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arXiv:quant-ph/0407078 · Physical Review A

Pure-state informationally complete and “really” complete measurements

J. Finkelstein

Published Nov 12, 2004• 77 citations
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Abstract

I construct a positive-operator-valued measure (POVM) which has $2d$ rank-1 elements and which is informationally complete for generic pure states in $d$ dimensions, thus confirming a conjecture made by Flammia, Silberfarb, and Caves (e-print quant-ph∕0404137). I show that if a rank-1 POVM is required to be informationally complete for all pure states in $d$ dimensions, it must have at least $3d\ensuremath{-}2$ elements. I also show that, in a POVM which is informationally complete for all pure states in $d$ dimensions, for any vector there must be at least $2d\ensuremath{-}1$ POVM elements which do not annihilate that vector.

DOI
10.1103/physreva.70.052107

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We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension d^2-1 corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC-POVMs contains weak SIC-POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC-POVM.
We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC POVMs contains weak SIC POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC POVM.
We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension d^2-1 corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC-POVMs contains weak SIC-POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC-POVM.
Quantum state tomography seeks to reconstruct an unknown state from measurement statistics. A finite measurement (POVM) is \emph{pure-state informationally complete} (PSI-Complete) if the outcome probabilities determine any pure state up to a global phase. We study \emph{rank-one} POVMs that are minimally sufficient for this task. We call such a POVM \emph{vital} if it is PSI-Complete but every proper subcollection is not PSI-Complete. We prove sharp upper bounds on the size of vital rank-one POVMs in dimension nnn: the size is at most (n+12)\binom{n+1}{2}(2n+1​) over R\mathbb{R}R and at most n2n^{2}n2 over C\mathbb{C}C, and we give constructions that attain these bounds. In the real case, we further exhibit a connection to block designs: whenever w∣n(n−1)w \mid n(n-1)w∣n(n−1), an (n,w,w−1)(n,w,w-1)(n,w,w−1) design produces a vital rank-one POVM with n+n(n−1)/wn + n(n-1)/wn+n(n−1)/w outcomes. We provide explicit constructions for w=2,n−1w=2,n-1w=2,n−1, and nnn.
There has been much interest in so-called SIC-POVMs, i.e., rank 1 symmetric informationally complete positive operator valued measures. In this paper we discuss the larger class of POVMs that are symmetric and informationally complete, but not necessarily rank 1. This class of POVMs is of some independent interest. In particular it includes a POVM that is closely related to the discrete Wigner function. However, it is interesting mainly because of the light it casts on the problem of constructing rank 1, symmetric, informationally complete POVMs. In this connection we derive an extremal condition alternative to the one derived by Renes et al.
Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1) every POVM element is rank one, (2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and (3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two, we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.