Randomization of Quantum Measurements¶

Convexity¶

Following Ref. [1], the set of \(n\)-outcome POVMs on the Hilbert space \(\mathcal{H}\) is denoted by \(\mathcal{M}(\mathcal{H},n)\). If only the dimension \(d\) of the Hilbert space is relevant, we can denote this set by \(\mathcal{M}(d,n)\). Note that for \(m<n\), an \(m\)-outcome POVM can be seen as an \(n\)-outcome POVM with \((n-m)\) null effects. Hence we have \(\mathcal{M}(d,m) \subset \mathcal{M}(d,n)\) for all \(m<n\).

Example:: Consider a \(3\)-outcome POVM on a single-qubit system : \(\mathbf{M} = \{M_1,M_2,M_3\} \in \mathcal{M}(2,3)\). It can be considered as a 4-outcome POVM with a null effect, \(\mathbf{M} = \{M_1,M_2,M_3,0\} \in \mathcal{M}(2,4)\). Note that \(\{M_1,0,M_2,M_3\}\) is an equivalent POVM up to the re-ordering of the effects.

Consider two POVMs \(\mathbf{M}_1,\mathbf{M}_2 \in \mathcal{M}(d,n)\), where \(\mathbf{M}_i = \{ M_{i,k}\}_k\), and a real number \(p \in \mathbb{R}\). The multiplication of a POVM by a scalar and the addition of two POVMs are defined element-wise, namely,

(7)¶\[\begin{split}\begin{aligned} p \mathbf{M}_1 &= \{p M_{1,k}\}_k \, , \\ \mathbf{M}_1 + \mathbf{M}_2 &= \{ M_{1,k} + M_{2,k}\}_k \, , \end{aligned}\end{split}\]

respectively. Crucially, these operations do not preserve the set of POVMs. That is, the effects resulting from these operations do not form POVMs in general. Indeed, it is easy to check that the resulting effects do not sum to the identity (except for the scalar multiplication by \(p=1\)). However, by taking convex combinations we arrive at the following result.

Lemma [2]:

The set \(\mathcal{M}(d,n)\) is convex. That is, for any \(\mathbf{M}_1, \mathbf{M}_2 \in \mathcal{M}(d,n)\) and any \(p \in [0,1]\), we have \(\mathbf{M} = p\mathbf{M}_1+ (1-p)\mathbf{M}_2 \in \mathcal{M}(d,n)\).

Proof:

The elements of \(\mathbf{M}\), given by \(M_k = p M_{1,k} + (1-p) M_{2,k}\) for \(k\in \{1,\dots,n\}\), are positive semi-definite by the convexity of the set of positive semi-definite operators \(\mathrm{Pos}(\mathcal{H})\). Moreover,

(8)¶\[\sum_k M_k = p \sum_k M_{1,k} + (1-p) \sum_k M_{2,k} = p \mathbb{I}_\mathcal{H} + (1-p) \mathbb{I}_\mathcal{H} = \mathbb{I}_\mathcal{H} \, ,\]

which concludes the proof. ◻

Such convex combinations can be extended to more than two POVMs. Consider a set of POVMs \(\mathcal{S} = \{\mathbf{M}_i\} \subset \mathcal{M}(d,n)\) and a probability distribution \(\{q_i\}\), then

(9)¶\[\mathbf{M} = \sum_i q_i \mathbf{M}_i \, \in \mathcal{M}(d,n)\]

as a corollary of Lemma [2].

The set of all convex combinations generated by a set \(\mathcal{S} = \{\mathbf{M}_i\}\), i.e. its convex hull, is denoted by \(\mathcal{S}^\mathrm{conv}\).

In practice¶

These convex combinations can be achieved in practice by the so-called randomization of quantum measurements procedure, which simply consists of two steps for each measurement shot:

Randomly pick \(\mathbf{M}_i\) with probability \(q_i\),
Perform the measurement associated with \(\mathbf{M}_i\).

Indeed, the probability of obtaining the outcome \(k\) with this procedure is

(10)¶\[\begin{split}\begin{split} \Pr(\textrm{outcome is }k) & = \sum_i \Pr(\textrm{outcome is } k \mid i \textrm{ was picked}) \Pr(i \textrm{ was picked}) \\ & = \sum_i \mathrm{Tr}[M_{i,k}\rho] q_i = \mathrm{Tr}[\sum_i q_i M_{i,k}\rho] \\ & = \mathrm{Tr}[M_k\rho] \, , \end{split}\end{split}\]

which is exactly equivalent to directly performing the measurement associated with \(\mathbf{M}\).

Example:

To illustrate the usefulness of this procedure, imagine one has two measurement apparatuses associated with the POVMs \(\mathbf{M}_1\) and \(\mathbf{M}_2\) respectively. To perform the measurement associated with \(\mathbf{M} = p\mathbf{M}_1+ (1-p)\mathbf{M}_2\), one does not need to build a new apparatus but only needs to use each of the two existing apparatuses with probability \(p\) and \(1-p\) respectively.

Example:

Consider two POVMs \(\mathbf{M},\mathbf{N} \in \mathcal{M}(2,2)\). We have

(11)¶\[\frac{1}{3} \mathbf{M} + \frac{2}{3}\mathbf{N} = \{ \, \frac{1}{3} M_1 + \frac{2}{3} N_1\, , \; \frac{1}{3} M_2 + \frac{2}{3} N_2 \, \} \in \mathcal{M}(2,2)\, ,\]

and also

(12)¶\[\frac{1}{3} \{M_1, M_2,0,0\} + \frac{2}{3} \{0,0,N_1, N_2\} = \{\frac{1}{3} M_1 \, , \; \frac{1}{3} M_2 \, , \; \frac{2}{3} N_1 \, , \; \frac{2}{3} N_2\} \in \mathcal{M}(2,4)\, ,\]

where in the second example \(\mathbf{M}\) and \(\mathbf{N}\) are considered as 4-outcome POVMs with 2 null effects: \(\{M_1, M_2,0,0\}, \{0,0, N_1, N_2\} \in \mathcal{M}(2,4)\) respectively.

We will use the latter construction very often. Therefore, given two POVMs \(\mathbf{M}_1\) and \(\mathbf{M}_2\), we introduce the notation

(13)¶\[\mathbf{M}_1 \uplus \mathbf{M}_2 = \{ M_{1,k}\}_k \uplus \{M_{2,k}\}_k = \{ M_{i,k}\}_{i,k}\]

where \(\uplus\) is the multiset sum [3]. More generally, for any set of POVMs \(\{\mathbf{M}_i \mid \mathbf{M}_i \in \mathcal{M}(d, n_i)\}\) and any probability distribution \(\{q_i\}\), we have

(14)¶\[\mathbf{M} = \biguplus_i q_i \mathbf{M}_i = \left\{ q_i M_{i,k} \right\}_{i,k} \ \in \mathcal{M}(d, {\textstyle\sum_i} n_i)\]

where the outcomes are now denoted by the pair \((i,k)\).

References

Footnotes