Mutually unbiased measurements

Mutually unbiased bases (MUBs) correspond to measurements in quantum theory that are complementary: if a measurement in a basis yields a definite outcome on a given quantum state, then a measurement in a basis unbiased to the first one yields a uniformly random outcome on the same state. Simple examples of MUBs are photon polarisation measurements in the horizontal and vertical directions, or spin measurements in the z and x directions of a spin-1/2 particle. Their complementary property makes MUBs highly useful in various quantum information processing tasks, such as quantum state tomography, communication tasks, Bell inequalities, and quantum cryptography.

In this talk---after an introduction to MUBs and their use in quantum information---I will introduce a generalisation of MUBs termed mutually unbiased measurements (MUMs). MUMs retain the complementary property of MUBs in a "device-independent" manner: in order to define MUMs, one does not need to refer to the Hilbert space dimension (the number of degrees of freedom, which is not an observable property), only to the outcome number of the measurements (an operational property). I will discuss the mathematical characterisation and constructions of MUMs, and the fundamental similarities and differences between MUBs and MUMs. Then, I will introduce a family of Bell inequalities tailored to MUMs, and show how to use these inequalities for device-independent quantum cryptography, as well as how to use these Bell inequalities to tackle a long-standing open problem on the number of MUBs in a given Hilbert space dimension.

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Room D or Zoom (Link)

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