Measurement and Open-System Formula Sheet

These formulas connect measurement probabilities, conditional states, channels, and continuous-time dynamics. Each group states the convention needed to use it; the linked lessons supply derivations and examples. Unless otherwise stated, Hilbert spaces are finite dimensional, $\rho\ge0$, and $\operatorname{Tr}\rho=1$.

Required background. Closed vs Open Quantum Systems, State-Update Rules, and Kraus Representation explain the objects summarized here.

Measurement probabilities and conditional states

A discrete POVM has effects $E_x\ge0$ with $\sum_xE_x=I$ and probabilities

$$p_x=\operatorname{Tr}(E_x\rho).$$

An instrument specifies more:

$$\begin{gathered} \mathcal I_x(\rho)=\sum_\alpha M_{x\alpha}\rho M_{x\alpha}^\dagger,\\ E_x=\sum_\alpha M_{x\alpha}^\dagger M_{x\alpha},\\ \rho_x=\frac{\mathcal I_x(\rho)}{p_x}\quad(p_x>0),\\ p_x=\operatorname{Tr}\mathcal I_x(\rho),\\ \Phi(\rho)=\sum_x\mathcal I_x(\rho). \end{gathered}$$

$\alpha$ labels alternatives not resolved by the recorded outcome $x$. The normalized conditional update is generally nonlinear in the input; the unnormalized operation and the average channel are linear. A zero-probability outcome has no conditional state defined by this formula.

For an orthogonal PVM $\{P_x\}$, the Lüders instrument is the specific choice $\mathcal I_x(\rho)=P_x\rho P_x$. Its unread channel is $\sum_xP_x\rho P_x$. A PVM alone, especially a degenerate one, does not force this instrument. See POVMs, Quantum Instruments, and Degenerate Measurements.

Channels, adjoints, and the Choi matrix

A Kraus channel and its observable adjoint are

$$\begin{gathered} \Phi(\rho)=\sum_rK_r\rho K_r^\dagger, \qquad\sum_rK_r^\dagger K_r=I_{\rm in},\\ \Phi^\dagger(A)=\sum_rK_r^\dagger A K_r,\\ \operatorname{Tr}[A\Phi(\rho)] =\operatorname{Tr}[\Phi^\dagger(A)\rho]. \end{gathered}$$

Thus trace preservation means $\Phi^\dagger(I_{\rm out})=I_{\rm in}$. Unitality instead means $\Phi(I_{\rm in})=I_{\rm out}$, or $\sum_rK_rK_r^\dagger=I_{\rm out}$; it is a separate condition. Two minimal Kraus sets for the same map differ by a unitary mixing of their labels. See Kraus Representation.

Use the output-first, unnormalized Choi convention:

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

For a linear map,

$$\begin{gathered} \text{complete positivity}\ \Longleftrightarrow\ J_\Phi\ge0,\\ \text{trace preservation}\ \Longleftrightarrow\ \operatorname{Tr}_{\rm out}J_\Phi=I_{\rm in},\\ \Phi(\rho)=\operatorname{Tr}_{\rm in} \bigl[J_\Phi(I_{\rm out}\otimes\rho^{\mathsf T})\bigr]. \end{gathered}$$

The transpose uses the same input basis as $J_\Phi$. For a channel, $J_\Phi/d_{\rm in}$ is a density operator whose input marginal is $I_{\rm in}/d_{\rm in}$. Total trace $\operatorname{Tr}J_\Phi=d_{\rm in}$ alone does not imply trace preservation.

Explicitly defining $|K\rangle\!\rangle=\sum_{a,i}K_{ai}|a\rangle_{\rm out}|i\rangle_{\rm in}$ gives $J_\Phi=\sum_r|K_r\rangle\!\rangle\langle\!\langle K_r|$. Do not assume a software vectorization has this tensor ordering; compare its index definition. See Choi Matrix.

Reduced dynamics and decoherence

For a fixed environment state $\rho_E$, independent of the chosen system input,

$$\Phi_t(\rho)=\operatorname{Tr}_E \bigl[U_t(\rho\otimes\rho_E)U_t^\dagger\bigr]$$

is a channel at every time. This does not imply the semigroup property or CP-divisible intermediate maps. Correlated initial preparations require their own preparation rule and domain; see Reduced Dynamics and Initial Correlations.

For orthonormal $|s_i\rangle$ and normalized environment vectors, a pure joint state gives

$$\begin{gathered} |\Psi\rangle=\sum_i c_i|s_i\rangle|e_i\rangle,\\ \rho_S=\sum_{i,j}c_ic_j^*\langle e_j|e_i\rangle |s_i\rangle\langle s_j|. \end{gathered}$$

The environment overlap controls reduced coherence. A mixed environment can also dephase a system without entangling with it; the displayed pure-state formula is not a universal entanglement criterion. See Environment-Induced Decoherence.

Generators and short-time operations

For constant $H$ and rate-absorbed operators $L_\mu$, the finite-dimensional GKLS generator is

$$\begin{gathered} \mathcal L(\rho)=-\frac{i}{\hbar}[H,\rho] +\sum_\mu\mathcal D[L_\mu]\rho,\\ \mathcal D[L]\rho=L\rho L^\dagger-\frac12\{L^\dagger L,\rho\}. \end{gathered}$$

Its observable adjoint is

$$\mathcal L^\dagger(A)={}\frac{i}{\hbar}[H,A] +\sum_\mu L_\mu^\dagger A L_\mu -\frac12\sum_\mu\{L_\mu^\dagger L_\mu,A\}.$$

The exact finite-time channel is $e^{t\mathcal L}$. A first-order jump construction uses

$$\begin{gathered} K_\mu=\sqrt{\Delta t}\,L_\mu,\\ K_0=I-\Delta t\left(\frac{iH}{\hbar} +\frac12\sum_\mu L_\mu^\dagger L_\mu\right). \end{gathered}$$

It reproduces the generator to first order, with $\sum_rK_r^\dagger K_r=I+O(\Delta t^2)$. These truncated operators are not an exactly trace-preserving finite-step algorithm. Likewise the Euler map $I+\Delta t\mathcal L$ need not be positive. See Lindblad–GKSL Equation, Quantum-Jump Trajectories, and Solving Lindblad Equations.

Relaxation rates and ordered noise

For constant qubit rates $\Gamma_\downarrow,\Gamma_\uparrow\ge0$ and pure dephasing $(\Gamma_\phi/2)\mathcal D[Z]$,

$$\begin{gathered} T_1^{-1}=\Gamma_\downarrow+\Gamma_\uparrow, \qquad T_2^{-1}=(2T_1)^{-1}+\Gamma_\phi,\\ p_e^*=\frac{\Gamma_\uparrow}{\Gamma_\downarrow+\Gamma_\uparrow} \quad(\Gamma_\downarrow+\Gamma_\uparrow>0). \end{gathered}$$

For a stationary Hermitian energy-valued $B$ and interaction $H_I=A\otimes B$ with dimensionless $A$, subtract the bath mean from $B$ and include $A\langle B\rangle$ in the system Hamiltonian. Then

$$\begin{gathered} S(\omega)=\int_{-\infty}^{\infty}e^{i\omega t} \langle B(t)B(0)\rangle\,dt,\\ \Gamma_\downarrow=|A_{ge}|^2S(+\omega_0)/\hbar^2,\\ \Gamma_\uparrow=|A_{ge}|^2S(-\omega_0)/\hbar^2. \end{gathered}$$

These weak-coupling rates use the ordered spectrum, not its symmetrized version. Equilibrium gives $S(-\omega)=e^{-\beta\hbar\omega}S(+\omega)$. For response to $H'=-f(t)B$, define $\chi^R(t)=(i/\hbar)\theta(t)\langle[B(t),B(0)]\rangle$ with the same Fourier sign. At nonzero frequency,

$$S_{\rm sym}(\omega)=\hbar\coth\!\left(\frac{\beta\hbar\omega}{2}\right) \operatorname{Im}\chi^R(\omega).$$

The regular part may have a zero-frequency limit, but an elastic contribution $2\pi C_0\delta(\omega)$ from conserved or nondecaying fluctuations must be specified separately. Centering $B$ removes its mean, not necessarily $C_0$. That weight cancels from the commutator and cannot be recovered by multiplying a zero response by a divergent thermal factor. See Fluctuation–Dissipation Relation for the scope and the general theorem for the static-response distinction.

See Common Master Equations, Common Lindblad Operators, Common Channels, and Common Noise Spectra for model-specific formulas.

Exercises

Noise with zero dissipative response. An isolated qubit has $H=\Delta|e\rangle\langle e|$, with gap $\Delta>0$, and is prepared in its Gibbs state at $0<T<\infty$. For the centered energy $B=H-\langle H\rangle$, find $C(t)$, $S_{\rm sym}(\omega)$, and $\chi^R(t)$. Does a vanishing commutator imply a vanishing energy variance?

Solution

Since $[H,B]=0$, $B(t)=B$. With $p_e=(1+e^{\beta\Delta})^{-1}$ and $p_g=1-p_e$,

$$\begin{gathered} C(t)=\operatorname{Var}(H) =\Delta^2p_ep_g =\frac{\Delta^2}{4\cosh^2(\beta\Delta/2)},\\ S_{\rm sym}(\omega)=2\pi\Delta^2p_ep_g\,\delta(\omega), \qquad \chi^R(t)=0. \end{gathered}$$

The Gibbs ensemble has energy fluctuations even though each realization conserves its energy. A drive proportional to $B$ cannot change the populations of this isolated qubit, so its dynamical response vanishes. Allowing the populations to reequilibrate with a reservoir is a different experiment. This zero-frequency line is not a finite white-noise level to insert into a Markovian rate formula.

A first-order trace error. Set $H=0$ and $L=\sqrt\Gamma|g\rangle\langle e|$. Compute $K_0^\dagger K_0+K_1^\dagger K_1$ for the truncated jump construction.

Solution

It is $I+(\Gamma\Delta t)^2|e\rangle\langle e|/4$. The first-order cancellation works, but exact normalization needs a better finite-step construction. Renormalizing after each step changes the numerical method and must itself be checked.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Reduced dynamics and generators.
• Clerk, A. A., et al. “Introduction to quantum noise, measurement, and amplification.” Reviews of Modern Physics 82, 1155–1208 (2010). doi:10.1103/RevModPhys.82.1155.
• Watrous, J. The Theory of Quantum Information. Cambridge University Press, 2018. Chapter 2, operations and Choi representations. Author manuscript.
• Wiseman, H. M., and G. J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2010. Instruments and conditional evolution.