Small sets of complementary observables

Two observables are called complementary if preparing a physical object in an eigenstate of one of them yields a completely random result in a measurement of the other. We investigate small sets of complementary observables that cannot be extended by yet another complementary observable. We construct explicit examples of unextendible sets up to dimension 16 and conjecture certain small sets to be unextendible in higher dimensions. Our constructions provide three complementary measurements, only one observable away from the ultimate minimum of two. Almost all our examples in finite dimensions are useful for discriminating pure states from some mixed states, and they help to shed light on the complex topology of the Bloch space of higher-dimensional quantum systems.

Figure 1

The problem studied here can be visualized as follows. A physical system prepared initially in a completely mixed state is interrogated by a sequence of $n$ players. Each of them chooses to perform the corresponding projective measurement $M_j$ or forwards the system unperturbed to the next player. It is required that all the measurements have completely random outcomes, independent of the results of the previous measurement and which of the measurements have been performed. We ask about the smallest number of players such that no further player could guarantee that the measurement he or she performs will possess random outcomes, starting with a $d$-level input state. See Table 1 for a summary in small dimensions and the main text for other dimensions.