Noisy frequency estimation with noisy probes

Each of the n + 1 qubits making up the probe can, in general, be in a different mixed state initially, with the ith qubit in state ρ_i. The overall initial state is the product

$$\begin{eqnarray}{{\rm{\Omega }}}_{0}=\underset{i=0}{\overset{n}{\otimes }}{\rho }_{i},\quad {\rm{where}}\quad {\rho }_{i}=\left(\begin{array}{cc}{\lambda }_{0}^{(i)} & 0\\ 0 & {\lambda }_{1}^{(i)}\end{array}\right),\quad {\lambda }_{0}^{(i)}=\displaystyle \frac{1+{p}_{i}}{2}\quad {\rm{and}}\quad {\lambda }_{1}^{(i)}=\displaystyle \frac{1-{p}_{i}}{2}.\end{eqnarray} \tag{ 1 }$$

Here, ${\lambda }_{0}^{(i)}$ and ${\lambda }_{1}^{(i)}$ are the probabilities of the ith qubit being in state $| 0\rangle$ and $| 1\rangle$, respectively. This setup could describe an experiment which senses magnetic or gravitational fields using an ensemble of atoms [20, 39]. Such an ensemble can be initially aligned with the direction an external magnetic field so that the Bloch vector of each qubit will be pointing in the same direction. In that case we have ${p}_{i}\in [0,1]$ and ${\lambda }_{0}^{(i)}\geqslant {\lambda }_{1}^{(i)}$.

This initial state may be fully described by a vector ${\bf{p}}=({p}_{0},{p}_{1},\,...,{p}_{n})$. By setting these parameters to different values, we can obtain any combination of diagonal mixed states ranging from fully mixed (${p}_{i}$ = 0) to pure (${p}_{i}$ = 1). In section 4.1 we will show a simple relationship between the sensitivity of our probe and ${\bf{p}}$. And in section 4.2, we will examine two special cases, which we dub uniform, where all ${p}_{i}=p$ for $i\in [0,n]$ and tilted, where p₀ = 1 and _{${p}_{i}$} = 0 for all i > 0.

As the 0th qubit is used to make control operation on the rest, we call it the special probe qubit (SPQ), while the latter n qubits are called register probe qubits (RPQs). Throughout this article we represent the state of the RPQs in the matrix basis of the SPQ:

$$\begin{eqnarray}{{\rm{\Omega }}}_{0}=\left(\begin{array}{cc}{\lambda }_{0}\underset{i=1}{\overset{n}{\otimes }}{\rho }_{i} & 0\\ 0 & {\lambda }_{1}\underset{i=1}{\overset{n}{\otimes }}{\rho }_{i}\end{array}\right),\end{eqnarray} \tag{ 2 }$$

where for simplicity we have dropped the superscript 0 from the λ's in the state ρ₀.

Parameter estimation with a pure state is optimal when the probe state is taken to be a GHZ state $\tfrac{1}{\sqrt{2}}(| 0...0\rangle +| 1...1\rangle )$. To attain optimal performance in this limit, we adopt a preparation procedure which creates a GHZ state in the case ${\bf{p}}=(1,\,\ldots ,\,1);$ more generally, we end up with a mixed GHZ-diagonal state, i.e., a mixture of state of the form $\tfrac{1}{\sqrt{2}}(| {\bf{k}}\rangle +| {\bf{1}}-{\bf{k}}\rangle )$, where ${\bf{k}}:= ({k}_{1},\,...,\,{k}_{n})$ with k_j = 0,1. Here the superscript denotes the jth qubit, and ${\bf{1}}:= (1,\,...,\,1)$.

In fact, we conjecture that GHZ diagonal states lead to optimal performance for any configuration of noisy qubits in equation (1). The preparation involves three steps: first we apply a set of controlled-not (Cnot) gates to all RPQs with the SPQ as the control. The Cnot gate applies the Pauli matrix X to the target if the control qubit SPQ is in the state $| 1\rangle$, and the identity operator ${\bf{I}}$ if the control qubit is in the state $| 0\rangle$. Next, a Hadamard gate is applied to the SPQ, followed by another set of Cnot gates on the RPQs, again controlled by the SPQ.

The state after the full preparation procedure has the following form:

$$\begin{eqnarray}{{\rm{\Omega }}}_{\mathrm{probe}}=\displaystyle \frac{{\lambda }_{0}}{2}\left(\begin{array}{cc}\underset{i=1}{\overset{n}{\otimes }}{\rho }_{i} & \underset{i=1}{\overset{n}{\otimes }}{\rho }_{i}{X}_{i}\\ \underset{i=1}{\overset{n}{\otimes }}{X}_{i}{\rho }_{i} & \underset{i=1}{\overset{n}{\otimes }}{X}_{i}{\rho }_{i}{X}_{i}\end{array}\right)+\displaystyle \frac{{\lambda }_{1}}{2}\left(\begin{array}{cc}\underset{i=1}{\overset{n}{\otimes }}{X}_{i}{\rho }_{i}{X}_{i} & -\underset{i=1}{\overset{n}{\otimes }}{X}_{i}{\rho }_{i}\\ -\underset{i=1}{\overset{n}{\otimes }}{\rho }_{i}{X}_{i} & \underset{i=1}{\overset{n}{\otimes }}{\rho }_{i}\end{array}\right).\end{eqnarray} \tag{ 3 }$$

The subscript i in X_i denotes that the X operator acts on the ith qubit. Note here that the matrix ${X}_{i}{\rho }_{i}{X}_{i}$ is diagonal in the same basis as ρ_i, but with the eigenvalues switched:

$$\begin{eqnarray}{X}_{i}{\rho }_{i}{X}_{i}=\left(\begin{array}{cc}{\lambda }_{1}^{(i)} & 0\\ 0 & {\lambda }_{0}^{(i)}\end{array}\right).\end{eqnarray} \tag{ 4 }$$

This prepared state is a GHZ diagonal state, which is easily seen by tracking how its eigenvectors are built up. After the application of the first Cnot gate and the Hadamard gate, the eigenvalues and eigenvectors of the probe state are

$$\begin{eqnarray}&&\{{\lambda }_{0}{\lambda }_{{k}_{1}}^{(1)}...{\lambda }_{{k}_{n}}^{(n)},| +,\,{\bf{k}}\rangle \}\quad {\rm{and}}\quad \{{\lambda }_{1}{\lambda }_{{k}_{1}}^{(1)}...{\lambda }_{{k}_{n}}^{(n)},| -,\,{\bf{1}}-{\bf{k}}\rangle \},\end{eqnarray} \tag{ 5 }$$

where $| \pm \rangle =\tfrac{1}{\sqrt{2}}(| 0\rangle \pm | 1\rangle )$. After the second Cnot gate, the eigenvectors and eigenvalues of the probe state are

$$\begin{eqnarray}&&\left\{{g}_{{\bf{k}}}^{(+)},| {G}_{{\bf{k}}}^{(+)}\rangle := \displaystyle \frac{| 0,{\bf{k}}\rangle +| 1,{\bf{1}}-{\bf{k}}\rangle }{\sqrt{2}}\right\},\quad \left\{{g}_{{\bf{1}}-{\bf{k}}}^{(-)},| {G}_{{\bf{1}}-{\bf{k}}}^{(-)}\rangle := \displaystyle \frac{| 0,{\bf{1}}-{\bf{k}}\rangle -| 1,{\bf{k}}\rangle }{\sqrt{2}}\right\},\end{eqnarray} \tag{ 6 }$$

where ${g}_{{\bf{k}}}^{(+)}={\lambda }_{0}{\lambda }_{{k}_{1}}^{(1)}...{\lambda }_{{k}_{n}}^{(n)}$ and ${g}_{{\bf{1}}-{\bf{k}}}^{(-)}:= {\lambda }_{1}{\lambda }_{{k}_{1}}^{(1)}...{\lambda }_{{k}_{n}}^{(n)}$. While the eigenvalue corresponding to $| {G}_{{\bf{k}}}^{(-)}\rangle$ is ${g}_{{\bf{k}}}^{(-)}={\lambda }_{1}{\lambda }_{1-{k}_{1}}^{(1)}...{\lambda }_{1-{k}_{n}}^{(n)}$, which we will make use of a little later.

The application of two Cnot gates may seem redundant. However, it is well known that this probe outperforms the probe where the first Cnot is omitted [17, 18]. We will show in section 4.3 that the first Cnot gate also introduces a nice symmetry in our protocol.

We are now ready to encode the parameter we want to estimate onto our prepared state. The parameter is encoded via free evolution of the probe. However, as we encode the desired parameter, the free evolution also introduces noise, which can be described by a phenomenological quantum master equation [40–42].

For simplicity, we restrict ourselves to a pure dephasing process that commutes with the encoding process. This means we can treat the parameter encoding and noise as occurring sequentially. With minor modifications we can consider a more general phase-covariant form for the noisy process³ . However, the case we consider is sufficient to show that 'Zeno scaling' is possible even when the probe state is noisy.

The parameter ω stems from a classical field, e.g. magnetic field, and it is applied identically to all qubits through the unitary operation ${U}^{\otimes n+1}$, with $U=\exp [-{\rm{i}}\omega {tZ}/2]$, where Z is a Pauli matrix and t is the time of free evolution. The system just after the free evolution is in the following state:

$$\begin{eqnarray}{{\rm{\Omega }}}_{\omega } & = & \displaystyle \frac{{\lambda }_{0}}{2}\left(\begin{array}{cc}\underset{i=1}{\overset{n}{\otimes }}{\rho }_{i} & {{\rm{e}}}^{-{\rm{i}}\omega t}\underset{i=1}{\overset{n}{\otimes }}{U}_{i}{\rho }_{i}{X}_{i}{U}_{i}^{\dagger }\\ {{\rm{e}}}^{{\rm{i}}\omega t}\underset{i=1}{\overset{n}{\otimes }}{U}_{i}{X}_{i}{\rho }_{i}{U}_{i}^{\dagger } & \underset{i=1}{\overset{n}{\otimes }}{X}_{i}{\rho }_{i}{X}_{i}\end{array}\right)\\ & & +\displaystyle \frac{{\lambda }_{1}}{2}\left(\begin{array}{cc}\underset{i=1}{\overset{n}{\otimes }}{X}_{i}{\rho }_{i}{X}_{i} & -{{\rm{e}}}^{-{\rm{i}}\omega t}\underset{i=1}{\overset{n}{\otimes }}{U}_{i}{X}_{i}{\rho }_{i}{U}_{i}^{\dagger }\\ -{{\rm{e}}}^{{\rm{i}}\omega t}\underset{i=1}{\overset{n}{\otimes }}{U}_{i}{\rho }_{i}{X}_{i}{U}_{i}^{\dagger } & \underset{i=1}{\overset{n}{\otimes }}{\rho }_{i}\end{array}\right).\end{eqnarray} \tag{ 7 }$$

The unitary operator on the SPQ has already been applied implicitly. Note that the states ρ_i and ${X}_{i}{\rho }_{i}{X}_{i}$ commute with the unitary matrix U_i. As before, the subscript i in U_i denotes the unitary operator action on the state ρ_i.

While the parameter is encoded, the probe is also subjected to dephasing noise. We can describe such a noisy process on the probe by the action of a superoperator Λ that consists of two Kraus operators L₀ and L₁:

$$\begin{eqnarray}&&{\rm{\Lambda }}[\rho ]=\displaystyle \sum _{i=0,1}{L}_{i}\rho {L}_{i}^{\dagger },\\ &&\quad {\rm{where}}\quad {L}_{0}=\sqrt{\displaystyle \frac{1+\exp (-{{gt}}^{\alpha })}{2}}{\bf{I}}\quad {\rm{and}}\quad {L}_{1}=\sqrt{\displaystyle \frac{1-\exp (-{{gt}}^{\alpha })}{2}}Z,\end{eqnarray} \tag{ 8 }$$

with the rate of dephasing g a positive real number and α a constant which determines the type of noise we face. When α = 1, the noisy process is described by a semigroup [43, 44], i.e., ${{\rm{\Lambda }}}_{\tau }\,\circ \,{{\rm{\Lambda }}}_{t}={{\rm{\Lambda }}}_{\tau +t}$. On the other hand, $\alpha \ne 1$ corresponds to a time-inhomogeneous indivisible process.

Although the encoding and the noisy process are happening simultaneously, we can consider them as sequential processes, because the actions of Λ and U commute. Using the fact that ${{\rm{\Lambda }}}_{i}[{\rho }_{i}]={{\rm{\Lambda }}}_{i}[{X}_{i}{\rho }_{i}{X}_{i}]={\rho }_{i}$, when ρ_i is diagonal, and ${U}_{i}{{\rm{\Lambda }}}_{i}[\rho ]{U}_{i}^{\dagger }={{\rm{\Lambda }}}_{i}[{U}_{i}\rho {U}_{i}^{\dagger }]$ for any ρ, we can compute the state after the noisy encoding:

$$\begin{eqnarray}{{\rm{\Omega }}}_{\omega ,g} & = & \displaystyle \frac{{\lambda }_{0}}{2}\left(\begin{array}{cc}\underset{i=1}{\overset{n}{\otimes }}{\rho }_{i} & {{\rm{e}}}^{-{\rm{i}}\omega t}{{\rm{e}}}^{-{{gt}}^{\alpha }}\underset{i=1}{\overset{n}{\otimes }}{{\rm{\Lambda }}}_{i}[{U}_{i}{\rho }_{i}{X}_{i}{U}_{i}^{\dagger }]\\ {{\rm{e}}}^{{\rm{i}}\omega t}{{\rm{e}}}^{-{{gt}}^{\alpha }}\underset{i=1}{\overset{n}{\otimes }}{{\rm{\Lambda }}}_{i}[{U}_{i}{X}_{i}{\rho }_{i}{U}_{i}^{\dagger }] & \underset{i=1}{\overset{n}{\otimes }}{X}_{i}{\rho }_{i}{X}_{i}\end{array}\right)\\ & & +\displaystyle \frac{{\lambda }_{1}}{2}\left(\begin{array}{cc}\underset{i=1}{\overset{n}{\otimes }}{X}_{i}{\rho }_{i}{X}_{i} & -{{\rm{e}}}^{-{\rm{i}}\omega t}{{\rm{e}}}^{-{{gt}}^{\alpha }}\underset{i=1}{\overset{n}{\otimes }}{{\rm{\Lambda }}}_{i}[{U}_{i}{X}_{i}{\rho }_{i}{U}_{i}^{\dagger }]\\ -{{\rm{e}}}^{{\rm{i}}\omega t}{{\rm{e}}}^{-{{gt}}^{\alpha }}\underset{i=1}{\overset{n}{\otimes }}{{\rm{\Lambda }}}_{i}[{U}_{i}{\rho }_{i}{X}_{i}{U}_{i}^{\dagger }] & \underset{i=1}{\overset{n}{\otimes }}{\rho }_{i}\end{array}\right).\end{eqnarray} \tag{ 9 }$$

The measurement procedure consists of two steps: a final Cnot gate applied to all RPQs with the SPQ as the control followed by a measurement of the RPQs in the Z basis and the SPQ in the X basis. In the appendix, we consider a different readout strategy, where only the SPQ is measured and all RPQs are discarded. In the case we are treating here, the final (pre-measurement) state, after the third Cnot gate, takes the following form

$$\begin{eqnarray}{{\rm{\Omega }}}_{\mathrm{final}} & = & \displaystyle \frac{{\lambda }_{0}}{2}\left(\begin{array}{cc}\underset{i=1}{\overset{n}{\otimes }}{\rho }_{i} & {{\rm{e}}}^{-{\rm{i}}\omega t}{{\rm{e}}}^{-(n+1){{gt}}^{\alpha }}\underset{i=1}{\overset{n}{\otimes }}{U}_{i}^{2}{\rho }_{i}\\ {{\rm{e}}}^{{\rm{i}}\omega t}{{\rm{e}}}^{-(n+1){{gt}}^{\alpha }}\underset{i=1}{\overset{n}{\otimes }}{\rho }_{i}{{U}_{i}^{\dagger }}^{2} & \underset{i=1}{\overset{n}{\otimes }}{\rho }_{i}\end{array}\right)\\ & & +\displaystyle \frac{{\lambda }_{1}}{2}\left(\begin{array}{cc}\underset{i=1}{\overset{n}{\otimes }}{X}_{i}{\rho }_{i}{X}_{i} & -{{\rm{e}}}^{-{\rm{i}}\omega t}{{\rm{e}}}^{-(n+1){{gt}}^{\alpha }}\underset{i=1}{\overset{n}{\otimes }}{U}_{i}^{2}{X}_{i}{\rho }_{i}{X}_{i}\\ -{{\rm{e}}}^{{\rm{i}}\omega t}{{\rm{e}}}^{-(n+1){{gt}}^{\alpha }}\underset{i=1}{\overset{n}{\otimes }}{X}_{i}{\rho }_{i}{X}_{i}{{U}_{i}^{\dagger }}^{2} & \underset{i=1}{\overset{n}{\otimes }}{X}_{i}{\rho }_{i}{X}_{i}\end{array}\right).\end{eqnarray} \tag{ 10 }$$

Going from equations (9) to (10), we have used several properties of the superoperator Λ. We will describe the transformation for the ith qubit, and will therefore drop the subscript i for the moment. Since the Kraus operator L₀ in equation (8) is proportional to the identity matrix, it commutes with everything. The L₁ Kraus operator is proportional to the Z Pauli matrix, therefore ${L}_{1}X=-{{XL}}_{1}$. From this, one can show that the superoperator Λ acting on an arbitrary 2 × 2 matrix η has the following properties: ${\rm{\Lambda }}[\eta ]X={L}_{0}\eta {{XL}}_{0}\,-{L}_{1}\eta {{XL}}_{1}$. Setting $\eta =U\rho {{XU}}^{\dagger }$, the upper left element of the first matrix of equation (9) becomes ${\rm{\Lambda }}[U\rho {{XU}}^{\dagger }]X={L}_{0}U\rho {{XU}}^{\dagger }{{XL}}_{0}-{L}_{1}U\rho {{XU}}^{\dagger }{{XL}}_{1}$. Next, we have that $U={{XU}}^{\dagger }X$ and the fact that $U,\rho ,{L}_{0},{\rm{and}}\ {L}_{1}$ all commute with each other, which yields ${\rm{\Lambda }}[U\rho {{XU}}^{\dagger }]X={L}_{0}^{2}{U}^{2}\rho -{L}_{1}^{2}{U}^{2}\rho \,=\exp [-{{gt}}^{\alpha }]{U}^{2}\rho$. The derivation of the other off-diagonal terms above follow in the same manner.

Next, the RPQs are measured in the Z basis. Each measurement can result in a qubit being either in state ${| 0\rangle }_{i}$ or ${| 1\rangle }_{i}$, and the given outcome of the measurement is fully defined by a vector ${\bf{k}}=({k}_{1},\,...,\,{k}_{n})$, where k_i = 0,1 corresponds to state ${| 0\rangle }_{i}$ or ${| 1\rangle }_{i}$, respectively, for the ith RPQ. The overall phase factor coming from the action of the U² operator depends only on the Hamming weight m_k of k. Every time we measure state ${| 0\rangle }_{i}$, we get a factor of $\exp [-{\rm{i}}\omega t]$, while for state ${| 1\rangle }_{i}$ we get a factor of $\exp [{\rm{i}}\omega t]$. Let us assume that for a given permutation of k we measure ${m}_{{\bf{k}}}={\sum }_{i=1}^{n}{k}_{i}$ times state $| 1\rangle ;$ this means that we measured state $| 0\rangle$ $n-{m}_{{\bf{k}}}$ times. Therefore, we get a phase factor of $\exp [{{\rm{i}}{m}}_{{\bf{k}}}\omega t]\times \exp [-(n-{m}_{{\bf{k}}}){\rm{i}}\omega t]=\exp [-{\rm{i}}(n-2{m}_{{\bf{k}}})\omega t]$ for this permutation.

The SPQ state after observing a particular string of RPQ outcomes k is

$$\begin{eqnarray}{\rho }_{{\rm{SPQ}}}^{({m}_{{\bf{k}}})} & = & \displaystyle \frac{{\lambda }_{0}}{2}\left(\begin{array}{cc}1 & {{\rm{e}}}^{-{\rm{i}}(n+1-2{m}_{{\bf{k}}})\omega t}{{\rm{e}}}^{-(n+1){{gt}}^{\alpha }}\\ {\rm{h}}{\rm{.}}{\rm{c}}. & 1\end{array}\right)\displaystyle \prod _{i=1}^{n}{\lambda }_{{k}_{i}}^{(i)}\\ & & +\displaystyle \frac{{\lambda }_{1}}{2}\left(\begin{array}{cc}1 & -{{\rm{e}}}^{-{\rm{i}}(n+1-2{m}_{{\bf{k}}})\omega t}{{\rm{e}}}^{-(n+1){{gt}}^{\alpha }}\\ {\rm{h}}{\rm{.}}{\rm{c}}. & 1\end{array}\right)\displaystyle \prod _{i=1}^{n}{\lambda }_{1-{k}_{i}}^{(i)}.\end{eqnarray} \tag{ 11 }$$

Next, we measure the SPQ in the X basis to get the probabilities ${q}_{m}^{(\pm )}:= \langle \pm | {\rho }_{{\rm{SPQ}}}^{({m}_{{\bf{k}}})}| \pm \rangle$, which we can expand to get

$$\begin{eqnarray}{q}_{{m}_{{\bf{k}}}}^{(\pm )} & = & \displaystyle \frac{{\lambda }_{0}}{2}\displaystyle \prod _{i=1}^{n}{\lambda }_{{k}_{i}}^{(i)}\{1\pm {{\rm{e}}}^{-(n+1){{gt}}^{\alpha }}\cos [(n+1-2{m}_{{\bf{k}}})\omega t]\}\\ & & +\displaystyle \frac{{\lambda }_{1}}{2}\displaystyle \prod _{i=1}^{n}{\lambda }_{1-{k}_{i}}^{(i)}\{1\mp {{\rm{e}}}^{-(n+1){{gt}}^{\alpha }}\cos [(n+1-2{m}_{{\bf{k}}})\omega t]\}.\end{eqnarray} \tag{ 12 }$$

In the next section, we will use these probabilities to compute the CFI of this distribution. But first, in order to determine the optimality of this measurement scheme, we compute the quantum Fisher Information (QFI) for our protocol.

To compute the QFI, we write the probe state in equation (3) in its eigenbasis ${{\rm{\Omega }}}_{\mathrm{probe}}={\sum }_{{\bf{k}},\pm }{g}_{{\bf{k}}}^{(\pm )}| {G}_{{\bf{k}}}^{(\pm )}\rangle \langle {G}_{{\bf{k}}}^{(\pm )}|$. The eigenvectors and eigenvalues of our probe are given in equation (6). If our encoding process were noiseless, we could compute the QFI using the formula [47, 50]

$$\begin{eqnarray}&&{{ \mathcal F }}_{Q}=2\displaystyle \sum _{{\bf{k}},{\bf{k}}^{\prime} }\displaystyle \sum _{r^{\prime} ,r\in \{\pm \}}\displaystyle \frac{{({g}_{{\bf{k}}}^{(r)}-{g}_{{\bf{k}}^{\prime} }^{(r^{\prime} )})}^{2}}{{g}_{{\bf{k}}}^{(r)}+{g}_{{\bf{k}}^{\prime} }^{(r^{\prime} )}}| \langle {G}_{{\bf{k}}}^{(r)}| { \mathcal G }| {G}_{{\bf{k}}^{\prime} }^{(r^{\prime} )}\rangle {| }^{2},\end{eqnarray} \tag{ 14 }$$

where ${ \mathcal G }$ is the generator encoding the parameter. Unfortunately, our process is not noiseless. However, since the noisy process and the unitary process commute, we can first subject the probe to noise to arrive at ${\tilde{{\rm{\Omega }}}}_{\mathrm{probe}}:= {{\rm{\Lambda }}}^{\otimes n+1}[{{\rm{\Omega }}}_{\mathrm{probe}}]$, before using equation (14), provided we know the eigen-decomposition of ${\tilde{{\rm{\Omega }}}}_{\mathrm{probe}}$.

Let us consider the action of the noise on the operator ${{\rm{\Gamma }}}_{{\bf{k}}}={g}_{{\bf{k}}}^{(+)}| {G}_{{\bf{k}}}^{(+)}\rangle \langle {G}_{{\bf{k}}}^{(+)}| +{g}_{{\bf{k}}}^{(-)}| {G}_{{\bf{k}}}^{(-)}\rangle \langle {G}_{{\bf{k}}}^{(-)}|$ for some choice of k. We expand the states in the computational basis to get

$$\begin{eqnarray}{{\rm{\Lambda }}}^{\otimes n+1}[{{\rm{\Gamma }}}_{{\bf{k}}}] & = & ({g}_{{\bf{k}}}^{(+)}+{g}_{{\bf{k}}}^{(-)}){{\rm{\Lambda }}}^{\otimes n+1}[| 0,{\bf{k}}\rangle \langle 0,{\bf{k}}| +| 1,{\bf{1}}-{\bf{k}}\rangle \langle 1,{\bf{1}}-{\bf{k}}| ]\\ & & +({g}_{{\bf{k}}}^{(+)}-{g}_{{\bf{k}}}^{(-)}){{\rm{\Lambda }}}^{\otimes n+1}[| 0,{\bf{k}}\rangle \langle 1,{\bf{1}}-{\bf{k}}| +| 1,{\bf{1}}-{\bf{k}}\rangle \langle 0,{\bf{k}}| ]\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&=\,{\tilde{g}}_{{\bf{k}}}^{(+)}| {G}_{{\bf{k}}}^{(+)}\rangle \langle {G}_{{\bf{k}}}^{(+)}| +{\tilde{g}}_{{\bf{k}}}^{(-)}| {G}_{{\bf{k}}}^{(-)}\rangle \langle {G}_{{\bf{k}}}^{(-)}| ,\,\end{eqnarray} \tag{ 16 }$$

where we have used ${{\rm{\Lambda }}}^{\otimes n+1}[| 0,{\bf{k}}\rangle \langle 1,{\bf{1}}-{\bf{k}}| ={{\rm{e}}}^{-(n+1){{gt}}^{\alpha }}[| 0,{\bf{k}}\rangle \langle 1,{\bf{1}}-{\bf{k}}|$. In other words, the noise maps the GHZ diagonal state into another GHZ diagonal state with new eigenvalues

$$\begin{eqnarray}&&{\tilde{g}}_{{\bf{k}}}^{(\pm )}=\displaystyle \frac{{g}_{{\bf{k}}}^{(+)}+{g}_{{\bf{k}}}^{(-)}}{2}\pm {{\rm{e}}}^{-(n+1){{gt}}^{\alpha }}\displaystyle \frac{{g}_{{\bf{k}}}^{(+)}-{g}_{{\bf{k}}}^{(-)}}{2}.\end{eqnarray} \tag{ 17 }$$

To compute the QFI, we write down the generator ${ \mathcal G }=\tfrac{t}{2}{\sum }_{i}{Z}^{(i)}\otimes {{\bf{I}}}^{(\bar{i})}$. Its action on an eigenvector yields

$$\begin{eqnarray}{ \mathcal G }| {G}_{{\bf{k}}}^{(\pm )}\rangle & = & \displaystyle \frac{t}{2\sqrt{2}}[(n+1-2{m}_{{\bf{k}}})| 0,{\bf{k}}\rangle \pm ({m}_{{\bf{k}}}-(n-{m}_{{\bf{k}}})+1)| 1,{\bf{1}}-{\bf{k}}\rangle ]\\ & = & t\displaystyle \frac{n\,+\,1-2{m}_{{\bf{k}}}}{2}| {G}_{{\bf{k}}}^{(\mp )}\rangle ,\end{eqnarray} \tag{ 18 }$$

where again ${m}_{{\bf{k}}}={\sum }_{i=1}^{n}{k}_{i}$. From this we have that $\langle {G}_{{\bf{k}}^{\prime} }^{(r^{\prime} )}| { \mathcal G }| {G}_{{\bf{k}}}^{(r)}\rangle =t\tfrac{n\,+\,1-2{m}_{{\bf{k}}}}{2}{\delta }_{{\bf{kk}}^{\prime} }{\delta }_{r,r^{\prime} +1}$. Immediately, we find the QFI to be

$$\begin{eqnarray}&&{{ \mathcal F }}_{Q}={t}^{2}\displaystyle \sum _{{\bf{k}}}\displaystyle \frac{{({\tilde{g}}_{{\bf{k}}}^{(+)}-{\tilde{g}}_{{\bf{k}}}^{(-)})}^{2}}{{\tilde{g}}_{{\bf{k}}}^{(+)}+{\tilde{g}}_{{\bf{k}}}^{(-)}}{(n+1-2{m}_{{\bf{k}}})}^{2}={t}^{2}{{\rm{e}}}^{-2(n+1){{gt}}^{\alpha }}\displaystyle \sum _{{\bf{k}}}{{ \mathcal P }}_{{\bf{k}}}{(n+1-2{m}_{{\bf{k}}})}^{2},\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{\rm{where}}\quad {{ \mathcal P }}_{{\bf{k}}}:= \displaystyle \frac{{({g}_{{\bf{k}}}^{(+)}-{g}_{{\bf{k}}}^{(-)})}^{2}}{{g}_{{\bf{k}}}^{(+)}+{g}_{{\bf{k}}}^{(-)}}=\displaystyle \frac{{({\lambda }_{0}{\displaystyle \prod }_{i=1}^{n}{\lambda }_{{k}_{i}}^{(i)}-{\lambda }_{1}{\displaystyle \prod }_{i=1}^{n}{\lambda }_{1-{k}_{i}}^{(i)})}^{2}}{{\lambda }_{0}{\displaystyle \prod }_{i=1}^{n}{\lambda }_{{k}_{i}}^{(i)}+{\lambda }_{1}{\displaystyle \prod }_{i=1}^{n}{\lambda }_{1-{k}_{i}}^{(i)}}.\end{eqnarray} \tag{ 20 }$$

We will shortly show that our readout scheme yields the same value for the CFI, i.e., it is experimentally possible to achieve the theoretical maximum precision.

The CFI to estimate parameter x is given by the following formula:

$$\begin{eqnarray}&&{{ \mathcal F }}_{C}=\displaystyle \sum _{i}\displaystyle \frac{{({\partial }_{x}{q}_{i})}^{2}}{{q}_{i}},\end{eqnarray} \tag{ 21 }$$

where the q_i are measurement outcomes, in our case taken from equation (12).

Our readout scheme begins with measuring m_k RPQs in state $| 1\rangle$ (and $n-{m}_{{\bf{k}}}$ in state $| 0\rangle$), followed by a measurement of the SPQ in the $| \pm \rangle$ basis. Since different qubits have different purity, i.e., different values of _{${p}_{i}$}, the CFI depends on the specific string of outcomes: ${{ \mathcal F }}_{C}={\sum }_{{\bf{k}}}{f}_{{\bf{k}}}$ with

$$\begin{eqnarray}&&{f}_{{\bf{k}}}=\displaystyle \frac{{({\partial }_{\omega }{q}_{{\bf{k}}}^{(+)})}^{2}}{{q}_{{\bf{k}}}^{(+)}}+\displaystyle \frac{{({\partial }_{\omega }{q}_{{\bf{k}}}^{(-)})}^{2}}{{q}_{{\bf{k}}}^{(-)}}={({\partial }_{\omega }{q}_{{\bf{k}}}^{(\pm )})}^{2}\displaystyle \frac{{q}_{{\bf{k}}}^{(+)}+{q}_{{\bf{k}}}^{(-)}}{{q}_{{\bf{k}}}^{(+)}{q}_{{\bf{k}}}^{(-)}}.\end{eqnarray} \tag{ 22 }$$

Here, the derivatives of the probabilities to measure $| +\rangle$ and $| -\rangle$ only differ by a sign; therefore, the squared derivatives are equal for both cases. We first rearrange equation (12) as

$$\begin{eqnarray}{q}_{\pm ,{\bf{k}}} & = & \left(\displaystyle \frac{{\lambda }_{0}}{2}\displaystyle \prod _{i=1}^{n}{\lambda }_{{k}_{i}}^{(i)}+\displaystyle \frac{{\lambda }_{1}}{2}\displaystyle \prod _{i=1}^{n}{\lambda }_{1-{k}_{i}}^{(i)}\right)\\ & & \pm \left(\displaystyle \frac{{\lambda }_{0}}{2}\displaystyle \prod _{i=1}^{n}{\lambda }_{{k}_{i}}^{(i)}-\displaystyle \frac{{\lambda }_{1}}{2}\displaystyle \prod _{i=1}^{n}{\lambda }_{1-{k}_{i}}^{(i)}\right){{\rm{e}}}^{-(n+1){{gt}}^{\alpha }}\cos [(n+1-2{m}_{{\bf{k}}})\omega t]\end{eqnarray} \tag{ 23 }$$

and substitute into equation (22) to get

$$\begin{eqnarray}{f}_{{\bf{k}}} & = & {t}^{2}{{\rm{e}}}^{-2(n+1){{gt}}^{\alpha }}{(n+1-2{m}_{{\bf{k}}})}^{2}{\sin }^{2}[(n+1-2{m}_{{\bf{k}}})\omega t]\\ & & \times \displaystyle \frac{({\lambda }_{0}{\displaystyle \prod }_{i=1}^{n}{\lambda }_{{k}_{i}}^{(i)}+{\lambda }_{1}{\displaystyle \prod }_{i=1}^{n}{\lambda }_{1-{k}_{i}}^{(i)}){\left(\tfrac{{\lambda }_{0}}{2}{\displaystyle \prod }_{i=1}^{n}{\lambda }_{{k}_{i}}^{(i)}-\tfrac{{\lambda }_{1}}{2}{\displaystyle \prod }_{i=1}^{n}{\lambda }_{1-{k}_{i}}^{(i)}\right)}^{2}}{{\left(\tfrac{{\lambda }_{0}}{2}{\displaystyle \prod }_{i=1}^{n}{\lambda }_{{k}_{i}}^{(i)}+\tfrac{{\lambda }_{1}}{2}{\displaystyle \prod }_{i=1}^{n}{\lambda }_{1-{k}_{i}}^{(i)}\right)}^{2}-{\left(\tfrac{{\lambda }_{0}}{2}{\displaystyle \prod }_{i=1}^{n}{\lambda }_{{k}_{i}}^{(i)}-\tfrac{{\lambda }_{1}}{2}{\displaystyle \prod }_{i=1}^{n}{\lambda }_{1-{k}_{i}}^{(i)}\right)}^{2}{ \mathcal E }},\end{eqnarray} \tag{ 24 }$$

where ${ \mathcal E }={{\rm{e}}}^{-2(n+1){{gt}}^{\alpha }}{\cos }^{2}[(n+1-2{m}_{{\bf{k}}})\omega t]$.

We want to look at the specific case where the Fisher information is maximised for any k. If one assumes that n is an even number (if it is odd, one of the RPQs can be discarded at the beginning of the protocol) and $\omega t=\tfrac{\pi }{2}$, then $(n+1-2{m}_{{\bf{k}}})$ is an odd number and the argument of the $\sin$ and $\cos$ functions has the form $\tfrac{\pi }{2}+l\pi$, where l is an integer. The $\sin$ function then gives one, and the $\cos$ function is equal to zero. This simplifies equation (24) to

$$\begin{eqnarray}&&{f}_{{\bf{k}}}={t}^{2}{{\rm{e}}}^{-2(n+1){{gt}}^{\alpha }}{{ \mathcal P }}_{{\bf{k}}}{(n+1-2{m}_{{\bf{k}}})}^{2},\end{eqnarray} \tag{ 25 }$$

where ${{ \mathcal P }}_{{\bf{k}}}$ is defined in equation (20). To calculate the total CFI, one has to sum over all possible strings of measurement outcomes, characterised by k:

$$\begin{eqnarray}&&{{ \mathcal F }}_{C}^{\sharp }=\displaystyle \sum _{{\bf{k}}}{f}_{{\bf{k}}}={t}_{\sharp }^{2}{{\rm{e}}}^{-2(n+1){{gt}}_{\sharp }^{\alpha }}\displaystyle \sum _{{\bf{k}}}{{ \mathcal P }}_{{\bf{k}}}{(n+1-2{m}_{{\bf{k}}})}^{2}.\end{eqnarray} \tag{ 26 }$$

Here, ${{ \mathcal F }}_{C}^{\sharp }$ is the time-optimised CFI, with optimal time for the correlated probes denoted by t_♯. The last equation is identical to equation (19). This means that the readout procedure introduced here achieves optimal sensitivity for estimating frequency. We no longer need to differentiate between QFI and CFI, but for concreteness we only use the abbreviation CFI from here on.

In the appendix, we derive the CFI for the case when only the SPQ is measured and all RPQs are discarded. Even then, we find that the CFI scales very much like in equation (26).

Before discussing our main results, let us compute the Fisher information for the case where the probe does not exploit correlations. Eventually, we want to compare the performance of the probe in the GHZ diagonal state to that of the uncorrelated state, i.e., the product state ${{\rm{\Omega }}}_{0}={\otimes }_{i=0}^{n}{\rho }_{i}$.

For the ith qubit, the QFI can be easily computed to be ${{ \mathcal F }}_{Q}^{(i)}={{\rm{e}}}^{-2{{gt}}_{\parallel }}{p}_{i}^{2}$ [18], which can be achieved with an X basis measurement on each qubit, hence ${{ \mathcal F }}_{Q}^{(i)}={{ \mathcal F }}_{C}^{(i)}$. In general, the run time for the uncorrelated probe will be different from that of the correlated probe; we denote the optimal run time for the former by ${t}_{\parallel }$.

As the Fisher information is additive, the total Fisher information for all qubits in Ω₀ is

$$\begin{eqnarray}&&{{ \mathcal F }}_{C}^{\parallel }=\displaystyle \sum _{i}{{ \mathcal F }}_{C}^{(i)}={{\rm{e}}}^{-2{{gt}}_{\parallel }}(n+1)\langle {{\bf{p}}}^{2}\rangle ,\end{eqnarray} \tag{ 27 }$$

where $\langle {{\bf{p}}}^{2}\rangle := \tfrac{1}{n+1}{\sum }_{i=0}^{n}{p}_{i}^{2}$ is the normalised length squared. This quantity is closely related to the average purity of all qubits, which has the form $\tfrac{1}{n+1}{\sum }_{i=0}^{n}\tfrac{1+{p}_{i}^{2}}{2}=\tfrac{1}{2}(1+\langle {{\bf{p}}}^{2}\rangle )$.

We have given an expression for the CFI in the last section that is rather complicated and does not shed much light on how well our probe performs The complexity entirely lies in the final term of equation (26). It turns out that this term can be approximated, in terms of the vector ${\bf{p}}$ for any initial state, by ${\sum }_{{\bf{k}}}{{ \mathcal P }}_{{\bf{k}}}{(n+1-2{m}_{{\bf{k}}})}^{2}\approx \langle {{\bf{p}}}^{2}\rangle {(n+1)}^{2}$. This approximation holds remarkably well, as numerically evidenced in figure 2. There, we plot ${\sum }_{{\bf{k}}}{{ \mathcal P }}_{{\bf{k}}}{(n+1-2{m}_{{\bf{k}}})}^{2}$ against $\langle {{\bf{p}}}^{2}\rangle {(n+1)}^{2}$ for randomly chosen $1\leqslant n\leqslant 11$ (number of RPQs) and uniformly sampled ${p}_{i}\in [0,1]$. For 10⁴ realisations, the Pearson correlation coefficient is 99.3%. We have performed ∼10⁶ calculations for a larger range of the parameter $n\in [1,15]$, and the results hold. For clarity we show the reduced range results. This shows that for any vector p the CFI scales as (n + 1)²:

$$\begin{eqnarray}&&{{ \mathcal F }}_{C}^{\sharp }\approx {t}_{\sharp }^{2}{{\rm{e}}}^{-2(n+1){{gt}}_{\sharp }^{\alpha }}\langle {{\bf{p}}}^{2}\rangle {(n+1)}^{2}.\end{eqnarray} \tag{ 28 }$$

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To compare the sensitivity of a correlated probe to that of an uncorrelated probe, we fix the total sensing time to ${ \mathcal T }$, as well as the number of probes n, and compute the total Fisher information acquired. Each run of sensing with a correlated probe takes time t_♯, while sensing with an uncorrelated probe takes time ${t}_{\parallel }$. The total number of runs is given by ${ \mathcal T }/{t}_{\sharp }$ and ${ \mathcal T }/{t}_{\parallel }$ in the two cases, respectively, and the corresponding total CFIs are

$$\begin{eqnarray}{F}_{C}^{\sharp } & = & \displaystyle \frac{{ \mathcal T }{{ \mathcal F }}_{C}^{\sharp }}{{t}_{\sharp }}\approx { \mathcal T }{t}_{\sharp }{{\rm{e}}}^{-2(n+1){{gt}}_{\sharp }^{\alpha }}\langle {{\bf{p}}}^{2}\rangle {(n+1)}^{2}\\ {\rm{and}}\qquad {F}_{C}^{\parallel } & = & \displaystyle \frac{{ \mathcal T }{{ \mathcal F }}_{C}^{\parallel }}{{t}_{\parallel }}={ \mathcal T }{t}_{\parallel }{{\rm{e}}}^{-2{{gt}}_{\parallel }^{\alpha }}\langle {{\bf{p}}}^{2}\rangle (n+1).\end{eqnarray} \tag{ 29 }$$

We then look at the case where the amount of noise in the system is the same after the experiment is concluded, i.e., we set $\exp [-2{{gt}}_{\parallel }^{\alpha }]=\exp [-2(n+1){{gt}}_{\sharp }^{\alpha }]$. Solving this for t_♯, we get ${t}_{\sharp }={t}_{\parallel }/\sqrt[\alpha ]{n+1}$. Another way to motivate this choice is by finding the optimal values for t_♯ and ${t}_{\parallel }$ independently, and we find the same relationship between the two times.

By substituting this into the total Fisher information and taking the ratio of the two CFIs in the last equation, we get the quantum advantage

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{C}^{\sharp }}{{F}_{C}^{\parallel }}\sim \displaystyle \frac{(n+1)}{\sqrt[\alpha ]{n+1}}.\end{eqnarray} \tag{ 30 }$$

It is clear that the correlations in the probe give us an enhancement in the Fisher information that scales with the number of qubits. For the simplest time inhomogeneous noise model, where α = 2, one gets a CFI that scales as ${(n+1)}^{\tfrac{3}{2}}$. While for the semigroup case, with α = 1, no advantage is drawn from the correlated probe. In other words, we have shown that the Fisher information scales exactly as for the case where a pure probe state is used [26–28]. This finding generalises the result of [18], where mixed state probes with ${p}_{i}$ = p for noiseless encoding were considered.

The sensing times ${t}_{\parallel }$ and t_♯ do not depend on the mixedness of the probe. The only difference is that we have an overhead of $\langle {{\bf{p}}}^{2}\rangle$, which can be overcome by repeating the experiment $\tfrac{1}{\langle {{\bf{p}}}^{2}\rangle }$ times. The correlations in the state of the probe lead to this enhancement for any $\langle {{\bf{p}}}^{2}\rangle \ne 0$. This includes highly mixed states that live in the separable ball [51] and thus cannot be entangled. Therefore, our work further highlights the importance of correlations beyond quantum entanglement in quantum metrology. This is one of our main result.

As mentioned before, in the appendix we consider the case where only the SPQ is measured and all RPQs are discarded. Even in this case, the 'Zeno' scaling survives at the expense of an adaptive measurement protocol. We now examine two special cases of this general results.

We now examine two special cases of our protocol. In the first case, we set all _{${p}_{i}$} = p and call this the uniform protocol. For the second case, we set ${p}_{0}\,=\,1$ and ${p}_{i}=0\forall i\ne 0$ and call it the tilted protocol. Here all RPQs are maximally mixed and the SPQ is pure. These two protocols were previously studied for noiseless encoding in [18, 19], respectively.

The initial probe state in the uniform protocol has a high degree of degeneracy, and its eigenvalues have the form ${\left(\tfrac{1+p}{2}\right)}^{n+1-m}{\left(\tfrac{1-p}{2}\right)}^{m}$. This simplifies the sum over the possible values of k in equation (26), and it can be rewritten as:

$$\begin{eqnarray}{{ \mathcal F }}_{C\star }^{\sharp } & = & {t}_{\sharp }^{2}{{\rm{e}}}^{-2(n+1){{gt}}_{\sharp }^{\alpha }}\displaystyle \sum _{{m}_{{\bf{k}}}=0}^{n}\displaystyle \left(\genfrac{}{}{0em}{}{n}{{m}_{{\bf{k}}}}\right){{ \mathcal P }}_{{m}_{{\bf{k}}}}{(n+1-2{m}_{{\bf{k}}})}^{2}\\ & \geqslant & {t}_{\sharp }^{2}{{\rm{e}}}^{-2(n+1){{gt}}_{\sharp }^{\alpha }}{p}^{2}{(n+1)}^{2}.\end{eqnarray} \tag{ 31 }$$

The inequality was numerically shown to hold in [18], with the difference between the left and the right hand sides to be in relatively small. This is in perfect correspondence with our result in equation (28), since $\langle {{\bf{p}}}^{2}\rangle ={p}^{2}$ in this case.

In contrast, the state of the probe in the tilted protocol has eigenvalues $\left\{\tfrac{1}{{2}^{n}},0\right\}$, which simplifies the term ${{ \mathcal P }}_{{\bf{k}}}$ in equation (26) to $\displaystyle \frac{1}{{2}^{n}}$. There are exactly $\displaystyle \left(\genfrac{}{}{0em}{}{{n}}{m}\right)$ vectors k that have m 1s and n − m 0s; therefore, the CFI can be rewritten as:

$$\begin{eqnarray}&&{{ \mathcal F }}_{C\$}^{\sharp }={t}_{\sharp }^{2}{{\rm{e}}}^{-2(n+1){{gt}}_{\sharp }^{\alpha }}\displaystyle \frac{1}{{2}^{n}}\displaystyle \sum _{{m}_{{\bf{k}}}=0}^{n}\displaystyle \left(\genfrac{}{}{0em}{}{n}{{m}_{{\bf{k}}}}\right){(n+1-2{m}_{{\bf{k}}})}^{2}={t}_{\sharp }^{2}{{\rm{e}}}^{-2(n+1){{gt}}_{\sharp }^{\alpha }}(n+1).\end{eqnarray} \tag{ 32 }$$

This is in exact agreement with equation (28), for which $\langle {{\bf{p}}}^{2}\rangle =\tfrac{1}{n+1}$ here.

Now, consider the CFI for the tilted protocol for the case α = 2. We can simplify equation (29) to get ${F}_{C\$}^{\sharp }={ \mathcal T }{t}_{\parallel }\exp [-2{{gt}}_{\parallel }^{2}]\sqrt{n\,+\,1}$. The scaling of $\sqrt{n\,+\,1}$ may give the impression that our quantum correlated probe performs worse than the SQL. However, if we were to use an uncorrelated probe consisting of one pure qubit and n fully mixed qubits, after a time ${ \mathcal T }$ the CFI would be ${F}_{C\$}^{\parallel }={ \mathcal T }{t}_{\parallel }\exp [-2{{gt}}_{\parallel }^{2}]$. Thus, the correlations are still giving us an enhancement that grows with the number of qubits in the probe.

The constant terms of the CFIs for the uniform and the tilted protocols are identical. However, in terms of the probe size the former scales as ${p}^{2}{(n+1)}^{2}$, and the latter scales as n + 1. Thus, when p $\geqslant 1/\sqrt{n\,+\,1}$ the uniform protocol fares better and ${p}\leqslant 1/\sqrt{n\,+\,1}$ the tilted protocol fares better. This relation gives us a way to understand the trade-offs between the two extreme choices for the initial state of the probe. That is, when is it favourable for the coherence to be concentrated in a single qubit, and when is it favourable for it to be spread amongst all qubits.

Finally, in [19], where the tilted protocol was originally introduced, the state preparation did not include the first Cnot gate (see figure 1). This is of course because the SPQ is pure and initially in state $| 0\rangle$. The first Cnot gate would simply do nothing to the probe. However, it turns out that with the inclusion of this gate we need not care which qubit serves as the SPQ.

We now show that CFI does not change under permutations of qubits or equivalently the elements of vector ${\bf{p}}$. This means our protocol does not depend on which qubit is used as SPQ. First, note that the term ${{ \mathcal P }}_{{\bf{k}}}$ in equation (26) is symmetric under permutations of the RPQs and thus changing their order does not change the value of CFI. Now, we consider swapping one RPQ with the SPQ. Without loss of generality, we consider a pair of vectors ${\bf{p}}:= ({p}_{0},{p}_{1},\,\ldots ,\,{p}_{n})$ and ${\bf{q}}:= ({p}_{1},{p}_{0},\,\ldots ,\,{p}_{n})$, where the first two entries have been switched.

We write the CFI given in equation (26), omitting the constant terms, for these two vectors:

$$\begin{eqnarray}&&{{ \mathcal F }}_{C,{\bf{p}}}\sim {\displaystyle \sum }_{{\bf{k}}}\tfrac{{({\lambda }_{0}^{(0)}{\lambda }_{{k}_{1}}^{(1)}{\displaystyle \prod }_{i=2}^{n}{\lambda }_{{k}_{i}}^{(i)}-{\lambda }_{1}^{(0)}{\lambda }_{1-{k}_{1}}^{(1)}{\displaystyle \prod }_{i=2}^{n}{\lambda }_{1-{k}_{i}}^{(i)})}^{2}}{{\lambda }_{0}^{(0)}{\lambda }_{{k}_{1}}^{(1)}{\displaystyle \prod }_{i=2}^{n}{\lambda }_{{k}_{i}}^{(i)}+{\lambda }_{1}^{(0)}{\lambda }_{1-{k}_{1}}^{(1)}{\displaystyle \prod }_{i=2}^{n}{\lambda }_{1-{k}_{i}}^{(i)}}{(n+1-2{m}_{{\bf{k}}})}^{2},\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&{{ \mathcal F }}_{C,{\bf{q}}}\sim {\displaystyle \sum }_{{\bf{k}}}\tfrac{{({\lambda }_{0}^{(1)}{\lambda }_{{k}_{0}}^{(0)}{\displaystyle \prod }_{i=2}^{n}{\lambda }_{{k}_{i}}^{(i)}-{\lambda }_{1}^{(1)}{\lambda }_{1-{k}_{0}}^{(0)}{\displaystyle \prod }_{i=2}^{n}{\lambda }_{1-{k}_{i}}^{(i)})}^{2}}{{\lambda }_{0}^{(1)}{\lambda }_{{k}_{0}}^{(0)}{\displaystyle \prod }_{i=2}^{n}{\lambda }_{{k}_{i}}^{(i)}+{\lambda }_{1}^{(1)}{\lambda }_{1-{k}_{0}}^{(0)}{\prod }_{i=2}^{n}{\lambda }_{1-{k}_{i}}^{(i)}}{(n+1-2{m}_{{\bf{k}}})}^{2}.\end{eqnarray} \tag{ 34 }$$

Let us look at one term in the sum. The vector describing this term (excluding first RPQ) is ${\bf{k}}^{\prime} =({k}_{2},\,\ldots ,\,{k}_{n})$, with Hamming weight ${m}_{{\bf{k}}^{\prime} }={\displaystyle \sum }_{i=2}^{n}{k}_{i}$. We will show that for each term in the sum in ${{ \mathcal F }}_{C,{\bf{p}}}$, there is an equivalent term in ${{ \mathcal F }}_{C,{\bf{q}}}$. For a given vector ${\bf{k}}^{\prime}$ and ${{ \mathcal F }}_{C,{\bf{p}}}$ there are two options: ${k}_{1}=0$ or ${k}_{1}=1$.

When k₁ = 0 the corresponding term in ${{ \mathcal F }}_{C,{\bf{q}}}$ will be term where k₀ = 0. For both p and q the Hamming weight of the vector k = (k₁, k₂, ..., k_n) will be the same and equal to m_k', hence the multiplying factor ${(n+1-2{m}_{{{\bf{k}}}^{{\prime} }})}^{2}$ will be the same.

When k₁ = 1 then the Hamming weight of vector p will be equal to ${m}_{{\bf{k}}^{\prime} }+1$ and the multiplying factor will be ${(n+1-2(m{{\prime} }_{{\bf{k}}}+1))}^{2}={(n-1-2{m}_{{\bf{k}}})}^{2}$. This term matches the term in ${{ \mathcal F }}_{C,{\bf{q}}}$ with ${k}_{0}\,=\,1$ and vector ${\bf{k}}^{\prime\prime} =(1-{k}_{2},\,\ldots ,\,1-{k}_{n})$. This makes the numerator for equation (34) ${({\lambda }_{0}^{(1)}{\lambda }_{1}^{(0)}\displaystyle \prod _{i=2}^{n}{\lambda }_{1-{k}_{i}}^{(i)}-{\lambda }_{1}^{(1)}{\lambda }_{0}^{(0)}\displaystyle \prod _{i=2}^{n}{\lambda }_{{k}_{i}}^{(i)})}^{2}$. The sign will be cancelled by squaring this term and the denominator will be the same as in ${{ \mathcal F }}_{C,{\bf{p}}}$. The Hamming weight of the vector q in this case will be $n-{m}_{{\bf{k}}^{\prime} }$, because the Hamming weight is calculated for vector k describing the product that goes with ${\lambda }_{0}^{(0)}$ eigenvalue of SPQ. The multiplying factor in this case will be the same: ${(n+1-2(n-{m}_{{\bf{k}}^{\prime} }))}^{2}={(-1)}^{2}{(n-1-2{m}_{{\bf{k}}^{\prime} })}^{2}$.

We can repeat this argument for all terms in the sum. Also, since CFI is invariant under permutations of RPQs, the proof holds for switching SPQ with any of the RPQs.

The symmetry of our protocol does not simply stem from the fact that the prepared state of the probe is GHZ-diagonal. To illustrate this, consider the case where the first Cnot gate in our protocol is omitted, while keeping the remainder of the procedure identical (as in figure 1). Using the arguments in section 2.2, it is easy to show that the prepared state in this case is also GHZ-diagonal. For concreteness we now work with three qubits and write down the CFI. Ignoring the constant terms that contain variables like t,g, etc, we find that CFI is proportional to the term with the sum, i.e., ${p}_{0}(8{\lambda }_{0}^{(1)}{\lambda }_{0}^{(2)}\,+\,1)$. This quantity is clearly not symmetric under exchange of qubits. Therefore, the first Cnot gate, while seemingly useless, is responsible for the symmetry of our protocol. Moreover, as mentioned above, the first Cnot gate also leads to better sensitivity. To see this, consider the fact that for a small value of p₀ this CFI will be very small if the first Cnot gate is omitted. In contrast, the CFI for our protocol will be proportional to $3({p}_{0}^{2}+{p}_{1}^{2}+{p}_{2}^{2})$.

Our approximation for the CFI in equation (28) is clearly monotonic in each of the elements of vector p. We will now argue, again with numerical support, that the exact CFI in equation (26) is indeed monotonic in this way. Since any permutation of the elements in vector p gives the same value of CFI, we can focus on changing just one parameter by adding a vector ${\varepsilon }_{j}=(0,0,\,\ldots ,\,{\epsilon }_{j},\,\ldots ,\,0)$, where ${\epsilon }_{j}\gt 0$ and ${p}_{j}+{\epsilon }_{j}\leqslant 1$. We first show that CFI grows as any one parameter of vector p becomes larger: ${{ \mathcal F }}_{C,{\bf{p}}+{\varepsilon }_{j}}\geqslant {{ \mathcal F }}_{C,{\bf{p}}}$.

We have tested the last inequality numerically for 10⁴ realisations; the results are presented in the figure 3(a). The plot shows that the difference in CFIs corresponding to randomly chosen ${\bf{p}}+{\varepsilon }_{j}$ and ${\bf{p}}$, subject to constrains from the last paragraph, is non-negative and grows with the size _j. The implications of the numerics is that the CFI is monotonic in each elements of p. Note that in this demonstration we are only interested whether the CFI grows and that the said difference is non-negative. The monotonicity stems from the monotonic nature of the sum in equation (26), and does not depend on ${t}_{\sharp }$ and g. For concreteness, we have taken ${t}_{\sharp }=1$ and $g=0$ for the purpose of the plot. These constants only contribute to a positive factor in the CFI and will be the same for any vector p. Different values of ${t}_{\sharp }$ and g will not change the sign of the difference, just its magnitude.

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Building on this result, we can see that if ${\epsilon }_{j}\gt {\epsilon }_{k}\gt 0$ than ${{ \mathcal F }}_{C,{\bf{p}}+{\varepsilon }_{j}}\geqslant {{ \mathcal F }}_{C,{\bf{p}}+{\varepsilon }_{k}}$, since we can change the order of the qubits and ${{ \mathcal F }}_{C,{\bf{p}}+{\varepsilon }_{j}}\geqslant {{ \mathcal F }}_{C,{\bf{p}}+\varepsilon {{\prime} }_{j}}$ for ${\epsilon }_{j}\gt \epsilon {{\prime} }_{j}={\epsilon }_{k}$. Moreover, we can order the CFI for different vectors: ${{ \mathcal F }}_{C,{\bf{p}}+{\varepsilon }_{j}+{\varepsilon }_{k}}\geqslant {{ \mathcal F }}_{C,{\bf{p}}+{\varepsilon }_{j}}\geqslant {{ \mathcal F }}_{C,{\bf{p}}+{\varepsilon }_{k}}\geqslant {{ \mathcal F }}_{C,{\bf{p}}}$ for ${\epsilon }_{j}\gt {\epsilon }_{k}\gt 0$. The figure 3(b) shows the difference between CFIs for arbitrary vector $\varepsilon =({\epsilon }_{0},{\epsilon }_{1},\,\ldots ,\,{\epsilon }_{n})$, where _j > 0 for all j, as a function of normalised length squared of this vector $\langle {\varepsilon }^{2}\rangle =\displaystyle \frac{\varepsilon \cdot \varepsilon }{n+1}$. The difference grows with the $\langle {\varepsilon }^{2}\rangle$. The two figures present strong numerical evidence for monotnicity of CFI with respect to p.

Majorisation. Finally, we present a negative result. Numerically we tested whether majorisation [52] has an affect on the Fisher information. Specifically, we considered how the majorisation of the two normalised vectors $\hat{p}$ and $\hat{q}$ first. Here $\hat{p}$ and $\hat{q}$ describe two sets of n + 1 qubits. We found that despite of majorisation the corresponding Fisher information do not have any hierarchy. We also considered the majorisation conditions for the spectrum of the initial state of the probe. In this case as well, we found no hierarchy for the corresponding Fisher information.