Conventions for Channels, Noise, and Decay Rates

Open-system formulas often differ by factors of two, tensor order, or the sign assigned to heat. This page explains how to translate those conventions before comparing results. Symbols are defined locally: a letter such as $\gamma$ does not identify a decay rate until the equation shows what decays.

Required background. The volume introduction, One Interaction, Different Measurement Records, and Closed versus Open Systems distinguish conditional states, channels, and reduced dynamics. General state notation is in Density Matrix Conventions.

States, effects, and maps

Symbol	Meaning
$\rho_S,\rho_{SE}$	System state and joint system–environment state
$\operatorname{Tr}_E$	Partial trace over the environment; the subscript names the factor removed
$P_r$ or $\Pi_r$	A projector for a sharp measurement outcome
$E_r$ or $F_r$	A positive POVM effect; its trace pairing with $\rho$ gives a probability
$K_{r\alpha}$	A Kraus operator for outcome $r$ and unresolved index $\alpha$
$\mathcal I_r$	An outcome-resolved completely positive operation
$\Phi$	A channel acting on density operators
$\mathcal L$	A generator acting on operators, with units of inverse time
$\rho_c$	A state conditioned on a specified measurement record

Effects and Kraus operators are related by $E_r=\sum_\alpha K_{r\alpha}^\dagger K_{r\alpha}$, but are not interchangeable. The operation is

$$\mathcal I_r(\rho)=\sum_\alpha K_{r\alpha}\rho K_{r\alpha}^\dagger.$$

Its trace is $p_r$; divide by $p_r$ only when $p_r>0$ to obtain a conditional density operator. The sum over all recorded outcomes must preserve trace. A missing no-click outcome can therefore turn an apparently incomplete detector model into a complete one. See Quantum Instruments.

The adjoint channel acts on observables:

$$\operatorname{Tr}[A\Phi(\rho)] =\operatorname{Tr}[\Phi^\dagger(A)\rho].$$

Thus $\Phi$ is trace preserving exactly when $\Phi^\dagger(I)=I$. This does not imply $\Phi(I)=I$: trace preservation and unitality are different properties.

Rates inside or outside jump operators

We use the dissipator

$$\mathcal D[L]\rho=L\rho L^\dagger -\tfrac12\{L^\dagger L,\rho\}.$$

The terms $\gamma\mathcal D[A]\rho$ and $\mathcal D[\sqrt\gamma A]\rho$ are identical for $\gamma\ge0$. If $A$ is dimensionless, $\gamma$ has units of inverse time and the rate-absorbed operator has units of inverse square-root time. A source defining its dissipator with twice this expression assigns a correspondingly different coefficient.

Two qubit examples make the conversion concrete:

Generator term	Observable decay in this convention
$\gamma_1\mathcal D[\sigma_-]\rho$	Excited population decays at $\gamma_1$; coherence decays at $\gamma_1/2$
$(\gamma_\phi/2)\mathcal D[Z]\rho$	Coherence decays at $\gamma_\phi$; populations are fixed

Here $\sigma_-=|0\rangle\langle1|$ and $Z=|0\rangle\langle0|-|1\rangle\langle1|$. With both terms and an energy-diagonal Hamiltonian, $1/T_2=1/(2T_1)+\gamma_\phi$ for this simple Markovian qubit model. More general nonexponential dynamics need not have a single $T_1$ or $T_2$.

For Brownian motion, the friction term $-i\gamma[q,\{p,\rho\}]/(2\hbar)$ damps mean momentum at rate $\gamma$. In this convention the high-temperature diffusion coefficient is $D_{pp}=M\gamma k_BT$ when its term is $-D_{pp}[q,[q,\rho]]/\hbar^2$. A source using twice the friction term often writes twice this diffusion coefficient. Compare the full equation, not either coefficient alone; see Quantum Brownian Motion.

Tensor order, Choi normalization, and vectorization

For an input space of dimension $d$, take the output factor first and the reference copy of the input second, as in the channel articles:

$$J_\Phi=\sum_{ij}\Phi(|i\rangle\langle j|) \otimes|i\rangle\langle j|.$$

A channel satisfies $J_\Phi\ge0$ and $\operatorname{Tr}_{\rm out}J_\Phi=I_{\rm ref}$. Consequently $\operatorname{Tr}J_\Phi=d$. Some sources call $J_\Phi/d$ the Choi state; its reference marginal is $I/d$. Other sources put the reference first, requiring the tensor order to be translated. Positivity and total trace alone do not impose trace preservation. Choi Matrix gives the reconstruction formula and examples.

For column-stacking of matrices,

$$\operatorname{vec}(AXB) =(B^{\mathsf T}\otimes A)\operatorname{vec}(X).$$

The transpose is not an adjoint. A row-stacking implementation changes the Kronecker order. In particular, the output-first Choi vector $\sum_{ai}K_{ai}|a\rangle_{\rm out}|i\rangle_{\rm ref}$ lists matrix entries in a different tensor order from ordinary column stacking. Define both index conventions explicitly before reshaping a Choi matrix or copying a matrix Liouvillian into code.

Noise spectra and stochastic records

For a stationary bath, the ordered two-sided angular-frequency spectrum is

$$S_{BB}(\omega)=\int_{-\infty}^{\infty} e^{i\omega t}\langle B(t)B(0)\rangle\,dt.$$

Subtract the mean of $B$ if the intended spectrum describes fluctuations alone. Ordered and symmetrized spectra are different quantum quantities. Frequency in hertz, one-sided spectra, and alternate Fourier signs require explicit conversion; see Noise Spectra.

Diffusive equations here use Itô Wiener increments with $\mathbb E[dW]=0$ and $dW^2=dt$. For counting records, $dN^2=dN$ and $\mathbb E[dN\mid\rho_c]=r_c\,dt$ to leading order, where $r_c$ is the conditional count rate. Efficiency $\eta$ lies between zero and one, but its operational meaning—collection, detection, or total monitoring efficiency—must be specified. These rules are developed in Stochastic Master Equations.

Heat and work signs

Unless a protocol says otherwise, heat $Q$ and work $W$ are positive into the system, so $\Delta U=Q+W$. For a closed two-point energy measurement protocol, $W=E_m^\tau-E_n^0$. Work extracted from the system has the opposite sign. At strong coupling, allocating interaction energy requires more than a sign choice; see Energy, Heat, and Work.

Exercises

Translate a dephasing coefficient. A paper writes $\dot\rho=c(Z\rho Z-\rho)$. What is the coherence decay rate?

Solution

The $01$ element of $Z\rho Z$ is $-\rho_{01}$, so $\dot\rho_{01}=-2c\rho_{01}$. The coherence decay rate is $2c$, and $c=\gamma_\phi/2$ in the table above.

A Choi trace is not enough. On a qubit let $K=\sqrt2|0\rangle\langle0|$ and $\Phi(\rho)=K\rho K^\dagger$. Its Choi operator is positive with trace two. Why is this not a channel?

Solution

$K^\dagger K=2|0\rangle\langle0|\ne I$. The map sends $|0\rangle\langle0|$ to an operator of trace two and $|1\rangle\langle1|$ to zero. Its Choi reference marginal is $2|0\rangle\langle0|$, not $I$. Complete positivity and the correct total Choi trace do not guarantee trace preservation.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Generators, rates, and bath correlations.
• Watrous, J. The Theory of Quantum Information. Cambridge University Press, 2018. Channel adjoints, Choi representations, and normalization. Author manuscript.
• Wiseman, H. M., and G. J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2010. Stochastic records and monitoring conventions.