.. _sec:mop: Minimization over purifications ------------------------------- The MOP method :cite:`Fujiwara2008, Escher2011, Demkowicz2012, kurdzialek2024` is based on an observation that the QFI of a state :math:`\rho_\theta\in\mathcal{L}(\mathcal{H})` is equal to the minimum of QFI over its purifications: .. math:: F_Q(\rho_\theta) = \min_{\ket{\Psi_\theta}} F_Q(\ket{\Psi_\theta}\bra{\Psi_\theta}), where :math:`\ket{\Psi_\theta} \in \mathcal{H} \otimes \mathcal{R}` is a purification of :math:`\rho_\theta`, :math:`\rho_\theta=\mathrm{Tr}_{\mathcal{R}} \ket{\Psi_\theta}\bra{\Psi_\theta}`. Consider now a parameter-encoding channel :math:`\Lambda_\theta: \mathcal{L}(\mathcal{I}) \mapsto \mathcal{L}(\mathcal{O})`. The idea of minimization over purifications leads to an efficiently computable formula for the channel QFI :cite:`Fujiwara2008, Escher2011, Demkowicz2012, kurdzialek2024` .. math:: :label: eq:minkraus F_Q(\Lambda_\theta) = 4\min_{\{K_{\theta, k}\}} \left\| \sum_{k=1}^r \dot{K}_{\theta,k}^\dagger \dot{K}_{\theta,k} \right\|, where :math:`\| \cdot \|` is the operator norm and the minimization is performed over all equivalent Kraus representations of the channel, such that: .. math:: \Lambda_\theta(\cdot)= \sum_{k=1}^r K_{\theta,k} \cdot K_{\theta,k}^\dagger. This optimization becomes feasible, as one can effectively parametrize the derivatives of the Kraus operators corresponding to all equivalent Kraus representations as: .. math:: \dot K_{\theta, k}(h) = \dot{K}_{\theta, k}- i \sum_l h_{kl} K_{\theta, l}, where :math:`h \in \mathbb{C}^{ r\times r}` is a Hermitian matrix and :math:`K_{\theta,k}` is any fixed Kraus representation of :math:`\Lambda_\theta`, e.g. the canonical representation obtained from eigenvectors of the corresponding Choi-Jamiołkowski (CJ) operator :cite:`Bengtsson2006`, see Appendix :ref:`sec:link_prod` for the definition of the CJ operator. As a result, one may rephrase the apparently difficult minimization problem :eq:`eq:minkraus` as: .. math:: F_Q(\Lambda_\theta) = 4 \min_{h} \| \alpha(h)\|, where .. math:: :label: eq:alpha \alpha(h) = \sum_{k} \dot K_{\theta, k}(h)^\dagger \dot K_{\theta, k}(h). Importantly, this problem may be effectively written as a simple semidefinite program (SDP) :cite:`Demkowicz2012, kurdzialek2024` .. math:: :label: eq:mop_sdp F_Q(\Lambda_\theta) = 4 \min_{\substack{\lambda \in \mathbb{R},\\ h=h^\dagger}} \lambda ~~\textrm{s.t.}~~ A \succeq 0, where .. math:: A = \left( \begin{array}{c|ccc} \lambda \mathbb{1}_\mathcal{I} & \dot{{ K}}_{\theta,1}^\dagger(h) & ... & \dot{{ K}}_{\theta,r}^\dagger(h) \\ \hline \dot{{ K}}_{\theta,1}(h) & & & \\ \vdots & & \mathbb{1}_{d \cdot r} & \\ \dot{{ K}}_{\theta, r}(h) & & & \end{array} \right), :math:`d= \textrm{dim} \mathcal{O}` and :math:`r` is the number of Kraus operators. Provided the dimension of the relevant Hilbert space is small enough, this problem may be solved efficiently using widely available SDP solvers, such as :cite:`mosek`. This optimization is implemented in the :py:func:`mop_channel_qfi ` function. .. _fig:linkadaptive: .. figure:: /_static/img/adptive_rho.drawio.svg :width: 90% :align: center Final state resulting from the action of an adaptive protocol on :math:`N` parameter encoding channels, written via a formal link product operation between the corresponding CJ operators—quantum combs. Notice that the dimension of ancilla :math:`\mathcal{A}` is not included in any of the constraints in :eq:`eq:mop_sdp`. This is because MOP gives the maximal QFI without any restriction on ancilla sizes. Furthermore, in order to identify the optimal input state :math:`\ket\psi` one needs to solve an additional SDP problem, see :cite:`Zhou2021, kurdzialek2024`. MOP can also be used to optimize QFI in case of multiple channel uses. For the parallel strategy, :numref:`fig:intro` (B), one can simply apply the single-channel method, replacing the channel :math:`\Lambda_\theta` with :math:`\Lambda_\theta^{(N)}=\Lambda_\theta^{\otimes N}`—this is implemented in the :py:func:`mop_parallel_qfi ` function. In order to address an adaptive strategy, as in :numref:`fig:intro` (C), one should first use the fact that a concatenated sequence of quantum channels :math:`\mathrm{C}_i` where the free inputs and free output spaces are respectively :math:`\mathcal{H}_{2i}`, :math:`\mathcal{H}_{2i+1}`, :math:`i \in \{0, \dots, N-1\}`, can be represented as a quantum comb :cite:`Chiribella2009` .. math:: C\in \mathrm{Comb}[ (\mathcal{H}_0, \mathcal{H}_1), \dots, (\mathcal{H}_{2N-2}, \mathcal{H}_{2N-1}) ], which is a CJ operator satisfying: .. math:: :label: eq:comb_cond C \succeq 0, \quad C^{(N)}=C, \quad C^{(0)}=1, \\ \mathrm{Tr}_{2k-1} C^{(k)} = \mathbb{1}_{2k-2} \otimes C^{(k-1)}, \quad k\in \{1, \dots, N\}. The above conditions guarantee that such quantum comb :math:`C` can always be regarded as a concatenation of quantum channels :math:`C_i`, namely :math:`C=C_0* \dots *C_{N-1}`, where :math:`*` denotes the link product :cite:`Chiribella2009`, see Appendix :ref:`sec:link_prod` for the definition of the link product. In the context of metrological adaptive scenario, given a series of channels :math:`\Lambda_{\theta, i}: \mathcal{L}(\mathcal{I}_i)\mapsto \mathcal{L}(\mathcal{O}_i)`, we can write the combined action of all the channels as: .. math:: :label: eq:uncorchan \Lambda^{(N)}_\theta = \Lambda_{\theta, 0} \otimes \dots \otimes \Lambda_{\theta, N-1}. We may now represent the general adaptive strategy that includes the input state as well as the control operations in the form of a quantum comb .. math:: C \in \mathrm{Comb}[ (\mathcal{\emptyset}, \mathcal{I}_0), (\mathcal{O}_0, \mathcal{I}_1), \dots, (\mathcal{O}_{N-3}, \mathcal{I}_{N-2}), (\mathcal{O}_{N-2}, \mathcal{I}_{N-1}\otimes\mathcal{A}) ] and concatenate the operations using the link product in order to obtain the output state .. math:: \rho_\theta = C * \mathit{\Lambda}_{\theta}^{(N)} \in \mathcal{L}(\mathcal{O}_{N-1}\otimes \mathcal{A}), where :math:`\mathit{\Lambda}_{\theta}^{(N)}` (italics) formally represents the CJ operator of :math:`\Lambda_{\theta}^{(N)}`, see :numref:`fig:linkadaptive`. Here :math:`\mathit{\Lambda}^{(N)}_\theta` does not necessarily need to represent the action of uncorrelated channels as in :eq:`eq:uncorchan`, but may also be a general parameter-dependent quantum comb and in this way represent models with correlated noise or non-local character of estimated parameter. Optimization over adaptive protocols is thus equivalent to maximizing the final QFI over quantum combs :math:`C`. Using again the MOP idea, this problem may be effectively written down as an SDP (this time slightly more involved) of the following form :cite:`Altherr2021, Liu2023, kurdzialek2024`: .. math:: F_{\mathrm{Q}}^{(N)}\!\left(\Lambda_\theta\right) = 4 \min_{\substack{\lambda \in \mathbb{R}, Q^{(k)}\\ h=h^\dagger}} \lambda ~~\textrm{s.t.}~~ A \succeq 0, \underset{2\leq k \leq N-1}{\forall} \mathrm{Tr}_{\mathcal{O}_{k-1}} Q^{(k)} = \mathbb{1}_{\mathcal{I}_{k-1}} \otimes Q^{(k-1)}, \quad Q^{(0)}= 1, where :math:`Q^{(k)} \in \mathcal{L}(\mathcal{O}_0 \otimes \mathcal{I}_0 \otimes \dots \otimes \mathcal{O}_{k-1} \otimes \mathcal{I}_{k-1})` are optimization variables and :math:`A=` .. math:: \left( \begin{array}{c|c} \mathbb{1}_{\mathcal{I}_{N-1}} \otimes Q^{(N-1)} & \ket{\dot{\tilde{K}}_{1,1}(h)} ... \ket{\dot{\tilde{K}}_{r,d}(h)} \\ \hline \bra{\dot{\tilde{K}}_{1,1}(h)} & \\ \vdots & \lambda \mathbb{1}_{d r} \\ \bra{\dot{\tilde{K}}_{r,d}(h)} & \end{array} \right) with .. math:: \ket{\dot{\tilde{K}}_{i,k}(h)} = {_{\mathcal{O}_{N-1}}}\!\! \braket{i|\dot{K}_{\theta,k}^{(N)}(h)}, where .. math:: \ket{\dot{{K}}_{i,k}^{(N)}(h)} \in \mathcal{O}_0 \otimes \mathcal{I}_0 \otimes \dots \otimes \mathcal{O}_{N-1} \otimes \mathcal{I}_{N-1}, are derivatives of vectorized Kraus operators of :math:`\Lambda_{\theta}^{(N)}`, :math:`r` is the number of Kraus operators of :math:`\Lambda_\theta^{(N)}`, :math:`d = \textrm{dim} \mathcal{O}_{N-1}`. This optimization is implemented in the :py:func:`mop_adaptive_qfi ` function. Similarly to the single channel case, optimization of parallel and adaptive strategies with MOP does not allow for the control of the ancilla dimension and finding optimal input state or optimal comb requires solving an additional SDP problem :cite:`Liu2023, kurdzialek2024`. Moreover, since the algorithm requires optimization over Kraus representations of :math:`\Lambda^{(N)}_\theta` its time and memory complexity is exponentially large in :math:`N`.