Measurement and Open-System Glossary

This glossary distinguishes the mathematical objects and physical processes used in measurement and open-system theory. Entries are alphabetical, with links to worked explanations. Use the Formula Sheet for equations and One Interaction, Different Measurement Records for one experiment connecting the terms.

Required background. Closed vs Open Quantum Systems introduces reduced states and the choice of a system boundary.

Backaction

Changes in the system’s state or future statistics associated with coupling to a measuring device. Backaction includes both the conditional change given a record and the disturbance visible when records are averaged. It need not be an apparatus error, and acquiring little information does not guarantee little disturbance. An eigenstate can also be unchanged by a particular sharp measurement. See Measurement Backaction.

Bath

An environment described through statistical properties such as a reference state and correlation functions. In many approximations it is large enough to remain close to that reference state during the experiment. “Bath” does not imply thermal equilibrium, short memory, or weak coupling. See Baths, Reservoirs, and Environments.

Channel

A completely positive, trace-preserving linear map from input operators to output operators. It specifies a deterministic state transformation. Channels can describe discarded records, but an enlarged output can also retain a classical record. A normalized update conditioned on one outcome is generally nonlinear and is not itself a channel on arbitrary inputs. See Quantum Operations and Common Channels.

Choi matrix

An operator representing a linear map through its action on one half of a maximally entangled vector. With $|\Omega\rangle=\sum_i|i\rangle|i\rangle$, use $J_\Phi=(\Phi\otimes\mathrm{id})(|\Omega\rangle\langle\Omega|)$, output factor first. Complete positivity means $J_\Phi\ge0$; trace preservation means $\operatorname{Tr}_{\rm out}J_\Phi=I_{\rm in}$. A normalized entangled vector changes the dimension factors. See Choi Matrix.

Complete positivity

A linear map is completely positive if $\Phi\otimes\mathrm{id}_n$ maps positive operators to positive operators for every untouched reference dimension $n$. The reference may be entangled with the input. Positivity on isolated system states is weaker: matrix transposition is positive but not completely positive. See Completely Positive Maps.

Continuous measurement

Gradual acquisition of a time-resolved record, such as photon counts or a homodyne current. The state conditioned on the available record is stochastic; averaging it gives the unconditional state for the same physical experiment. Weak coupling over each short step can produce strong accumulated information and disturbance. See Continuous Monitoring and Measurement Records.

Decoherence

Suppression of accessible interference in a specified reduced description, often through interactions with unobserved degrees of freedom. It can be partial, basis dependent, and reversible in a larger controlled system. Pure-system–environment examples use entanglement, but mixed environments can also dephase a system without becoming entangled with it. Reduced decoherence alone does not select a unique measurement outcome. See What Is Decoherence?.

Dephasing

Suppression of phase coherence in a specified basis. Pure dephasing preserves populations in that basis while changing off-diagonal entries. It preserves energy populations when that basis is the energy basis, a qualification that matters when the Hamiltonian and monitored observable do not commute. See Dephasing vs Dissipation and Dephasing Channel.

Dissipation

Relaxation or damping associated with exchange between a system and its surroundings, often energy relaxation. A thermal stationary state can have zero net energy current even while upward and downward transitions continue. The mathematical term “dissipator” is broader: $\mathcal D[Z]$ can describe pure dephasing without energy exchange for a Hamiltonian diagonal in $Z$. See Dephasing vs Dissipation.

Dynamical map

A map or family of maps propagating a state from a specified preparation time. For a fixed uncorrelated environment, each initial-to-final reduced map is a channel. Intermediate propagators need not be channels, and correlated preparations can restrict the allowed input domain. With preparation time zero, a time-homogeneous semigroup obeys $\Phi_{t+s}=\Phi_t\circ\Phi_s$ and $\Phi_0=\mathrm{id}$. See Reduced Dynamics and Quantum Dynamical Semigroups.

Environment

Degrees of freedom omitted from the retained system description: a radiation field, substrate, detector, auxiliary qubit, or another subsystem. The boundary is a modeling choice. Retaining a strongly coupled mode as part of the system can make a previously memory-dependent reduced description simpler. See System–Bath Hamiltonians.

Instrument

An outcome-resolved collection of completely positive operations $\mathcal I_x$ whose sum is trace preserving. It gives $p_x=\operatorname{Tr}\mathcal I_x(\rho)$ and, for $p_x>0$, the conditional state $\mathcal I_x(\rho)/p_x$. Its associated POVM fixes probabilities, but several instruments can have the same POVM and different state changes. See Quantum Instruments.

Kraus operator

One member of an operator-sum representation $\Phi(\rho)=\sum_rK_r\rho K_r^\dagger$. For a channel, $\sum_rK_r^\dagger K_r=I$. An individual operator is not automatically an observed outcome: an outcome may group several Kraus terms, and the representation is nonunique. See Kraus Operators and Kraus Representation.

Lindblad operator

An operator in a GKLS generator,

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

Here rates are absorbed into $L_\mu$. Their normalization and the chosen monitoring scheme determine how to interpret decay rates or jumps. Different operator lists can give the same generator. See Lindblad Operators and Common Lindblad Operators.

Markov approximation

A short-memory approximation replacing a history-dependent bath influence with a local evolution on the timescale being resolved. In weak-coupling derivations, the bath correlation time is compared with the slow reduced-state evolution in the relevant interaction picture. Born factorization and secular frequency averaging are separate approximations. Short memory alone does not ensure a completely positive generator. See Markov Approximation.

Nonselective measurement

A measurement description in which outcomes are averaged: $\rho\mapsto\sum_x\mathcal I_x(\rho)$. The interaction can disturb the system even when no observer reads its outcome. Merely forgetting a stored record does not apply an additional physical disturbance; it changes which conditional prediction is available. See Selective and Nonselective Measurements.

POVM

A positive operator-valued measure, represented for discrete outcomes by $E_x\ge0$, $\sum_xE_x=I$. It gives $p_x=\operatorname{Tr}(\rho E_x)$. It describes outcome statistics, including noise and inefficiency, while the instrument determines state changes. A PVM is the special case of mutually orthogonal projectors. See POVMs.

Pointer state

A state or approximate family that remains relatively robust under the relevant environment coupling and system dynamics. Robust states can move, as a damped coherent state does; they need not be stationary energy eigenstates or an exact orthogonal basis. Diagonalizing one reduced density matrix does not by itself identify pointer states. See Pointer States.

Quantum trajectory

A stochastic sequence of conditional states for a specified record, or a simulation representation whose average reproduces an unconditional evolution. Photon counting commonly gives jumps; homodyne detection gives diffusive records. It is not a unique detector-independent path possessed by the system. See Quantum-Jump Trajectories and Diffusive Trajectories.

Reservoir

An environment idealized as a stable source or sink for quantities such as energy or particles. Temperature and chemical potential can characterize equilibrium reservoirs; engineered reservoirs require a different specification. The approximation that these properties remain fixed can fail for a small or strongly disturbed environment. See Baths, Reservoirs, and Environments.

Selective measurement

Conditioning on a known measurement outcome and normalizing the corresponding operation, provided its probability is nonzero. It describes a selected subensemble or the prediction available to an observer with that record. Comparing it with an unconditional state compares different available information. See State-Update Rules.

Stinespring dilation

For a channel, a representation $\Phi(\rho)=\operatorname{Tr}_E(V\rho V^\dagger)$ with an isometry $V^\dagger V=I$. A general completely positive map has a similar operator representation, but $V$ need not be an isometry unless the map is trace preserving. Minimal channel dilations have a precise equivalence under transformations of the auxiliary space; a dilation does not identify a unique actual environment. See Stinespring Dilation.

Unraveling

A stochastic representation of an unconditional master equation. A physical monitoring scheme can realize a particular unraveling, but an algorithmic unraveling need not correspond to the detector used in an experiment. The same averaged state can arise from different record statistics and conditional states. See Unravelings.

Weak measurement

A measurement regime with weak coupling or little information acquired in a typical elementary step, defined through its interaction and instrument. It does not guarantee a small conditional change for every rare postselected outcome, and many weak steps can have a large total effect. Weak values and weak disturbance are related through specific assumptions, not interchangeable definitions. See Weak Measurements and Weak Values.

Exercises

Probabilities or state changes? Two detectors have the same POVM. Must they have the same instrument or averaged channel?

Solution

No. They give the same outcome probabilities for every input, but can apply different conditional operations. Their averaged channels can consequently differ. One Interaction, Different Measurement Records also gives the converse kind of example: distinct instruments sharing one averaged channel.

One equation, two records. A photon-counting simulation and a homodyne simulation average to the same density matrix. Does one record invalidate the other?

Solution

No. They can be different unravelings of the same unconditional equation. Which record describes an experiment depends on its monitoring apparatus. Agreement of ensemble averages is an essential consistency check, but does not make record statistics identical.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Dynamical maps, baths, and generators.
• Schlosshauer, M. Decoherence and the Quantum-to-Classical Transition. Springer, 2007. Decoherence and pointer states.
• Watrous, J. The Theory of Quantum Information. Cambridge University Press, 2018. Channels, measurements, and dilation theorems. Author manuscript.
• Wiseman, H. M., and G. J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2010. Instruments, trajectories, and monitoring.