nLab mutually unbiased bases

Idea

In quantum information theory a pair of orthonormal linear bases of a given (usually finite-dimensional) Hilbert space of quantum states is called mutually unbiased if the absolute value of the inner products of any one element in one basis with any in the other basis all have the same value.

Examples

In the 2-dimensional Hilert space of qbits $\simeq \mathbb{C}^2$ the pair consisting of

• the “measurement basis” (Pauli-Z-eigenstates)

  $$\Big\{ \left\vert 0 \right\rangle ,\, \left\vert 1 \right\rangle \Big\}$$

• the Hadamard basis (Pauli-X-eigenstates)

  $$\Big\{ \tfrac{1}{\sqrt{2}} \big( \left\vert 0 \right\rangle \pm \left\vert 1 \right\rangle \big) \Big\}$$

are (each orthonormal and) mutually unbiased: The absolute value-square of the mutual inner product is

$$\tfrac{1}{\sqrt{2}} \left\vert \left\langle b \vert 0 \right\rangle \,\pm\, \left\langle b \vert 1 \right\rangle \right\vert \;\;\; = \;\;\; \tfrac{1}{\sqrt{2}}$$

for all $b \,\in\, \{0,1\}$ and $\pm \in \{+,-\}$.

Related concepts

• quantum measurement

• ZX-calculus

References

Original articles:

• William K. Woooters?, Brian D. Fields?, Optimal state-determination by mutually unbiased measurements, Annals of Physics 191 2 (1989) 363-381 [doi:10.1016/0003-4916(89)90322-9]

Review:

• Ingemar Bengtsson, Three ways to look at mutually unbiased bases, AIP Conference Proceedings 889 (2007) 40-51 [arXiv:quant-ph/0610216, doi:10.1063/1.2713445]

See also:

• Wikipedia, Mutually unbiased bases