\eta +(1-\eta )/{{2}^{N}}+(1-\eta )/{{2}^{N}}}\mid {{\langle {{\bar{G}}^{N}}\mid {{H}_{N}}\mid {{G}^{N}}\rangle }\mid ^{2}}\end{eqnarray} \tag{ B.2 }$$

$$\begin{eqnarray}&&\quad \,=\frac{{{\eta }^{2}}}{\eta +2(1-\eta )/{{2}^{N}}}\mid \left( \langle 0{{\mid }^{\otimes N}}-\langle 1{{\mid }^{\otimes N}} \right)N\mid 1{{\rangle }^{\otimes N}}\mid ^{2}\end{eqnarray} \tag{ B.3 }$$

$$\begin{eqnarray}&&=\frac{{{2}^{N}}}{{{2}^{N}}\eta +2(1-\eta )}{{N}^{2}}{{\eta }^{2}}.\end{eqnarray} \tag{ B.4 }$$

Now let us compute the QFI for adaptive readout. For N qubits the adaptive strategy reduces to making (probabilistic) projective measurements. We begin by considering the measurement by the first qubit. Since the local state of this qubit is maximally mixed, the local Fisher information vanishes. Therefore, to maximize the Fisher information for adaptive readout, the Fisher information of the remaining $N-1$ qubits must be maximized.

After the parameter has been encoded the state of N qubits is

$$\begin{eqnarray}\begin{array}{rcl} \mathcal{W}_{\phi }^{N} & = & (1-\eta )\frac{\mathbb{I}}{{{2}^{N}}}+\eta \frac{1}{2}\left( \mid 0{{\rangle }^{\otimes N}}+{{{\rm e}}^{{\rm i}N\phi }}\mid 1{{\rangle }^{\otimes N}} \right) \\ {} & {} & \times \left( \langle 0{{\mid }^{\otimes N}}+{{{\rm e}}^{-{\rm i}N\phi }}\langle 1{{\mid }^{\otimes N}} \right). \\ \end{array}\end{eqnarray} \tag{ B.5 }$$

Let us imagine that the last party has a measurement outcome along some direction $\langle m\mid ={{m}_{0}}\langle 0\mid +{{m}_{1}}\langle 1\mid$, with $\mid {{m}_{0}}{{\mid }^{2}}+\mid {{m}_{1}}{{\mid }^{2}}=1$. The corresponding conditional state of the remaining qubits is

$$\begin{eqnarray}\begin{array}{rcl} \mathcal{W}_{\phi }^{N-1} & = & (1-\eta )\frac{\mathbb{I}}{{{2}^{N-1}}}+\eta \left( {{m}_{0}}\mid 0{{\rangle }^{\otimes N-1}}+{{m}_{1}}{{{\rm e}}^{{\rm i}N\phi }}\mid 1{{\rangle }^{\otimes N-1}} \right) \\ {} & {} & \times \left( m_{0}^{*}\langle 0{{\mid }^{\otimes N-1}}+m_{1}^{*}{{{\rm e}}^{-{\rm i}N\phi }}\langle 1{{\mid }^{\otimes N-1}} \right) \\ \end{array}\end{eqnarray} \tag{ B.6 }$$

We want to find values of m₀ and m₁ 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 $N-1$

$$\begin{eqnarray}\begin{array}{rcl} {{\mathcal{W}}^{N-1}} & = & (1-\eta )\frac{\mathbb{I}}{{{2}^{N-1}}}+\eta \left( {{m}_{0}}\mid 0{{\rangle }^{\otimes N-1}}+{{m}_{1}}\mid 1{{\rangle }^{\otimes N-1}} \right) \\ {} & {} & \times \left( m_{0}^{*}\langle 0{{\mid }^{\otimes N-1}}+m_{1}^{*}\langle 1{{\mid }^{\otimes N-1}} \right) \\ \end{array}\end{eqnarray} \tag{ B.7 }$$

and sent it through a process generated by the Hamiltonian $\frac{N}{N-1}\oplus {{_{i}^{{}}}^{{}}}\mid 1\rangle \langle 1\mid$. The coherent QFI of the last state can be computed to be

$$\begin{eqnarray}&&{{\mathcal{F}}_{{\rm co}}}({{\mathcal{W}}^{N-1}})=4\frac{{{2}^{N-1}}}{{{2}^{N-1}}-({{2}^{N-1}}-2)(1-\eta )}{{N}^{2}}\mid {{m}_{0}}{{m}_{1}}{{\mid }^{2}}{{\eta }^{2}},\end{eqnarray} \tag{ B.8 }$$

which is maximum for ${{m}_{0}}={{m}_{1}}=1/\sqrt{2}$. Therefore we can conclude that the first measurement is in the $\mid \pm \rangle$ basis. The resultant conditional state of the $N-1$ qubits is

$$\begin{eqnarray}\begin{array}{rcl} \mathcal{W}_{\phi }^{N-1} & = & (1-\eta )\frac{\mathbb{I}}{{{2}^{N-1}}}+\eta \frac{1}{2}\left( \mid 0{{\rangle }^{\otimes N-1}}\pm {{{\rm e}}^{{\rm i}N\phi }}\mid 1{{\rangle }^{\otimes N-1}} \right) \\ {} & {} & \times \left( \langle 0{{\mid }^{\otimes N-1}}\pm {{{\rm e}}^{-{\rm i}N\phi }}\langle 1{{\mid }^{\otimes N-1}} \right). \\ \end{array}\end{eqnarray} \tag{ B.9 }$$

But this state looks exactly like the N qubit state we started with in equation (B.5). Therefore by carrying out the exact same analysis we find that the next measurement also has to be in the $\mid \pm \rangle$ basis.

After all but the last party has measured the final qubit has the state

$$\begin{eqnarray}&&\mathcal{W}_{\phi }^{1}=(1-\eta )\frac{\mathbb{I}}{2}+\eta \frac{1}{2}\left( \mid 0\rangle \pm {{{\rm e}}^{{\rm i}N\phi }}\mid 1\rangle \right)\left( \langle 0\mid \pm {{{\rm e}}^{-{\rm i}N\phi }}\langle 1\mid \right).\end{eqnarray} \tag{ B.10 }$$

Let us denote the number of parties that observe $\mid -\rangle$ with k. If k is odd then a ${{\sigma }_{z}}$ is applied to change the minus sign in the superposition. This state is exactly as if we had prepared the state

$$\begin{eqnarray}&&\mathcal{W}_{\phi }^{1}=(1-\eta )\frac{\mathbb{I}}{2}+\eta \frac{1}{2}\left( \mid 0\rangle +\mid 1\rangle \right)\left( \langle 0\mid +\langle 1\mid \right)\end{eqnarray} \tag{ B.11 }$$

and sent it through a process generated by Hamiltonian $N\mid 1\rangle \langle 1\mid$. The Fisher information for this Hamiltonian and the last state is the Fisher information for adaptive readout:

$$\begin{eqnarray}&&{{\mathcal{F}}_{{\rm ad}}}({{\mathcal{W}}^{N}})=\frac{{{\eta }^{2}}}{\eta +(1-\eta )}{{N}^{2}}={{\eta }^{2}}{{N}^{2}}.\end{eqnarray} \tag{ B.12 }$$

As in any practical scenario we will restrict our analysis to some class of POVMs. Note that in a given experimental setup the kind of operations that we can perform without adding ancillary systems is limited by the nature of the setup. We will start from the optimal POVM for the global coherent strategy in the the noiseless case (outlined in figure C1 (i)). We will further see that this choice of POVM is optimal for a specific value of ϕ. Indeed, this POVM is the set of projectors on the Bell basis state (note that ${{\mathcal{W}}^{N}}$ is a Bell diagonal state). Such a readout can be constructed by means of a sequence of coherent interactions (C-not gates) as indicated in figure C1(i). We can compactly represent the N C-not gates in the readout strategy as

$$\begin{eqnarray}&&{\rm C}_{{\rm NOT}}^{N}=\mid 0\rangle \langle 0\mid \otimes \mathbb{I}_{2}^{\otimes N-1}+\mid 1\rangle \langle 1\mid \otimes \sigma _{x}^{\otimes N-1},\end{eqnarray} \tag{ C.2 }$$

so that

$$\begin{eqnarray}\begin{array}{rcl} {\rm C}_{{\rm NOT}}^{N}\mathcal{W}_{\phi }^{N}{\rm C}_{{\rm NOT}}^{N} & = & \left[ \eta \mid {{G}_{\phi }}\rangle \langle {{G}_{\phi }}\mid +(1-\eta )\frac{{{\mathbb{I}}_{2}}}{{{2}^{N}}} \right]\otimes \mid 0\rangle \langle 0{{\mid }^{\otimes N-1}} \\ {} & {} & +(1-\eta )\frac{{{\mathbb{I}}_{2}}}{{{2}^{N}}}\otimes \left( \mathbb{I}_{2}^{\otimes N-1}-\mid 0\rangle \langle 0{{\mid }^{\otimes N-1}} \right), \\ \end{array}\end{eqnarray} \tag{ C.3 }$$

with $\mid {{G}_{\phi }}\rangle =\left( \mid 0\rangle +{{{\rm e}}^{{\rm i}N\phi }}\mid 1\rangle \right)/\sqrt{2}$. The global POVM set can be written as $\{\Pi _{k}^{G}={\rm C}_{{\rm NOT}}^{N}H\mid {{k}_{({\rm bin})}}\rangle \langle {{k}_{({\rm bin})}}\mid H{\rm C}_{{\rm NOT}}^{N}\}$, where $k=0,1,2,...,({{2}^{N}}-1)$ and ${{k}_{({\rm bin})}}$ is the binary representation of k with N-bits. In this readout scheme the obtained probability distribution reads

$$\begin{eqnarray}&&{{p}_{\phi }}(k)={\rm Tr}\;\left( \mathcal{W}_{\phi }^{N}\Pi _{k}^{G} \right).\end{eqnarray} \tag{ C.4 }$$

Since $\frac{\partial }{\partial \phi }{{p}_{\phi }}(k)=0$ for $k\ne 0,{{2}^{N-1}}$, the global Fisher information, in this case, is given by

$$\begin{eqnarray}&&{{\mathcal{F}}_{{\rm co}}}=\mathop{\sum }\limits_{k}\frac{{{[{{\partial }_{\phi }}{{p}_{\phi }}(k)]}^{2}}}{{{p}_{\phi }}(k)}=\frac{{{[{{\partial }_{\phi }}{{p}_{\phi }}(0)]}^{2}}}{{{p}_{\phi }}(0)}+\frac{{{[{{\partial }_{\phi }}{{p}_{\phi }}({{2}^{N-1}})]}^{2}}}{{{p}_{\phi }}({{2}^{N-1}})},\end{eqnarray} \tag{ C.5 }$$

where ${{p}_{\phi }}(0)=\eta {{{\rm cos} }^{2}}\left( N\frac{\phi }{2} \right)+\frac{1-\eta }{{{2}^{N}}}$ and ${{p}_{\phi }}({{2}^{N-1}})=\eta {{{\rm sin} }^{2}}\left( N\frac{\phi }{2} \right)+\frac{1-\eta }{{{2}^{N}}}$. From which it follows that

$$\begin{eqnarray}&&\mathop{{\rm max} }\limits_{\{\phi \}}{{\mathcal{F}}_{{\rm co}}}=\frac{{{2}^{N}}}{{{2}^{N}}\eta +2(1-\eta )}{{N}^{2}}{{\eta }^{2}}.\end{eqnarray} \tag{ C.6 }$$

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In the adaptive strategy, the set of local POVMs corresponding to the circuit displayed in figure C1(ii) is ${{\left\{ \Pi _{1\mid 2...N}^{({{x}_{1}})}\otimes \left(\otimes _{k=2}^{N}\Pi _{k}^{({{x}_{k}})} \right) \right\}}^{{}}}$. The measurement on the qubits $2,3,...,N$ is a projective measurement performed in the ${{\sigma }_{x}}$ basis and $\Pi _{k}^{({{x}_{k}})}=\mid {{x}_{k}}\rangle \langle {{x}_{k}}\mid$ for $k\geqslant 2$, where ${{x}_{k}}=+,-$. This is an optimal POVM for the noiseless scenario. A convenient form for these projectors is given by $\Pi _{k}^{({{x}_{k}})}=\left( {{\mathbb{I}}_{2}}+{{(-1)}^{f({{x}_{k}})}}{{\sigma }_{x}} \right)/2$ with $f({{x}_{k}})=0,1$. Therefore, the POVM set for the adaptive strategy can be written as

$$\begin{eqnarray}&&\left\{ {{\Pi }_{{\rm ad}}}=\sigma _{z}^{\mu }H\mid {{x}_{1}}\rangle \langle {{x}_{1}}\mid H\sigma _{z}^{\mu }\otimes \left( \mathop{\otimes }\limits_{k=2}^{N}\Pi _{k}^{({{x}_{k}})} \right) \right\},\end{eqnarray} \tag{ C.7 }$$

where ${{x}_{1}}=0,1$ and $\sum _{k=2}^{N}f({{x}_{k}})\equiv \mu ({\rm mod} \;\;2)$ is the parity of the outcomes $\{{{x}_{2}},...,{{x}_{N}}\}$.

Let us inspect the action of these measurements on the state $\mathcal{W}_{\phi }^{N}$ given in equation C.1. Considering the operation $\Pi _{N}^{({{x}_{N}})}$ on the Nth qubit, we have

$$\begin{eqnarray}&&\Pi _{N}^{({{x}_{N}})}\mathcal{W}_{\phi }^{N}\Pi _{N}^{({{x}_{N}})}=\left[ \eta \mid G_{\phi }^{N-1}\rangle \langle G_{\phi }^{N-1}\mid +(1-\eta )\frac{\mathbb{I}_{2}^{\otimes N-1}}{{{2}^{N-1}}} \right]\otimes \frac{\Pi _{N}^{({{x}_{N}})}}{2},\end{eqnarray} \tag{ C.8 }$$

where $\mid G_{\phi }^{N-1}\rangle =\left( \mid 0{{\rangle }^{\otimes N-1}}+{{(-1)}^{f({{x}_{N}})}}{{{\rm e}}^{{\rm i}N\phi }}\mid 1{{\rangle }_{\otimes N-1}} \right)/\sqrt{2}$ and x_N is the outcome of the measurement on the Nth qubit. Now, applying the operations ${{\otimes _{k=2}^{N-1}}^{{}}}\Pi _{k}^{({{x}_{k}})}$ on the $\{2,...,N-1\}$ remaining qubits, we have

$$\begin{eqnarray}&&\begin{array}{l} {{\mathbb{I}}_{2}}\otimes \left( \mathop{\otimes }\limits_{k=2}^{N}\Pi _{k}^{({{x}_{k}})} \right)\;{{\rho }_{\phi }}\;{{\mathbb{I}}_{2}}\otimes \left( \mathop{\otimes }\limits_{k=2}^{N}\Pi _{k}^{({{x}_{k}})} \right)=\left( \frac{\eta }{2}\mid G_{\phi }^{\mu }\rangle \langle G_{\phi }^{\mu }\mid +(1-\eta )\frac{{{\mathbb{I}}_{2}}}{2} \right) \\ \quad \otimes \left( \mathop{\otimes }\limits_{k=2}^{N}\Pi _{k}^{({{x}_{k}})} \right)\times \frac{1}{{{2}^{N-1}}}, \\ \end{array}\end{eqnarray} \tag{ C.9 }$$

where $\mid G_{\phi }^{\mu }\rangle =\left( \mid 0\rangle +{{(-1)}^{\mu }}{{{\rm e}}^{{\rm i}N\phi }}\mid 1\rangle \right)/\sqrt{2}$. At this stage the probe is an uncorrelated state and all the information about ϕ was transferred to the qubit 1.

The last step is to manifest the phase in a probability distribution via the classically conditioned operation $\Pi _{1\mid 2...N}^{({{x}_{1}})}={{({{\sigma }_{z}})}^{\mu }}H\mid {{x}_{1}}\rangle \langle {{x}_{1}}\mid H{{({{\sigma }_{z}})}^{\mu }}$ on the qubit ${{X}_{1}}$. This operation removes the extra phase ${{(-1)}^{\mu }}$ from $\mid G_{\phi }^{\mu }\rangle \langle G_{\phi }^{\mu }\mid$ resulting in

$$\begin{eqnarray*}&&\left[ \eta \mid \phi \rangle \langle \phi \mid +(1-\eta )\frac{{{\mathbb{I}}_{2}}}{2} \right]\otimes \left( \mathop{\otimes }\limits_{k=2}^{N}\Pi _{k}^{({{x}_{k}})} \right)\times \frac{1}{{{2}^{N-1}}},\end{eqnarray*}$$

where $\mid \phi \rangle ={\rm cos} \left( N\frac{\phi }{2} \right)\mid 0\rangle +{\rm i}\;{\rm sin} \left( N\frac{\phi }{2} \right)\mid 1\rangle$. The Fisher information turns out to be

$$\begin{eqnarray}&&{{\mathcal{F}}_{{\rm ad}}}=\mathop{\sum }\limits_{k=2}^{N}\mathcal{F}({{p}_{k}})+\mathop{\sum }\limits_{\{{{x}_{k}}\}}{{p}_{2}}({{x}_{2}})...{{p}_{N}}({{x}_{N}})\mathcal{F}({{p}_{1\mid {{x}_{2}}...{{x}_{N}}}}),\end{eqnarray} \tag{ C.10 }$$

with $\mathcal{F}({{p}_{k}})$ being the Fisher information for the local probability distribution, ${{p}_{k}}({{x}_{k}})$, associated to the measurement of $\Pi _{k}^{({{x}_{k}})}$ (on the kth qubit). Regarding the local distributions, we have ${{\partial }_{\phi }}{{p}_{k}}=0$ for $k\ne 1$ implying that the local Fisher information vanishes, $\sum _{k=2}^{N}\mathcal{F}({{p}_{k}})=0$. Moreover, the product ${{p}_{2}}({{x}_{2}})\times ...\times {{p}_{N}}({{x}_{N}})=1/{{2}^{N-1}}$ for any combination of outcomes $\{{{x}_{k}}\;\mid \;k=2,...,N\}$.

The Fisher information of the multipartite adaptive strategy reduces to ${{\mathcal{F}}_{{\rm ad}}}=\mathcal{F}({{p}_{1}})$, where ${{p}_{1}}({{x}_{1}})={\rm Tr}({{\rho }_{1}}\mid {{x}_{1}}\rangle \langle {{x}_{1}}\mid )$ (with ${{x}_{1}}=0,1$) and ${{\rho }_{1}}=\left[ \eta \mid \phi \rangle \langle \phi \mid +(1-\eta ){{\mathbb{I}}_{2}}/2 \right]$. Now, we have ${{p}_{\phi }}(0)=\eta {{{\rm cos} }^{2}}\left( N\frac{\phi }{2} \right)+\frac{1-\eta }{2}$ and ${{p}_{\phi }}(1)=\eta {{{\rm sin} }^{2}}\left( N\frac{\phi }{2} \right)+\frac{1-\eta }{2}$, and thus we have

$$\begin{eqnarray}&&\mathop{{\rm max} }\limits_{\{\phi \}}{{\mathcal{F}}_{{\rm ad}}}={{N}^{2}}{{\eta }^{2}}.\end{eqnarray} \tag{ C.11 }$$

We observe that the maxima in equations (C.6) and (C.11) occur at the same value, $\phi =\pi /(2N)$. Moreover, the Fisher informations in equations (C.6) and (C.11) that comes from the POVM implemented in circuit depicted in figure C1 are equal to the ones obtained from the QFI formula in the appendix B. This shows that the POVMs implemented here are optimal at $\phi =\pi /(2N)$. It is worthwhile to observe that, we are interested in small deviations of a given value of phase. Since we only measure relative phases, we can always calibrate the interferometer to measure a small deviation from $\pi /(2N)$. So, the optimal POVM presented above can be used to measure a small deviation around any value of ϕ by suitably calibrating the measurement apparatus in our example. This of course assumes that $\mathcal{F}(\phi )$ is an analytic function, which is trivially true here; it is a mild assumption that can be expected to break down in only the most pathological examples.