Mutually unbiased measurements with arbitrary purity

Abstract

Mutually unbiased measurements are a generalization of mutually unbiased bases in which the measurement operators need not to be rank one projectors. In a d-dimensional space, the purity of measurement elements ranges from 1/d for the measurement operators corresponding to maximally mixed states to 1 for the rank one projectors. In this contribution, we provide a class of MUM that encompasses the full range of purity. Similar to the MUB in which the operators corresponding to different outcomes of the same measurement commute mutually, our class of MUM possesses this sense of compatibility within each measurement. The spectra of these MUMs provide a way to construct a class of d-dimensional orthogonal matrices which leave the vector of equal components invariant. Based on this property, and by using the MUM-based entanglement witnesses, we examine the minimal number of measurements needed to detect entanglement of bipartite states. For a general bipartite pure state, we need only two MUMs: The first one assigns a zero mean value for all pure states; however, a complementary measurement is needed to give a negative mean value for entangled states. Interestingly, the amount of this negative value is proportional to an entanglement monotone.

Appendix A: The number of independent eigenvalues

To count the number of independent eigenvalues, we have to find the number of independent constraints on the set of eigenvalues. Clearly, Eq. (12) gives one independent relation. Equation (13), on the other hand, reads

$$\begin{aligned} \sum _{j=0}^{d-1} \mu ^2_{n\oplus j}=\kappa -1/d, \end{aligned}$$

(A1)

for $n=n^\prime$, and

$$\begin{aligned} \sum _{j=0}^{d-1} \mu _{n\oplus j}\mu _{n^\prime \oplus j}=-\left( \frac{\kappa -1/d}{d-1}\right) , \end{aligned}$$

(A2)

for $n\ne n^\prime$. Equation (A1) gives its own independent relation; however, Eq. (A2) provides independent relations only for $n^\prime =n\oplus 1,n\oplus 2,\cdots ,n\oplus \left\lfloor d/2\right\rfloor$, where $\left\lfloor d/2\right\rfloor$ denotes the integral part of d/2, i.e., $\left\lfloor d/2\right\rfloor =d/2$ if d is even and $\left\lfloor d/2\right\rfloor =(d-1)/2$ if d is odd. This follows from the fact the LHS of this equation is invariant under the change $\{n \rightarrow n+n_0,\; n^\prime \rightarrow n^\prime +n_0\}$ for any $n_0=0,\cdots d-1$. For $n^\prime =1,\cdots ,\left\lfloor (d-1)/2\right\rfloor$, the multiplicity is d; however, for $n^\prime =d/2$ (when d is even) the corresponding multiplicity is d/2. There is, however, a relation between Eqs. (A1) and (A2), following easily from

$$\begin{aligned} 2\sum _{n<n^\prime }^{d-1} \mu _{n}\mu _{n^\prime }=\left( \sum _{n=0}^{d-1} \mu _{n}\right) ^2-\sum _{n=0}^{d-1} \mu ^2_{n}=-\left( \kappa -1/d\right) , \end{aligned}$$

(A3)

which, using the multiplicity of each term, can be written as

$$\begin{aligned} d\sum _{i=1}^{\left\lfloor \frac{d-1}{2}\right\rfloor }\left[ \sum _{j=0}^{d-1} \mu _{n\oplus j}\mu _{n\oplus i+ j}\right] +\frac{d}{2}\mathcal {E}=-\frac{1}{2}\left( \kappa d-1\right) . \end{aligned}$$

(A4)

Above $\mathcal {E}=\sum _{j=0}^{d-1} \mu _{n\oplus j}\mu _{n\oplus d/2+ j}$ if d is even and it is zero if d is odd. Putting everything together, we find the number of independent eigenvalues as $N=d-(2+\left\lfloor d/2\right\rfloor -1)=\left\lfloor \frac{d-1}{2}\right\rfloor$.