Frame theory and dual space¶

POVMs as frames¶

As a reminder, given an IC-POVM \(\mathbf{M} = \{M_k\}_{k \in \{1, \dots, n \}}\) and an observable \(\mathcal{O}\), there exist \(\omega_k \in \mathbb{R}\) such that

(15)¶\[\mathcal{O} = \sum_{k=1}^{n} \omega_k M_k .\]

We will now outline a formal approach to obtain the coefficients \(\omega_k\) in Eq. (15) for a given observable \(\mathcal{O}\). First, we note that the minimal number of linearly independent POVM elements for an IC-POVM is \(n = d^2\). We call such POVMs minimally informationally complete. In that case, the coefficients \(\omega_k\) are unique. However, for POVMs with \(n > d^2\), such as those that arise from IC PM-simulable POVMs, the decomposition in Eq. (15) is not unique. This redundancy is described by frame theory, as outlined in Ref. [1].

Simply speaking, a frame is a generalization of the notion of the basis of a vector space, where the basis elements may be linearly dependent. The set of POVM operators \(\mathbf{M} = \{M_k\}_{k \in \{1, \dots, n \}}\) forms a frame for the space of Hermitian operators if and only if it is IC. For any frame, there exists at least one dual frame \(\mathbf{D} = \{D_k\}_{k \in \{1, \dots, n \}}\), such that

(16)¶\[\mathcal{O} = \sum_{k=1}^n \mathrm{Tr}[\mathcal{O} D_k] M_k\]

for any Hermitian operator \(\mathcal{O}\). Therefore, the coefficients \(\omega_k\) can simply be obtained from the duals \(\mathbf{D}\) as

(17)¶\[\omega_k = \mathrm{Tr}[\mathcal{O} D_k].\]

Notably, dual operators generalize the concept of classical shadows of a quantum state [2] (see section below for details), thus providing a direct connection to the popular randomized measurement toolbox [3].

Constructing dual frames¶

For a minimally IC POVM, only one dual frame exists. It can be constructed from the POVM elements as

(18)¶\[\left| D_k \right\rangle\kern-3mu\rangle = \mathcal{F}^{-1} \left| M_k \right\rangle\kern-3mu\rangle \, , \quad k =1,2,\dots,n\]

with the canonical frame superoperator

(19)¶\[\mathcal{F} = \sum_{k=1}^n \left| M_k \right\rangle\kern-3mu\rangle\kern-5mu\left\langle\kern-3mu\langle M_k \right|,\]

where we have used the widespread vectorized ‘double-ket’ notation. Thus, the frame superoperator can be used to transform between the POVM space and the dual space.

For an overcomplete POVM, the canonical frame superoperator creates one of infinitely many possible dual frames. Other valid dual frames can be obtained through a parametrized frame superoperator as follows:

(20)¶\[\left| D_k \right\rangle\kern-3mu\rangle = \alpha_k \mathcal{F}^{-1}_{\alpha} \left| M_k \right\rangle\kern-3mu\rangle \, , \quad \quad \text{with } \mathcal{F}_{\alpha} = \sum_{k=1}^n \alpha_k \left| M_k \right\rangle\kern-3mu\rangle\kern-5mu\left\langle\kern-3mu\langle M_k \right|,\]

for real parameters \(\{\alpha_k\}_k \subset \mathbb{R}\) such that \(\mathcal{F}_{\alpha}\) in invertible [4].

Relation to classical shadows¶

We now show the explicit connection to the technique of classical shadows [2]. The technique consists of rotating the state \(\rho\) by a unitary \(U_i\), sampled from a set \(\mathcal{U}\), and then performing a measurement in the computational basis. We show in the section PM-simulable measurements that this protocol is equivalent to performing the PM-simulable POVM \(\mathbf{M} = \biguplus_i q_i \mathbf{P}_i = \{q_i P_{i,k}\}_{(i,k)}\), where \(P_{i,k} = U_i^\dagger \ketbra{k}{k} U_i\) and the outcomes are labeled by \((i,k)\). It now appears that the measurement channel

\[\mathcal{M} : \rho \mapsto \mathbb{E}_{i \sim \{q_i\}} \sum_k \mathrm{Tr}[\rho P_{i,k}] P_{i,k} = \sum_{i,k} \frac{\mathrm{Tr}[\rho M_{i,k}]}{\mathrm{Tr}[M_{i,k}]} M_{i,k} \, , \qquad M_{i,k} = q_i P_{i,k} \, ,\]

is actually an \(\alpha\)-frame superoperator \(\mathcal{F}_{\alpha}\) associated with the POVM \(\mathbf{M}\), where the coefficients are given by \(\alpha_{i,k} = 1/\mathrm{Tr}[M_{i,k}] = 1/q_i\) for all \(i,k\). Most importantly, the elements of the dual frame given by this \(\alpha\)-parametrization are the classical shadows:

\[\hat{\rho}_{i,k} = \mathcal{M}^{-1}(P_{i,k}) = \frac{1}{q_i} \mathcal{M}^{-1}(M_{i,k}) = \alpha_{i,k} \mathcal{F}_{\alpha}^{-1}(M_{i,k}) = D_{i,k} \, .\]

In other words, the classical shadows technique consists of performing a PM-simulable POVM and choosing a specific dual frame. However, nothing prevents us from choosing another dual frame. Any dual frame defines an unbiased estimator of the state. More precisely, for any dual frame \(\mathbf{D} = \{D_{i,k}\}\) and any state \(\rho\), we have

\[\rho = \sum_{i,k} \mathrm{Tr}[ \rho M_{i,k}] D_{i,k} = \mathbb{E}_{i,k}[D_{i,k}] \, ,\]

which follows from the reciprocity of duality. That is, if \(\mathbf{D}\) is a dual frame to \(\mathbf{M}\), then \(\mathbf{M}\) is a dual frame to \(\mathbf{D}\) [5].

References