nLab bit-flip quantum channel



Quantum systems

quantum logic

quantum physics

quantum probability theoryobservables and states

quantum information

quantum computation


quantum algorithms:

quantum sensing

quantum communication




In quantum information theory, by the bit flip channel one means the quantum channel on a quantum system represented by a single q-bit whose effect is to “flip” the basis q-bit quantum states |0|1\left\vert 0 \right\rangle \leftrightarrow \left\vert 1 \right\rangle with some probability pp or else leave the system unchanged with probability 1p1 - p.

(This would more properly be called the q-bit flip channel, but this is not commonly used terminology.)

The tensor product of NN bit flip channels model the analogous process on NN q-bits, where every single one of them may flip with probability pp, independently of all the others. In this form, the spin flip channel serves as a simple but important model for quantum noise. Basic examples of quantum error correction codes may be used to provide partial correction of such bit flip errors, see at bit flip code.


Fix a real number p[0,1]p \in [0,1] serving as a measure of the probability for a spin flip to occur when data is sent through the channel.

We write:

Now, a common way to define the spin flip channel is by the following expression (e.g. Chuang (2011), p. 296):

flip :States(QBit)States(QBit) flip(ρ) (1p)ρ+p(XρX) \begin{aligned} flip &\colon\; States(QBit) \longrightarrow States(QBit) \\ flip(\rho) &\equiv\; (1-p) \cdot \rho \,+\, p \cdot (X \rho X) \end{aligned}

Since XX is a unitary operator, this makes the bit flip channel an example of a mixed unitary quantum channel.

One readily checks that on pure states in the q-bit-basis this acts as expected:

States(QBit) States(QBit) |00 (1p)|00|+p|11| |11 p|00|+(1p)|11| \begin{array}{ccl} States(QBit) &\longrightarrow& States(QBit) \\ \vert 0 \rangle \langle 0 \rangle &\mapsto& \mathrlap{ (1-p) \cdot \left\vert 0 \right\rangle \left\langle 0 \right\vert \;+\; \;\;\;\;\;\;\;\;\; p \cdot \left\vert 1 \right\rangle \left\langle 1 \right\vert } \\ \vert 1 \rangle \langle 1 \rangle &\mapsto& \mathrlap{ \;\;\;\;\;\;\;\;\; p \cdot \left\vert 0 \right\rangle \left\langle 0 \right\vert \;+\; (1-p) \cdot \left\vert 1 \right\rangle \left\langle 1 \right\vert } \end{array}


Equivalent formulations


The bit flip channel may be expressed as a unitary quantum channel followed by a quantum measurement channel:

flip:ρP 0U flipρU flipP 0+P 1U flipρU flipP 1, flip \;\colon\; \rho \;\mapsto\; P_0 \, U_{flip} \, \rho \, U_{flip} \, P_0 \;+\; P_1 \, U_{flip} \, \rho \, U_{flip} \, P_1 \,,

where the unitary operator U flipU_{flip} is:

U flip=[+1p p p 1p]:QBitQBit U_{flip} \;=\; \left[ \array{ + \sqrt{1-p} & \sqrt{p} \\ \sqrt{p} & - \sqrt{1-p} } \right] \;\colon\; QBit \multimap QBit


The bit-flip quantum channel has an alternative presentation as a uniformly mixed unitary quantum channel and hence as a unistochastic quantum channel, see the discussion there.


Last revised on September 29, 2023 at 09:22:52. See the history of this page for a list of all contributions to it.