nLab bit-flip quantum channel

Idea

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 $\left\vert 0 \right\rangle \leftrightarrow \left\vert 1 \right\rangle$ with some probability $p$ or else leave the system unchanged with probability $1 - p$.

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

The tensor product of $N$ bit flip channels model the analogous process on $N$ q-bits, where every single one of them may flip with probability $p$, 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.

Definition

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

We write:

• $QBit \,\simeq\, \mathbb{C}\cdot \left\vert 0 \right\rangle \,\oplus\, \mathbb{C} \cdot \left\vert 1 \right\rangle$ for the space of quantum states of a q-bit, regarded as a Hilbert space (Hermitian inner product space) with $\big\{\left\vert 0 \right\rangle, \left\vert 1 \right\rangle \big\}$ being an orthonormal linear basis;

• $States(QBit) \subset Herm(QBit)$ for the space of density matrices on this Hilbert space, hence the subspace of positive semi-definite Hermitian operators $QBit \multimap QBit$ of unit trace;

• $X \;\colon\; QBit \multimap QBit$ for the “Pauli X” (“NOT”) quantum logic gate given by $X \left\vert 0 \right\rangle \,=\, \left\vert 1 \right\rangle$, and $X \left\vert 1 \right\rangle \,=\, \left\vert 0 \right\rangle$.

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

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

Since $X$ 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:

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

Properties

Equivalent formulations

Example

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

$$flip \;\colon\; \rho \;\mapsto\; P_0 \, U_{flip} \, \rho \, U_{flip} \, P_0 \;+\; P_1 \, U_{flip} \, \rho \, U_{flip} \, P_1 \,,$$

where the unitary operator $U_{flip}$ is:

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

Example

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.

Related concepts

• bit flip code

References

• Michael A. Nielsen, Isaac L. Chuang, §8.1 and §8.3.3 in: Quantum computation and quantum information, Cambridge University Press (2000) [doi:10.1017/CBO9780511976667, pdf, pdf]

• Isaac L. Chuang, Quantum error correction, Chapter 7 in: Quantum Machines: Measurement and Control of Engineered Quantum Systems Lecture Notes of the Les Houches Summer School 96 (2011) 273–320 [doi:10.1093/acprof:oso/9780199681181.003.0007]

• Stéphane Attal, §6.1.6.1 in: Quantum Channels, Lecture 6 in: Lectures on Quantum Noises [pdf, webpage]