nLab mutually unbiased bases



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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.


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

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

    {|0,|1} \Big\{ \left\vert 0 \right\rangle ,\, \left\vert 1 \right\rangle \Big\}
  • the Hadamard basis (Pauli-X-eigenstates)

    {12(|0±|1)} \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

12|b|0±b|1|=12 \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{0,1}b \,\in\, \{0,1\} and ±{+,}\pm \in \{+,-\}.


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]


See also:

Created on March 7, 2023 at 16:12:14. See the history of this page for a list of all contributions to it.