Density-Matrix and Open-System Formulas

These cards collect formulas for quantum states that are mixed, reduced, noisy, measured without a retained outcome, or coupled to unobserved degrees of freedom. The first question is always which mathematical object is being used:

• a density operator describes a state;
• an effect describes a measurement event;
• a partial trace discards a subsystem;
• a channel describes a finite state transformation;
• a master equation describes a time-local evolution model;
• entropy and purity summarize spectral properties of a state.

These roles are related but not interchangeable.

Choose a card

Need	Card	Core statement
Predict an observable or outcome	Density-Matrix Expectation Values	$\langle A\rangle=\operatorname{Tr}(\rho A)$ and $p(a)=\operatorname{Tr}(\rho E_a)$
Diagnose pure versus mixed	Purity	$\gamma(\rho)=\operatorname{Tr}(\rho^2)$
Obtain a subsystem state	Partial Trace	$\rho_A=\operatorname{Tr}_B\rho_{AB}$
Apply a finite quantum operation	Kraus Map	$\mathcal E(\rho)=\sum_\alpha K_\alpha\rho K_\alpha^\dagger$
Evolve a Markovian open system	Lindblad Equation	$\dot\rho=-(i/\hbar)[H,\rho]+\sum_\mu\gamma_\mu\mathcal D[L_\mu]\rho$
Quantify spectral uncertainty	Von Neumann Entropy	$S(\rho)=-\operatorname{Tr}(\rho\log\rho)$

State layer

A finite-dimensional density operator satisfies

$$\rho^\dagger=\rho, \qquad \rho\geq0, \qquad \operatorname{Tr}\rho=1.$$

The trace rule gives predictions:

$$\langle A\rangle_\rho = \operatorname{Tr}(\rho A),$$ $$p(a) = \operatorname{Tr}(\rho E_a).$$

Purity and entropy inspect the spectrum:

$$\gamma(\rho) = \operatorname{Tr}(\rho^2),$$ $$S(\rho) = - \operatorname{Tr} \left( \rho\log\rho \right).$$

In dimension $d$,

$$\frac1d \leq \gamma(\rho) \leq 1,$$

while, for logarithm base $b$,

$$0 \leq S_b(\rho) \leq \log_b d.$$

Purity one and entropy zero both characterize a pure state, but the two quantities rank mixed spectra differently in general and have different operational uses.

Subsystem layer

For a bipartite state $\rho_{AB}$,

$$\rho_A = \operatorname{Tr}_B\rho_{AB}$$

is the unique operator satisfying

$$\operatorname{Tr}_{AB} \left[ \rho_{AB} (A\otimes I_B) \right] = \operatorname{Tr}_A(\rho_AA)$$

for every local observable $A$.

Partial trace is a quantum channel. It preserves positivity and trace but can turn a globally pure entangled state into a mixed local state.

Reduced-state impurity diagnoses entanglement only when the global bipartite state is pure. For a globally mixed state, classical preparation uncertainty can also make the subsystem mixed.

Transformation layer

A Kraus representation has the form

$$\mathcal E(\rho) = \sum_\alpha K_\alpha\rho K_\alpha^\dagger.$$

For a trace-preserving channel,

$$\sum_\alpha K_\alpha^\dagger K_\alpha = I.$$

For a trace-nonincreasing measurement branch,

$$\sum_\alpha K_\alpha^\dagger K_\alpha \leq I.$$

Its output trace is the branch probability. A Kraus representation is not unique; the channel is the map, not one preferred list of operators.

Trace preservation should not be confused with unitality. The latter requires

$$\mathcal E(I)=I \quad\Longleftrightarrow\quad \sum_\alpha K_\alpha K_\alpha^\dagger=I,$$

with the operator order reversed. A channel may satisfy either condition without satisfying the other.

A Lindblad equation describes continuous Markovian evolution:

$$\frac{d\rho}{dt} = - \frac{i}{\hbar} [H,\rho] + \sum_\mu \gamma_\mu \left( L_\mu\rho L_\mu^\dagger - \frac12 \left\lbrace L_\mu^\dagger L_\mu, \rho \right\rbrace \right).$$

Here $\gamma_\mu\geq0$ in a diagonal canonical representation. One may absorb $\sqrt{\gamma_\mu}$ into $L_\mu$, which gives the equally common convention with no explicit rate. This form preserves trace and complete positivity under its semigroup assumptions. It is not the universal equation for every environment, strong-coupling regime, memory effect, or initial system–environment correlation.

A finite CPTP map need not be one member of a time-homogeneous Lindblad semigroup. The Kraus card answers how a specified operation acts; the Lindblad card answers which generator produces a continuous Markovian family.

Object and assumption map

Object	Required check	What it does not determine
$\rho$	Positive, trace one, correct Hilbert space	Its preparation ensemble or dynamics
$E_a$	$0\leq E_a\leq I$ and measurement completeness	Conditional state update
$\rho_A$	Tensor split and subsystem order specified	Joint correlations by itself
$\mathcal E$	Complete positivity and stated trace condition; check unitality separately	A unique environment, Kraus list, or Lindblad embedding
Lindblad generator	Markovian semigroup assumptions and a positive semidefinite rate matrix	General non-Markovian dynamics
$\gamma(\rho)$	Valid normalized state and dimension	Full spectrum or noise mechanism
$S(\rho)$	Logarithm base and finite value	Closeness to a chosen target

State comparison lives nearby

Purity describes one state’s spectral concentration. It is not a distance to a target. For two states, use quantities such as:

$$F(\rho,\sigma) = \left[ \operatorname{Tr} \sqrt{ \sqrt\rho\,\sigma\sqrt\rho } \right]^2,$$ $$D(\rho,\sigma) = \frac12 \lVert\rho-\sigma\rVert_1.$$

Fidelity uses the squared Uhlmann convention and measures state overlap. Trace Distance is a metric with a one-shot discrimination meaning.

A high-purity state can have zero fidelity with a desired pure target. A small trace distance to one target says nothing about closeness to another target.

Exact formulas and model assumptions

The state validity conditions, trace rule, partial trace, and finite-channel Kraus representation are structural parts of standard quantum theory.

Open-system equations add modeling assumptions:

• weak coupling may justify a Born approximation;
• short environmental memory may justify a Markov approximation;
• separation of frequencies may justify secularization;
• coarse graining fixes the temporal resolution;
• initial factorization excludes system–environment correlations;
• a chosen set of jump operators reflects a representation of the generator.

Do not infer these assumptions merely because a differential equation has Lindblad form. Conversely, a non-Markovian process can still define a completely positive channel between two selected times.

Finite and infinite dimensions

In finite dimension, matrices make every bounded trace expression well-defined. In infinite dimension:

• density operators must be positive trace-class operators;
• bounded-observable expectations are well defined;
• unbounded-observable moments require spectral integrability and domain conditions;
• entropy can be infinite even for a valid state;
• the identity is not trace class, so no uniform maximally mixed state exists on the full Hilbert space;
• Kraus sums and generators may require convergence and domain control.

Finite-dimensional intuition is useful, but cutoffs and truncations must be reported when they carry physical meaning.

Numerical checks

For a candidate state:

1. Hermitize within a documented tolerance.
2. Check the trace.
3. Diagonalize and inspect the smallest eigenvalue.
4. Record any physicality projection rather than silently clipping.
5. Verify expected symmetries and subsystem ordering.

For a channel:

1. Check input and output dimensions.
2. Verify

$$\sum_\alpha K_\alpha^\dagger K_\alpha = I$$

for trace preservation. 3. Test positivity of the Choi matrix when a general superoperator is given. 4. Compare trace, Hermiticity, and positivity on representative states.

For time evolution:

1. Monitor $\operatorname{Tr}\rho$.
2. Monitor the minimum eigenvalue within solver tolerance.
3. Compare conserved quantities and known steady states.
4. Check step-size, basis-cutoff, and integration-time convergence.
5. Separate physical purity change from numerical loss of positivity.

Common routing mistakes

• Using a state formula as if it specified dynamics.
• Treating trace one as sufficient without positivity.
• Reading diagonal entries as probabilities in every basis.
• Using a POVM effect as if it fixed state update.
• Forgetting the tensor identity in a local observable.
• Trying to reconstruct joint correlations from reduced states.
• Treating one Kraus representation as physically unique.
• Equating trace preservation with unitality.
• Assuming every finite CPTP map admits a time-homogeneous Lindblad generator.
• Calling every time-local master equation Markovian without qualification.
• Assuming every channel decreases purity.
• Comparing entropy values without a logarithm base.
• Using purity as target fidelity.
• Applying finite-dimensional lower bounds in an unbounded Hilbert space.

Canonical explanations

• Density Operators
• Trace Rule for Expectation Values
• Pure versus Mixed States
• Reduced Density Matrices
• Entropy Overview
• Quantum Operations
• Kraus Representation
• Choi Matrix
• Lindblad–GKSL Equation
• Lindblad Theorem

References

• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer, 1983.
• G. Lindblad, “On the generators of quantum dynamical semigroups,” Communications in Mathematical Physics 48, 119–130 (1976).
• V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of N-level systems,” Journal of Mathematical Physics 17, 821–825 (1976).
• 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.