Glossary

This glossary defines the terms that recur across measurement theory, decoherence, quantum channels, and open-system master equations. It is an orientation page: it gives compact definitions, common warnings, and links to the canonical pages where derivations and examples live.

For equations in one place, see the Formula Sheet. For assumption checks in weak-coupling master equations, see the Approximation Checklist.

Backaction

Backaction is the change in a system’s state or future statistics caused by acquiring information about it or coupling it to a detector. In quantum mechanics it is not merely a technical imperfection: when different possible measurement records become correlated with different system states, conditional and unconditional state assignments generally change.

Backaction can be unitary, dissipative, dephasing, or state-selective depending on the measurement model. A sharp projective measurement has strong backaction in the measured basis; a weak measurement produces small conditional changes per step but can accumulate large effects over many repetitions. See Measurement Backaction and State-Update Rules.

Common mistake: treating backaction as synonymous with “error.” A measurement can be ideal and still have unavoidable backaction because it creates correlations that were not present before the measurement.

Bath

A bath is an idealized collection of degrees of freedom whose detailed state is not tracked, but whose statistical properties influence a system. In weak-coupling open-system theory one often specifies a bath by its Hamiltonian, stationary state, and correlation functions. The system–bath split is written schematically as

$$H = H_S+H_B+H_I.$$

A bath is usually large enough that the system does not noticeably change its macroscopic properties. Thermal baths, squeezed baths, structured electromagnetic environments, phonon baths, and engineered reservoirs are common examples. See Baths, Reservoirs, and Environments and System–Bath Hamiltonians.

Common mistake: assuming “bath” always means memoryless or thermal. A bath may be finite, structured, nonthermal, or strongly correlated, in which case Markovian master equations may fail.

Channel

A quantum channel is a completely positive trace-preserving linear map from input states to output states:

$$\rho \mapsto \Phi(\rho).$$

Channels describe finite-time state transformations when the measurement outcome, environment record, or microscopic degrees of freedom are not retained. Equivalently, in finite dimensions a channel can be represented by Kraus operators,

$$\Phi(\rho) = \sum_r K_r\rho K_r^\dagger, \qquad \sum_r K_r^\dagger K_r=I.$$

The canonical pages are Completely Positive Maps, Kraus Representation, and Common Noise Channels.

Common mistake: calling every reduced dynamical equation a channel. A channel is a map between states. A generator, stochastic trajectory, or outcome-resolved instrument contains more structure.

Choi Matrix

The Choi matrix is a state-like operator that represents a linear map by applying it to one half of a maximally entangled reference:

$$J_\Phi = (\Phi\otimes \operatorname{id}) \left( \lvert\Omega\rangle\langle\Omega\rvert \right).$$

In finite dimensions, $\Phi$ is completely positive exactly when $J_\Phi\ge 0$, with normalization conditions encoding trace preservation or unitality. This makes the Choi matrix a practical test object for quantum channels, process tomography, and semidefinite optimization. See Choi Matrix.

Common mistake: forgetting the convention. Some authors use normalized $\lvert\Omega\rangle$, others use unnormalized $\lvert\Omega\rangle$; traces and reconstruction formulas then differ by a dimension factor.

Complete Positivity

Complete positivity means a map remains positive after adjoining an arbitrary untouched ancilla:

$$X\ge 0 \quad\Longrightarrow\quad (\Phi\otimes \operatorname{id}_n)(X)\ge 0 \quad \text{for every } n.$$

This is stronger than ordinary positivity. The extra requirement matters because a system may be entangled with degrees of freedom outside the modeled process. Physical finite-time channels are therefore completely positive and trace preserving on the states for which they are meant to act. See Completely Positive Maps.

Common mistake: proving that $\Phi(\rho)$ is positive for isolated inputs and concluding that $\Phi$ is a valid channel on entangled inputs. Ordinary positivity is not enough.

Continuous Measurement

Continuous measurement is a time-resolved measurement in which information is acquired gradually and recorded as a stochastic signal. Instead of a single outcome $x$, one has a record over time, such as photon counts, homodyne currents, or repeated weak readouts. The conditioned state obeys a stochastic update, often written schematically as

$$d\rho_c(t) = \mathcal L[\rho_c(t)]\,dt + \text{record-dependent term}.$$

The average over records recovers the unconditional master equation when the monitoring model is consistent:

$$\mathbb E_{\text{records}}[\rho_c(t)] = \rho(t).$$

See Continuous Monitoring, Measurement Records, Stochastic Master Equations, Quantum-Jump Trajectories, and Diffusive Trajectories.

Common mistake: thinking “continuous” means nondisturbing. Continuous monitoring is often weak per unit time, but the accumulated record can strongly condition the state.

Decoherence

Decoherence is the loss of phase coherence between components of a quantum state due to correlations with unobserved degrees of freedom. In a reduced density matrix, it appears as suppression of off-diagonal terms in a preferred basis or approximate pointer structure:

$$\rho_{ij}(t) \to 0 \qquad (i\ne j)$$

in that basis, while the total system-plus-environment state may still evolve unitarily. See What Is Decoherence? and Pointer States.

Common mistake: saying decoherence by itself selects one outcome. Decoherence explains why interference becomes inaccessible in the reduced description, but it is not identical to a literal collapse postulate.

Dephasing

Dephasing is loss of relative phase coherence without necessarily changing populations in the relevant basis. For a two-level system in the energy basis, pure dephasing leaves $\rho_{00}$ and $\rho_{11}$ fixed while reducing $\rho_{01}$ and $\rho_{10}$.

Dephasing can arise from classical noise, entanglement with an environment, random frequency shifts, or continuous measurement of a compatible observable. See Dephasing vs Dissipation and Dephasing Channel.

Common mistake: using dephasing and decoherence as exact synonyms. Dephasing is a common mechanism or manifestation of decoherence, but decoherence can be broader and basis-dependent.

Dissipation

Dissipation is irreversible exchange of energy, particles, or other conserved quantities between the system and its surroundings. In master equations it is often represented by relaxation terms, damping rates, or jump operators. A simple amplitude-damping channel models decay from an excited state to a lower state.

Dissipation and dephasing often occur together, but they are conceptually distinct. A bath can randomize phase without energy relaxation, and a dissipative process can also introduce noise required by fluctuation-dissipation constraints. See Dephasing vs Dissipation and Amplitude-Damping Channel.

Common mistake: calling every nonunitary term “dissipation.” Some nonunitary terms represent measurement conditioning, coarse graining, dephasing, leakage, or mathematical approximations rather than energy loss.

Dynamical Map

A dynamical map is a family of maps that propagates a reduced state between times:

$$\rho_S(t) = \Phi_{t,t_0}\!\left[\rho_S(t_0)\right].$$

For a closed system, $\Phi_{t,t_0}$ is unitary conjugation. For an open system, it may be a channel, a map on a restricted preparation domain, or an approximate object derived from assumptions about the initial system-environment state.

If $\Phi_{t+s,t_0}=\Phi_{t,t_0}\Phi_{s,t_0}$ with time-translation invariance and complete positivity, one has the special case of a quantum dynamical semigroup. See Reduced Dynamics and Quantum Dynamical Semigroups.

Common mistake: assuming every open-system dynamical map has a time-local Lindblad–GKSL generator. Memory, initial correlations, or restricted domains can obstruct that conclusion.

Environment

The environment is everything outside the degrees of freedom retained as the system. It may be a physical surrounding, a detector, a substrate, a radiation field, an engineered ancilla, or simply the variables traced out in a model.

“Environment” is the broadest term in this glossary. A bath is usually an environment with idealized statistical properties. A reservoir is usually a bath that can absorb or supply conserved quantities while maintaining stable macroscopic parameters.

Common mistake: treating the system-environment split as unique. It is a modeling choice, and moving degrees of freedom across the boundary can change whether a process appears unitary, dissipative, Markovian, or non-Markovian.

Instrument

A quantum instrument is an outcome-resolved collection of completely positive maps $\{\mathcal I_x\}$ that gives both the outcome probabilities and the post-measurement states:

$$p(x) = \operatorname{Tr}\mathcal I_x(\rho), \qquad \rho_x = \frac{\mathcal I_x(\rho)}{p(x)}.$$

The sum $\sum_x\mathcal I_x$ is trace preserving when all outcomes are included. Instruments are the right language when probabilities and state updates must be specified together. See Quantum Instruments.

Common mistake: assuming a POVM determines the state update. A POVM gives outcome probabilities; an instrument gives the operations that implement those outcomes.

Kraus Operator

A Kraus operator is one operator in an operator-sum representation of a completely positive map:

$$\Phi(\rho) = \sum_r K_r\rho K_r^\dagger.$$

For a trace-preserving channel, the Kraus operators satisfy $\sum_rK_r^\dagger K_r=I$. For an outcome-resolved operation, the same form may be trace nonincreasing. Individual Kraus operators often acquire physical meaning only after specifying a detector record, environment basis, or measurement implementation. See Kraus Operators and Kraus Representation.

Common mistake: treating a particular Kraus list as unique. Kraus representations are nonunique; unitary rotations among Kraus operators can represent the same channel.

Lindblad Operator

A Lindblad operator is an operator that appears in a Lindblad–GKSL generator. With rates absorbed into $L_\mu$ or written separately as $\gamma_\mu$, a common form is

$$\frac{d\rho}{dt} = -i[H,\rho] + \sum_\mu \gamma_\mu \left( L_\mu\rho L_\mu^\dagger - \frac{1}{2}\{L_\mu^\dagger L_\mu,\rho\} \right).$$

Depending on context, a Lindblad operator may represent spontaneous emission, dephasing noise, particle loss, measurement backaction, or an effective coarse-grained process. See Lindblad–GKSL Equation and Lindblad Operators.

Common mistake: reading a Lindblad operator as a unique physical event. Different unravelings and operator bases can yield the same unconditional master equation.

Markov Approximation

The Markov approximation replaces environment memory by an effectively local-in-time influence on the system. In weak-coupling derivations, it is usually justified when bath correlations decay on a time scale much shorter than the system’s coarse-grained evolution time.

Markov approximation is not a single algebraic step. It may involve extending memory integrals, replacing $\rho_S(s)$ by $\rho_S(t)$ inside a kernel, choosing a stationary bath state, and sometimes combining with secular averaging. See Markov Approximation, Born Approximation, and Secular Approximation.

Common mistake: equating “Markovian” with “Lindblad form” without checking the domain, positivity, time dependence, and approximation history.

Nonselective Measurement

A nonselective measurement is a measurement whose physical interaction occurs but whose outcome is ignored or averaged over. If an instrument has operations $\mathcal I_x$, the nonselective state update is

$$\rho \mapsto \sum_x \mathcal I_x(\rho).$$

For projective measurements this usually removes coherences between the measured eigenspaces. See Selective and Nonselective Measurements and Projective Measurements.

Common mistake: confusing “unobserved” with “not performed.” A nonselective measurement can disturb the state even when the outcome is discarded.

POVM

A positive operator-valued measure, or POVM, is a collection of positive effects $\{E_x\}$ that sum to the identity:

$$E_x\ge 0, \qquad \sum_x E_x=I.$$

It gives measurement probabilities by

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

POVMs describe noisy, coarse-grained, inefficient, unsharp, and generalized measurements at the probability level. See POVMs.

Common mistake: assigning a unique collapse rule to a POVM effect. The same POVM can be implemented by different instruments with different backaction.

Pointer State

A pointer state is a system state or approximate family of states that remains relatively stable under monitoring by an environment or apparatus. Pointer states are selected by the interaction structure, not by a purely abstract preference for one basis.

In decoherence theory, superpositions of distinct pointer states rapidly become entangled with distinguishable environment states, suppressing interference in the reduced density matrix. See Pointer States.

Common mistake: assuming the pointer basis is always the energy basis or always the position basis. It depends on the dominant system-environment coupling and on the relevant time scales.

Quantum Trajectory

A quantum trajectory is a conditioned state evolution associated with a measurement record or with an unraveling of an unconditional master equation. Photon-counting records lead naturally to jump trajectories; homodyne or heterodyne records lead naturally to diffusive trajectories.

The ensemble average over trajectories recovers the unconditional state when the unraveling is consistent:

$$\rho(t) = \mathbb E[\rho_c(t)].$$

See Quantum-Jump Trajectories, Diffusive Trajectories, and Quantum Jump Simulation.

Common mistake: treating a trajectory as a detector-independent hidden path. It is a conditioned description tied to a record, monitoring scheme, or simulation representation.

Reservoir

A reservoir is an environment or bath treated as a large source or sink for energy, particles, phase coherence, angular momentum, or another quantity. In thermodynamics and transport, a reservoir is often characterized by stable parameters such as temperature, chemical potential, density of states, and spectral response.

A reservoir is typically assumed to remain close to a reference state despite exchange with the system. That assumption can fail for small environments, strongly driven environments, or feedback-controlled environments.

Common mistake: using reservoir language without specifying what is conserved or exchanged. For heat, work, and entropy production, the physical identification of reservoir exchange is part of the model.

Selective Measurement

A selective measurement is a measurement in which the outcome is known and the state is conditioned on that outcome. For an operation $\mathcal I_x$, the conditional state is

$$\rho_x = \frac{\mathcal I_x(\rho)} {\operatorname{Tr}\mathcal I_x(\rho)}.$$

Selective measurement is therefore a statement about a subensemble or conditioned observer state, not merely about the apparatus coupling. See Selective and Nonselective Measurements.

Common mistake: comparing a selective state to an unconditional master-equation solution as if they were predictions for the same data set.

Stinespring Dilation

Stinespring dilation represents a completely positive map as an isometry into a larger Hilbert space followed, for channels, by discarding an environment:

$$\Phi(\rho) = \operatorname{Tr}_E \left( V\rho V^\dagger \right), \qquad V^\dagger V=I.$$

In finite-dimensional physics language, this often becomes “attach an environment, evolve unitarily, then trace out the environment.” See Stinespring Dilation.

Common mistake: thinking the dilation is unique. It is unique only up to appropriate isometries under minimality conditions; different dilations can encode different physical implementations of the same channel.

Unraveling

An unraveling is a representation of an unconditional open-system evolution as an ensemble of conditioned stochastic evolutions. The same Lindblad master equation can have photon-counting, homodyne, heterodyne, or other unravelings depending on how the environment is monitored.

Unravelings are extremely useful for simulations and for interpreting measurement records, but they are not additional dynamics unless a monitoring scheme has been specified. See Stochastic Master Equations.

Common mistake: assuming the master equation uniquely determines the observed record. The master equation determines the averaged state; the record statistics require a measurement model.

Weak Measurement

A weak measurement obtains partial information in a single trial and produces a correspondingly small conditional disturbance in that trial. Repeated weak measurements, or a continuous limit of weak measurements, can still reveal strong information and produce substantial cumulative backaction.

Weak measurement is best defined operationally through an instrument or measurement model, not just by saying that a coupling constant is small. The meaning depends on the signal-to-noise ratio, the number of repetitions, and what postselection or feedback is performed. See Measurement Backaction and Quantum Instruments.

Common mistake: equating weak measurement with negligible disturbance. Weak per step does not mean disturbance-free over the full record.

Common Confusions

Bath, environment, reservoir. Environment is the broadest term. Bath emphasizes statistical influence on the system. Reservoir emphasizes large, stable capacity to exchange a conserved quantity.

Decoherence, dephasing, dissipation. Decoherence is loss of accessible interference through correlations and coarse graining. Dephasing is loss of phase coherence in a specified basis. Dissipation is exchange of conserved quantities, usually including energy relaxation or damping.

POVM, instrument, channel. A POVM gives probabilities. An instrument gives outcome-conditioned operations. A channel is the outcome-averaged state transformation when all records are ignored.

Trajectory, unraveling, master equation. A trajectory is conditioned on a record. An unraveling is a way to generate such trajectories for a given unconditional equation. A master equation describes the averaged state.

Markov approximation, complete positivity, Lindblad–GKSL form. A short-memory approximation does not automatically guarantee complete positivity. Lindblad–GKSL form is the standard structure for completely positive trace-preserving semigroups and many time-local Markovian models, but it must come from stated assumptions.

Self-Checks

1. A detector interacts with a qubit in the $\sigma_z$ basis, but the readout is lost. Is the resulting state update selective or nonselective?

Solution

It is nonselective. The system has interacted with the detector, so backaction may remain, but the observer averages over the possible outcomes. For an instrument $\{\mathcal I_x\}$ the update is $\rho\mapsto\sum_x\mathcal I_x(\rho)$.

2. A map is positive on every isolated qubit density matrix. Is that enough to call it a physical channel on one half of an entangled pair?

Solution

No. A physical channel must be completely positive on the stated input domain. Positivity on isolated states does not guarantee positivity after tensoring with an untouched reference system.

3. Two simulations use different jump operators but average to the same Lindblad master equation. Must they represent different unconditional dynamics?

Solution

Not necessarily. They may be different unravelings of the same unconditional dynamics. To assign operational meaning to the jumps, specify the monitoring scheme or detector record.

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, Cambridge University Press, 2010.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.
• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press, 2014.
• H. Carmichael, An Open Systems Approach to Quantum Optics, Springer, 1993.
• W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Reviews of Modern Physics 75, 715, 2003.
• V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of N-level systems,” Journal of Mathematical Physics 17, 821, 1976.
• G. Lindblad, “On the generators of quantum dynamical semigroups,” Communications in Mathematical Physics 48, 119, 1976.