Dephasing Channel

The dephasing channel is the basic channel for loss of phase coherence in a preferred basis. It suppresses off-diagonal density-matrix elements while leaving the corresponding populations unchanged.

For a qubit in the $Z$ basis, the defining action is

$$\begin{pmatrix} \rho_{00} & \rho_{01}\\ \rho_{10} & \rho_{11} \end{pmatrix} \mapsto \begin{pmatrix} \rho_{00} & \lambda\rho_{01}\\ \lambda^*\rho_{10} & \rho_{11} \end{pmatrix}, \qquad |\lambda|\le1.$$

The parameter $\lambda$ is the coherence factor. Its magnitude is the remaining coherence; its phase is a coherent $Z$ rotation that can often be separated from the irreversible dephasing.

What the Channel Means

Dephasing is not energy relaxation. In the dephasing basis, populations are unchanged:

$$\rho_{00}'=\rho_{00}, \qquad \rho_{11}'=\rho_{11}.$$

Only coherences are damped:

$$\rho_{01}'=\lambda\rho_{01}.$$

This is why pure dephasing is the canonical model for:

• fluctuating energy splittings,
• unread which-path information,
• elastic scattering that distinguishes alternatives,
• nonselective projective measurement in a basis,
• slow classical phase noise,
• environment records that become correlated with basis states.

The basis matters. A channel that is pure dephasing in one basis may change populations when viewed in another basis.

For the broader physical distinction, see Dephasing vs Dissipation.

Dephasing and Amplitude Damping owns the QI-facing conversion from protocol-qualified relaxation, coherence, and equilibrium-population records to a finite idle channel, including physicality, composition, and held-out audits; this page retains the dephasing-channel definition, Kraus and Choi forms, environment-record and random-phase pictures, higher-dimensional extension, and formal semigroup limit.

Phase Flip Kraus Form

For real $-1\le\lambda\le1$, the qubit dephasing channel can be written as a phase-flip mixture:

$$\Phi_\lambda(\rho) = \frac{1+\lambda}{2}\rho + \frac{1-\lambda}{2}Z\rho Z.$$

A Kraus representation is

$$K_0 = \sqrt{\frac{1+\lambda}{2}}\,I, \qquad K_1 = \sqrt{\frac{1-\lambda}{2}}\,Z.$$

The trace-preserving condition is

$$K_0^\dagger K_0+K_1^\dagger K_1=I.$$

Applying the channel to the density matrix gives

$$\Phi_\lambda(\rho) = \begin{pmatrix} \rho_{00} & \lambda\rho_{01}\\ \lambda\rho_{10} & \rho_{11} \end{pmatrix}$$

when $\lambda$ is real. If $\lambda$ is complex, write $\lambda=|\lambda|e^{-i\theta}$ and treat the phase as a $Z$ rotation together with damping by $|\lambda|$.

Equivalently, use an error probability

$$p=\frac{1-\lambda}{2}.$$

Then

$$\Phi_p(\rho) = (1-p)\rho+pZ\rho Z, \qquad 0\le p\le1.$$

This is the $Z$-axis special case of a Pauli channel.

The Markovian pure-dephasing semigroup has $0\le p(t)\le1/2$ and $\lambda(t)=e^{-\Gamma_\phi t}$; allowing $p>1/2$ is still a valid phase-flip channel but not the usual monotone Markovian decay convention.

Bloch Sphere Action

Write a qubit state as

$$\rho = \frac{1}{2} \left( I+r_xX+r_yY+r_zZ \right).$$

For real $\lambda$, the dephasing channel acts as

$$(r_x,r_y,r_z) \mapsto (\lambda r_x,\lambda r_y,r_z).$$

The Bloch sphere is contracted toward the $Z$ axis. The north and south pole states are fixed, while transverse superpositions lose contrast.

Repeated dephasing multiplies coherence factors, and the fixed states are exactly diagonal in the dephasing basis. See Channel Composition and Fixed Points for the general channel language.

Complete dephasing is the case $\lambda=0$:

$$(r_x,r_y,r_z) \mapsto (0,0,r_z).$$

It maps every state to its diagonal part in the $Z$ basis.

Choi Matrix and Complete Positivity

Using the output-input convention of Choi Matrix, the Choi matrix of the qubit dephasing channel is

$$J_{\Phi_\lambda} = \begin{pmatrix} 1 & 0 & 0 & \lambda\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \lambda^* & 0 & 0 & 1 \end{pmatrix}.$$

The nonzero block has eigenvalues

$$1+|\lambda|, \qquad 1-|\lambda|.$$

Therefore the channel is completely positive exactly when

$$|\lambda|\le1.$$

The Choi rank is $2$ for $0\lt|\lambda|\lt1$, rank $1$ for a unitary phase rotation with $|\lambda|=1$, and rank $2$ at complete dephasing for the qubit channel with $\lambda=0$.

Nonselective Measurement

An unread projective measurement in the $Z$ basis gives complete dephasing:

$$\Delta_Z(\rho) = P_0\rho P_0+P_1\rho P_1,$$

where

$$P_0=\lvert0\rangle\langle0\rvert, \qquad P_1=\lvert1\rangle\langle1\rvert.$$

In matrix form,

$$\Delta_Z \left( \begin{pmatrix} \rho_{00} & \rho_{01}\\ \rho_{10} & \rho_{11} \end{pmatrix} \right) = \begin{pmatrix} \rho_{00} & 0\\ 0 & \rho_{11} \end{pmatrix}.$$

Thus an unread ideal measurement is not “no measurement.” It is a channel that removes coherence in the measured decomposition. See Measurement Backaction for the operational distinction.

Environment-Record Picture

Suppose the environment becomes correlated with the dephasing basis:

$$\lvert0\rangle\lvert e\rangle \mapsto \lvert0\rangle\lvert e_0\rangle, \qquad \lvert1\rangle\lvert e\rangle \mapsto \lvert1\rangle\lvert e_1\rangle.$$

For an input coherence $\rho_{01}$, tracing out the environment gives

$$\rho_{01} \mapsto \langle e_1|e_0\rangle\,\rho_{01}.$$

Thus

$$\lambda=\langle e_1|e_0\rangle.$$

Orthogonal environment records give complete dephasing. Identical environment states give no dephasing. Partial distinguishability gives partial dephasing.

This is the simplest channel version of Environment-Induced Decoherence.

Random Phase Picture

Classical random phase noise gives another representation. Let

$$U_\phi = e^{-i\phi Z/2}$$

and average over phases with probability density $p(\phi)$:

$$\Phi(\rho) = \int d\phi\,p(\phi)\, U_\phi\rho U_\phi^\dagger.$$

The off-diagonal element transforms as

$$\rho_{01} \mapsto \left[ \int d\phi\,p(\phi)e^{-i\phi} \right] \rho_{01}.$$

Therefore

$$\lambda = \langle e^{-i\phi}\rangle.$$

For a Gaussian random phase with mean $0$ and variance $\sigma_\phi^2$,

$$\lambda=e^{-\sigma_\phi^2/2}.$$

Random-unitary phase averaging is a useful model for classical frequency noise. A quantum bath can produce the same channel form while carrying quantum correlations rather than a classical random label.

Markovian Master Equation Limit

A Markovian pure-dephasing master equation can be written

$$\frac{d\rho}{dt} = \frac{\Gamma_\phi}{2} \left( Z\rho Z-\rho \right).$$

The populations are constant:

$$\frac{d\rho_{00}}{dt}=0, \qquad \frac{d\rho_{11}}{dt}=0.$$

The coherence obeys

$$\frac{d\rho_{01}}{dt} = -\Gamma_\phi\rho_{01}.$$

Hence

$$\lambda(t)=e^{-\Gamma_\phi t}.$$

The corresponding phase-flip probability is

$$p(t) = \frac{1-e^{-\Gamma_\phi t}}{2}.$$

For the exact coherence-factor model behind nonexponential dephasing, see Pure Dephasing Model. For the generator-level assumptions, rate conventions, and $T_2$ relation, see Pure Dephasing Master Equation. In $T_2$ language, this pure-dephasing contribution is often written as $T_\phi^{-1}=\Gamma_\phi$, with relaxation contributions added separately.

Higher Dimensional Dephasing

For projectors $\{P_a\}$ onto a preferred orthogonal decomposition, complete dephasing is

$$\Delta(\rho) = \sum_a P_a\rho P_a.$$

More general partial dephasing can multiply blocks by coherence factors:

$$\Phi(\rho) = \sum_{a,b} \Gamma_{ab} P_a\rho P_b.$$

For rank-one projectors, complete positivity is equivalent to the matrix $\Gamma$ being positive semidefinite with

$$\Gamma_{aa}=1.$$

This $\Gamma$ can be interpreted as a Gram matrix of environment states:

$$\Gamma_{ab} = \langle e_b|e_a\rangle.$$

The qubit channel is the two-dimensional special case.

Common Mistakes

Treating dephasing as basis independent

Dephasing always refers to a basis or decomposition. Changing the basis changes what counts as population and coherence.

Confusing dephasing with depolarizing noise

Dephasing contracts only transverse directions relative to the dephasing axis. Depolarizing noise contracts all Bloch directions equally.

Confusing dephasing with amplitude damping

Pure dephasing does not transfer population between energy eigenstates in the dephasing basis. Amplitude damping does.

Assuming unread means reversible

If which-outcome information has leaked into an apparatus or environment and is then ignored, the reduced channel is dephasing. The coherence is not restored by merely forgetting to read the record.

Overinterpreting a negative coherence factor

A real $\lambda\lt0$ is allowed by complete positivity when $|\lambda|\le1$; it includes an additional phase flip. It is not the usual one-parameter Markovian pure-dephasing semigroup, where $\lambda(t)\ge0$.

Exercises

Kraus action

Using

$$K_0=\sqrt{\frac{1+\lambda}{2}}\,I, \qquad K_1=\sqrt{\frac{1-\lambda}{2}}\,Z,$$

show that $\rho_{01}\mapsto\lambda\rho_{01}$ for real $\lambda$.

Solution

The channel is

$$\Phi_\lambda(\rho) = \frac{1+\lambda}{2}\rho + \frac{1-\lambda}{2}Z\rho Z.$$

Since $Z=\operatorname{diag}(1,-1)$,

$$(Z\rho Z)_{01}=-\rho_{01}.$$

Therefore

$$(\Phi_\lambda(\rho))_{01} = \frac{1+\lambda}{2}\rho_{01} - \frac{1-\lambda}{2}\rho_{01} = \lambda\rho_{01}.$$

Bloch contraction

Show that real-$\lambda$ dephasing maps $(r_x,r_y,r_z)$ to $(\lambda r_x,\lambda r_y,r_z)$.

Solution

For

$$\rho = \frac12 \begin{pmatrix} 1+r_z & r_x-ir_y\\ r_x+ir_y & 1-r_z \end{pmatrix},$$

dephasing leaves the diagonal entries fixed and multiplies off-diagonal entries by $\lambda$. Hence

$$r_z' = r_z, \qquad r_x'-ir_y'=\lambda(r_x-ir_y).$$

For real $\lambda$ this gives

$$r_x'=\lambda r_x, \qquad r_y'=\lambda r_y.$$

Gaussian phase noise

Let $\phi$ be a zero-mean Gaussian random variable with variance $\sigma_\phi^2$. Show that $\langle e^{-i\phi}\rangle=e^{-\sigma_\phi^2/2}$.

Solution

The characteristic function of a zero-mean Gaussian with variance $\sigma_\phi^2$ is

$$\langle e^{ik\phi}\rangle = e^{-k^2\sigma_\phi^2/2}.$$

Set $k=-1$ to obtain

$$\langle e^{-i\phi}\rangle = e^{-\sigma_\phi^2/2}.$$

References

• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer (1983).
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2010).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, Springer (2007).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).