nLab maximally mixed quantum state

Idea

In quantum information theory, by the maximally mixed state of a quantum systems represented by a finite-dimensional Hilbert space $\mathscr{H}$ one means the mixed state whose density matrix is a multiple of the identity matrix on $\mathscr{H}$.

Hence if $\big(\left\vert w \right\rangle\big)_{w \colon W}$ is any orthonormal linear basis for $\mathscr{H}$, then the maximally mixed state is (in bra-ket notation) given by

$$\frac{1}{dim(\mathscr{H})} \, \underset{w}{\sum} \left\vert w \right\rangle \left\langle w \right\vert \;\;\colon\;\; \mathscr{H} \otimes \mathscr{H}^\ast \,,$$

representing the uniform probability distribution on $W$.

Properties

Relation to quantum channels

A quantum channel which preserves the maximally mixed state is called a unital quantum channel.

A quantum channel with an environmental representation given by coupling to a maximally mixed state of the environment is called a unistochastic quantum channel or (ambiguously) noisy quantum operation. Such operation on single qbits constitute the “DQC1-model” of quantum computation.

Related concepts

• open quantum systems

• quantum noise

References

See references at unistochastic quantum channel.