Randomization of Quantum Measurements

See also

This content is adapted from the work of Timothée Dao; “Informationally Complete Generalized Measurements for Estimating Quantum Expectation Values” [Unpublished master’s thesis]; ETH Zürich (2023).

Convexity

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

Example:

Consider a 3-outcome POVM on a single-qubit system : M={M1,M2,M3}M(2,3). It can be considered as a 4-outcome POVM with a null effect, M={M1,M2,M3,0}M(2,4). Note that {M1,0,M2,M3} is an equivalent POVM up to the re-ordering of the effects.

Consider two POVMs M1,M2M(d,n), where Mi={Mi,k}k, and a real number pR. The multiplication of a POVM by a scalar and the addition of two POVMs are defined element-wise, namely,

(7)pM1={pM1,k}k,M1+M2={M1,k+M2,k}k,

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 M(d,n) is convex. That is, for any M1,M2M(d,n) and any p[0,1], we have M=pM1+(1p)M2M(d,n).

Proof:

The elements of M, given by Mk=pM1,k+(1p)M2,k for k{1,,n}, are positive semi-definite by the convexity of the set of positive semi-definite operators Pos(H). Moreover,

(8)kMk=pkM1,k+(1p)kM2,k=pIH+(1p)IH=IH,

which concludes the proof. ◻

Such convex combinations can be extended to more than two POVMs. Consider a set of POVMs S={Mi}M(d,n) and a probability distribution {qi}, then

(9)M=iqiMiM(d,n)

as a corollary of Lemma [2].

The set of all convex combinations generated by a set S={Mi}, i.e. its convex hull, is denoted by Sconv.

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:

  1. Randomly pick Mi with probability qi,

  2. Perform the measurement associated with Mi.

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

(10)Pr(outcome is k)=iPr(outcome is ki was picked)Pr(i was picked)=iTr[Mi,kρ]qi=Tr[iqiMi,kρ]=Tr[Mkρ],

which is exactly equivalent to directly performing the measurement associated with M.

Example:

To illustrate the usefulness of this procedure, imagine one has two measurement apparatuses associated with the POVMs M1 and M2 respectively. To perform the measurement associated with M=pM1+(1p)M2, one does not need to build a new apparatus but only needs to use each of the two existing apparatuses with probability p and 1p respectively.

Example:

Consider two POVMs M,NM(2,2). We have

(11)13M+23N={13M1+23N1,13M2+23N2}M(2,2),

and also

(12)13{M1,M2,0,0}+23{0,0,N1,N2}={13M1,13M2,23N1,23N2}M(2,4),

where in the second example M and N are considered as 4-outcome POVMs with 2 null effects: {M1,M2,0,0},{0,0,N1,N2}M(2,4) respectively.

We will use the latter construction very often. Therefore, given two POVMs M1 and M2, we introduce the notation

(13)M1M2={M1,k}k{M2,k}k={Mi,k}i,k

where is the multiset sum [3]. More generally, for any set of POVMs {MiMiM(d,ni)} and any probability distribution {qi}, we have

(14)M=iqiMi={qiMi,k}i,k M(d,ini)

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

References

Footnotes