Math Needed for Open Systems

An open quantum system exchanges information or energy with degrees of freedom that we do not observe in full. The mathematical starting point is a density operator and a rule for changing it. Begin with finite matrices; the extra conditions for infinite-dimensional operators can wait until you work with oscillators or continuous variables. The physics is developed in Measurement and Open Quantum Systems.

Density operators and subsystem states

First learn to use a positive trace-one matrix,

$$\rho\ge0,\qquad \operatorname{Tr}\rho=1,\qquad \langle A\rangle=\operatorname{Tr}(\rho A).$$

Density operators explains this representation. Tensor products and the partial trace then let you compute the subsystem state $\rho_S=\operatorname{Tr}_E\rho_{SE}$. A pure joint state can have mixed subsystem states, so a mixed density operator need not describe ignorance about a pre-existing pure state of that subsystem.

For measurement outcomes, add conditional probability. An outcome probability and the corresponding state change are different pieces of a measurement model; a POVM specifies the first without uniquely specifying the second.

Channels, differential equations, and noise

Goal	Useful mathematics	What to check
Apply a finite-dimensional channel	Adjoints, matrix products, positivity, and trace identities	The Kraus operators obey the trace-preserving sum rule
Model system–environment interaction	Tensor products, unitary evolution, and partial trace	The initial correlations and environment state match the model
Solve a master equation	ODEs, commutators, and matrix exponentials	The generator’s assumptions and the state’s trace and positivity
Simulate conditional trajectories	Conditional probability, random variables, and Monte Carlo	Enough independent samples and the correct outcome probabilities
Relate noise correlations to spectra	Fourier transforms and convolution	Transform conventions, stationarity, and units

A channel’s Kraus form is

$$\mathcal E(\rho)=\sum_\alpha M_\alpha\rho M_\alpha^\dagger,\qquad \sum_\alpha M_\alpha^\dagger M_\alpha=I.$$

This guarantees complete positivity and trace preservation. Positivity alone is weaker: complete positivity requires positivity even when an unchanged auxiliary system is included. The Kraus matrices are not unique, so one representation does not identify a unique microscopic noise mechanism.

For a time-independent finite-dimensional generator, $\dot\rho=\mathcal L(\rho)$ gives $\rho(t)=e^{t\mathcal L}\rho(0)$. An arbitrary matrix exponential on operators is not automatically a physical channel. The open-systems volume explains the physical assumptions and the conditions on generators. With time-dependent generators, a single exponential of the integrated generator generally fails when generators at different times do not commute.

A first dephasing calculation

Take $M_0=\sqrt{1-p}\,I$ and $M_1=\sqrt p\,\sigma_z$, with $0\le p\le1$. The completeness sum is $(1-p)I+pI=I$. Acting on

$$\rho=\begin{pmatrix}a&c\\c^*&1-a\end{pmatrix}$$

gives

$$\mathcal E(\rho)=\begin{pmatrix}a&(1-2p)c\\(1-2p)c^*&1-a\end{pmatrix}.$$

The populations are unchanged. At $p=1/2$, the off-diagonal entries vanish in this basis. At $p=1$, the channel is the deterministic unitary $\sigma_z$, so increasing this parameter all the way to one does not mean increasing irreversible decoherence.

Exercise: averaging and conditioning

Apply the channel above to $\lvert+\rangle\langle+\rvert$ at $p=1/2$. Compare the output when the random phase-flip label is forgotten with the two possible conditional outputs when it is retained.

Solution

The averaged state is $I_2/2$. Conditional on label $0$, the state is $\lvert+\rangle$; conditional on label $1$, it is $\lvert-\rangle$. Each label has probability $1/2$. Averaging the two projectors gives the same $I_2/2$, although each conditional state is pure.

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.