Absolute Output Coherence

• Absolute output coherence is a measure that evaluates coherence based on a preset output representation, making it inherently basis, frame, or protocol dependent.
• It distinguishes between intrinsic state properties and output-conditioned descriptors by assessing absolute values, variances, or off-diagonal terms in specific decompositions.
• Applications span areas such as three-dimensional polarization, networked control systems, quantum resource theories, and thermodynamic protocols, each employing tailored methodologies and closed-form metrics.

“Absolute output coherence” is an Editor’s term rather than a uniform technical term. Across optics, networked control, and quantum information, it denotes quantities that assess coherence of specified outputs, or absolute-valued and unnormalized coherence attached to a chosen output decomposition, rather than a basis-free invariant of the underlying state or dynamics. In three-dimensional polarization theory it refers to the absolute pairwise degrees of mutual coherence between Cartesian field components and is explicitly frame dependent (Gil, 2020). In higher-order consensus with leaders it denotes the steady-state variance of the absolute output $y=x_1$ relative to a fixed desired trajectory (Mackin et al., 2019). In quantum resource theories and output-statistics formalisms it is realized through basis-dependent state coherence, absolute differences between output distributions, or dephasing losses of output states (Baumgratz et al., 2013, Sun et al., 2017).

1. Terminological scope and recurring structure

The literature does not supply a single canonical definition of absolute output coherence. Several of the relevant papers explicitly do not use the exact phrase, but provide the closest formal counterparts in their own domains: basis-fixed output-state coherence, absolute-valued off-diagonal coherence, absolute output variance, or unnormalized coherence functions of emitted fields (Wang et al., 2021, Singh et al., 2016, Daniel et al., 9 Sep 2025). The common pattern is that coherence is evaluated after specifying an output representation: a Cartesian field decomposition, a network output map, a reference basis, a measurement protocol, or an output field operator.

A second recurring feature is the tension between intrinsic descriptors and relative descriptors. In the three-dimensional polarization problem, the absolute values $|\mu_{12}|,|\mu_{13}|,|\mu_{23}|$ are not invariants of the polarization state; they vary under physically admissible rotations of the laboratory frame (Gil, 2020). In the quantum-coherence literature, coherence is defined relative to a fixed basis, so any output coherence $C(\Phi(\rho))$ inherits that basis dependence (Baumgratz et al., 2013). In output-statistics approaches, the quantity of interest is often the maximal absolute deviation between the distributions produced by a state and by its decohered counterpart, again relative to a preferred observable (Sun et al., 2017).

A common misconception is that the adjective “absolute” signals observer independence. In much of the relevant literature it means something narrower: absolute output amplitude, absolute value of an off-diagonal term, absolute variance of a non-relative output, or absolute difference between two coherence values. Taken together, these results suggest that absolute output coherence is best understood as a family of output-conditioned coherence descriptors whose meaning is fixed by a chosen frame, basis, monitoring scheme, or output channel.

2. Three-dimensional polarization and frame-dependent Cartesian output coherence

The most literal optical treatment is the analysis of a statistically stationary three-dimensional electromagnetic field at a fixed point $\mathbf r$, represented by the analytic-signal vector

$$\boldsymbol{\varepsilon}(t)= \begin{pmatrix} \varepsilon_1(t)\ \varepsilon_2(t)\ \varepsilon_3(t) \end{pmatrix},$$

with coherency matrix

$$\mathbf R=\left\langle \boldsymbol{\varepsilon}(t)\otimes \boldsymbol{\varepsilon}^\dagger(t)\right\rangle, \qquad R_{ij}=\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle.$$

The pairwise degrees of mutual coherence between Cartesian components are

$$\mu_{ij}= \frac{\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle} {\sqrt{\langle |\varepsilon_i(t)|^2\rangle\langle |\varepsilon_j(t)|^2\rangle}}, \qquad i\neq j,$$

so that

$|\mu_{ij}|=\frac{|R_{ij}|}{\sqrt{R_{ii}R_{jj}}}.$

The central result is that these absolute pairwise coherences are relative quantities: they depend on the laboratory axes, because both diagonal intensities and off-diagonal correlations change under real orthogonal rotations of the frame (Gil, 2020).

This dependence is structurally distinct from unitary diagonalization. Under a general unitary similarity transform $$\mathbf R\mapsto \mathbf U\mathbf R\mathbf U^\dagger$$, the coherency matrix can be diagonalized and all $\mu_{ij}$ vanish, but this generally changes the physical polarization structure rather than merely re-expressing the same state. The physically relevant transformation law is instead

$|\mu_{12}|,|\mu_{13}|,|\mu_{23}|$0

which corresponds to a rotation of the laboratory frame for the same three-dimensional state (Gil, 2020).

The minima of the absolute pairwise coherences occur in the intrinsic frame, where the coherency matrix takes the form

$|\mu_{12}|,|\mu_{13}|,|\mu_{23}|$1

For the genuinely three-dimensional case $|\mu_{12}|,|\mu_{13}|,|\mu_{23}|$2,

$|\mu_{12}|,|\mu_{13}|,|\mu_{23}|$3

The maxima occur in an equal-diagonal frame $|\mu_{12}|,|\mu_{13}|,|\mu_{23}|$4 with

$|\mu_{12}|,|\mu_{13}|,|\mu_{23}|$5

for which

$|\mu_{12}|,|\mu_{13}|,|\mu_{23}|$6

The paper states that $|\mu_{12}|,|\mu_{13}|,|\mu_{23}|$7 can be computed in practice via the Bendel–Mickey algorithm (Gil, 2020).

This analysis is explicitly contrasted with rotation-invariant state descriptors such as the total intensity $|\mu_{12}|,|\mu_{13}|,|\mu_{23}|$8, the degree of linear polarization $|\mu_{12}|,|\mu_{13}|,|\mu_{23}|$9, the degree of directionality $C(\Phi(\rho))$0, the degree of circular polarization $C(\Phi(\rho))$1, and the three-dimensional degree of polarimetric purity

$C(\Phi(\rho))$2

Those are genuine descriptors of the state; $C(\Phi(\rho))$3 are not (Gil, 2020).

The formalism reduces naturally to the two-dimensional case $C(\Phi(\rho))$4, equivalent to $C(\Phi(\rho))$5. In that limit the usual degree of mutual coherence obeys Wolf’s bound

$C(\Phi(\rho))$6

with the maximum attained in transverse frames where the two component intensities are equal. The three-dimensional analysis therefore establishes a precise distinction between intrinsic polarimetric purity and output-channel coherence in a chosen Cartesian decomposition (Gil, 2020).

3. Absolute-output coherence in higher-order consensus dynamics

In networked control, the closest formal analogue of absolute output coherence is the coherence of a leader-follower consensus system with absolute information. The network is a connected, undirected, weighted graph $C(\Phi(\rho))$7 with Laplacian $C(\Phi(\rho))$8. Each node $C(\Phi(\rho))$9 carries $\mathbf r$0 scalar states $\mathbf r$1, and the $\mathbf r$2-th order dynamics are

$\mathbf r$3

where $\mathbf r$4 is zero-mean white stochastic disturbance (Mackin et al., 2019).

A subset $\mathbf r$5 is selected as leaders. Leaders receive absolute information through the anchoring term $\mathbf r$6 in the control law, so the grounded matrix

$\mathbf r$7

governs both stability and performance. The output is chosen as the first-order state itself,

$\mathbf r$8

not a disagreement output or a deviation from the network average (Mackin et al., 2019).

The performance metric is

$\mathbf r$9

which the paper identifies with the squared $$\boldsymbol{\varepsilon}(t)= \begin{pmatrix} \varepsilon_1(t)\ \varepsilon_2(t)\ \varepsilon_3(t) \end{pmatrix},$$0 norm,

$$\boldsymbol{\varepsilon}(t)= \begin{pmatrix} \varepsilon_1(t)\ \varepsilon_2(t)\ \varepsilon_3(t) \end{pmatrix},$$1

where $$\boldsymbol{\varepsilon}(t)= \begin{pmatrix} \varepsilon_1(t)\ \varepsilon_2(t)\ \varepsilon_3(t) \end{pmatrix},$$2 solves

$$\boldsymbol{\varepsilon}(t)= \begin{pmatrix} \varepsilon_1(t)\ \varepsilon_2(t)\ \varepsilon_3(t) \end{pmatrix},$$3

This is “absolute” because the output is the absolute node state, the desired trajectory is fixed at zero, and leader grounding removes the translation invariance that normally forces relative outputs in consensus without leaders (Mackin et al., 2019).

The paper gives exact stability conditions. For $$\boldsymbol{\varepsilon}(t)= \begin{pmatrix} \varepsilon_1(t)\ \varepsilon_2(t)\ \varepsilon_3(t) \end{pmatrix},$$4, stability holds iff $$\boldsymbol{\varepsilon}(t)= \begin{pmatrix} \varepsilon_1(t)\ \varepsilon_2(t)\ \varepsilon_3(t) \end{pmatrix},$$5 and $$\boldsymbol{\varepsilon}(t)= \begin{pmatrix} \varepsilon_1(t)\ \varepsilon_2(t)\ \varepsilon_3(t) \end{pmatrix},$$6. For $$\boldsymbol{\varepsilon}(t)= \begin{pmatrix} \varepsilon_1(t)\ \varepsilon_2(t)\ \varepsilon_3(t) \end{pmatrix},$$7, the condition is

$$\boldsymbol{\varepsilon}(t)= \begin{pmatrix} \varepsilon_1(t)\ \varepsilon_2(t)\ \varepsilon_3(t) \end{pmatrix},$$8

For $$\boldsymbol{\varepsilon}(t)= \begin{pmatrix} \varepsilon_1(t)\ \varepsilon_2(t)\ \varepsilon_3(t) \end{pmatrix},$$9,

$$\mathbf R=\left\langle \boldsymbol{\varepsilon}(t)\otimes \boldsymbol{\varepsilon}^\dagger(t)\right\rangle, \qquad R_{ij}=\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle.$$0

It also proves that equal gains $$\mathbf R=\left\langle \boldsymbol{\varepsilon}(t)\otimes \boldsymbol{\varepsilon}^\dagger(t)\right\rangle, \qquad R_{ij}=\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle.$$1 cannot stabilize the system for $$\mathbf R=\left\langle \boldsymbol{\varepsilon}(t)\otimes \boldsymbol{\varepsilon}^\dagger(t)\right\rangle, \qquad R_{ij}=\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle.$$2 (Mackin et al., 2019).

Once stable, the absolute-output coherence admits closed forms. For second order,

$$\mathbf R=\left\langle \boldsymbol{\varepsilon}(t)\otimes \boldsymbol{\varepsilon}^\dagger(t)\right\rangle, \qquad R_{ij}=\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle.$$3

For third order,

$$\mathbf R=\left\langle \boldsymbol{\varepsilon}(t)\otimes \boldsymbol{\varepsilon}^\dagger(t)\right\rangle, \qquad R_{ij}=\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle.$$4

For fourth order,

$$\mathbf R=\left\langle \boldsymbol{\varepsilon}(t)\otimes \boldsymbol{\varepsilon}^\dagger(t)\right\rangle, \qquad R_{ij}=\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle.$$5

Leader placement enters entirely through $$\mathbf R=\left\langle \boldsymbol{\varepsilon}(t)\otimes \boldsymbol{\varepsilon}^\dagger(t)\right\rangle, \qquad R_{ij}=\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle.$$6, so output coherence reduction is a grounded-Laplacian optimization problem (Mackin et al., 2019).

The corresponding leader-selection objective is submodular. For $$\mathbf R=\left\langle \boldsymbol{\varepsilon}(t)\otimes \boldsymbol{\varepsilon}^\dagger(t)\right\rangle, \qquad R_{ij}=\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle.$$7, the paper constructs non-decreasing submodular functions $$\mathbf R=\left\langle \boldsymbol{\varepsilon}(t)\otimes \boldsymbol{\varepsilon}^\dagger(t)\right\rangle, \qquad R_{ij}=\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle.$$8 equivalent to minimizing $$\mathbf R=\left\langle \boldsymbol{\varepsilon}(t)\otimes \boldsymbol{\varepsilon}^\dagger(t)\right\rangle, \qquad R_{ij}=\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle.$$9, so a greedy algorithm has a constant-factor guarantee: $$\mu_{ij}= \frac{\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle} {\sqrt{\langle |\varepsilon_i(t)|^2\rangle\langle |\varepsilon_j(t)|^2\rangle}}, \qquad i\neq j,$$0 It reports empirical near-optimality of the greedy method and emphasizes that optimal leader sets need not coincide across dynamic orders (Mackin et al., 2019). In this literature, absolute output coherence is therefore an $$\mu_{ij}= \frac{\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle} {\sqrt{\langle |\varepsilon_i(t)|^2\rangle\langle |\varepsilon_j(t)|^2\rangle}}, \qquad i\neq j,$$1-type absolute variance of anchored outputs, not a basis-dependent off-diagonal state quantity.

4. Quantum-state, output-statistics, and cloning-based formulations

The foundational state-based resource theory fixes an incoherent basis $$\mu_{ij}= \frac{\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle} {\sqrt{\langle |\varepsilon_i(t)|^2\rangle\langle |\varepsilon_j(t)|^2\rangle}}, \qquad i\neq j,$$2 and defines incoherent states as diagonal density operators

$$\mu_{ij}= \frac{\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle} {\sqrt{\langle |\varepsilon_i(t)|^2\rangle\langle |\varepsilon_j(t)|^2\rangle}}, \qquad i\neq j,$$3

Two standard coherence measures are

$$\mu_{ij}= \frac{\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle} {\sqrt{\langle |\varepsilon_i(t)|^2\rangle\langle |\varepsilon_j(t)|^2\rangle}}, \qquad i\neq j,$$4

These measures are monotone under incoherent CPTP maps and directly induce output-state quantities of the form $$\mu_{ij}= \frac{\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle} {\sqrt{\langle |\varepsilon_i(t)|^2\rangle\langle |\varepsilon_j(t)|^2\rangle}}, \qquad i\neq j,$$5, although the framework itself is state-based rather than channel-based (Baumgratz et al., 2013).

Several later works attach “absolute” content to such state coherence in different ways. One route is absolute-valued matrix decomposition. The holographic measure

$$\mu_{ij}= \frac{\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle} {\sqrt{\langle |\varepsilon_i(t)|^2\rangle\langle |\varepsilon_j(t)|^2\rangle}}, \qquad i\neq j,$$6

is shown to be a bona fide coherence measure, and the same paper proves that a coherence witness is optimal iff all of its diagonal elements are zero (Wang et al., 2021). Another route is parameterization by Tsallis relative operator $$\mu_{ij}= \frac{\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle} {\sqrt{\langle |\varepsilon_i(t)|^2\rangle\langle |\varepsilon_j(t)|^2\rangle}}, \qquad i\neq j,$$7-entropy, yielding

$$\mu_{ij}= \frac{\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle} {\sqrt{\langle |\varepsilon_i(t)|^2\rangle\langle |\varepsilon_j(t)|^2\rangle}}, \qquad i\neq j,$$8

with $$\mu_{ij}= \frac{\langle \varepsilon_i(t)\varepsilon_j^*(t)\rangle} {\sqrt{\langle |\varepsilon_i(t)|^2\rangle\langle |\varepsilon_j(t)|^2\rangle}}, \qquad i\neq j,$$9, $|\mu_{ij}|=\frac{|R_{ij}|}{\sqrt{R_{ii}R_{jj}}}.$0, and the paper proves nonnegativity, monotonicity, strong monotonicity, and convexity (Guo et al., 2023).

A different interpretation of absolute output coherence is the absolute difference between coherence values of the same state in incompatible bases. For the relative entropy of coherence, if $|\mu_{ij}|=\frac{|R_{ij}|}{\sqrt{R_{ii}R_{jj}}}.$1 and $|\mu_{ij}|=\frac{|R_{ij}|}{\sqrt{R_{ii}R_{jj}}}.$2 are the dephased states in two bases and

$|\mu_{ij}|=\frac{|R_{ij}|}{\sqrt{R_{ii}R_{jj}}}.$3

then

$|\mu_{ij}|=\frac{|R_{ij}|}{\sqrt{R_{ii}R_{jj}}}.$4

The same paper also derives uncertainty-like lower bounds on sums of coherence values in incompatible bases, including bipartite versions tightened by negative conditional entropy $|\mu_{ij}|=\frac{|R_{ij}|}{\sqrt{R_{ii}R_{jj}}}.$5 (Singh et al., 2016).

Theory-independent approaches recast coherence directly as an output-statistics difference. For a preferred observable $|\mu_{ij}|=\frac{|R_{ij}|}{\sqrt{R_{ii}R_{jj}}}.$6, let $|\mu_{ij}|=\frac{|R_{ij}|}{\sqrt{R_{ii}R_{jj}}}.$7 be the decohered state obtained after a sharp measurement of $|\mu_{ij}|=\frac{|R_{ij}|}{\sqrt{R_{ii}R_{jj}}}.$8. The measurement-outcome-difference quantity is

$|\mu_{ij}|=\frac{|R_{ij}|}{\sqrt{R_{ii}R_{jj}}}.$9

and the Kolmogorov version is

$$\mathbf R\mapsto \mathbf U\mathbf R\mathbf U^\dagger$$0

In quantum mechanics this becomes

$$\mathbf R\mapsto \mathbf U\mathbf R\mathbf U^\dagger$$1

so absolute output coherence is the maximal absolute deviation between output distributions of the original and dephased states (Sun et al., 2017).

Cloning theory supplies a more literal output-state perspective. The Wootters–Zurek cloner maps every input to a reduced two-copy output diagonal in the computational basis, so the studied output coherence always vanishes. By contrast, the Buzek–Hillery family contains a subfamily with $$\mathbf R\mapsto \mathbf U\mathbf R\mathbf U^\dagger$$2 for which

$$\mathbf R\mapsto \mathbf U\mathbf R\mathbf U^\dagger$$3

for every input, and the choice $$\mathbf R\mapsto \mathbf U\mathbf R\mathbf U^\dagger$$4 yields

$$\mathbf R\mapsto \mathbf U\mathbf R\mathbf U^\dagger$$5

with the same reduced two-copy output for all inputs. Within that restricted reduced-state sense, the paper calls this an optimal universal quantum coherence machine (Goswami et al., 2017).

5. Field-output, continuous-basis, and non-equilibrium open-system notions

In quantum optics, the natural meaning of absolute output coherence is unnormalized coherence of the emitted field. In the Markovian input-output relation,

$$\mathbf R\mapsto \mathbf U\mathbf R\mathbf U^\dagger$$6

the path-integral generating functional gives direct access to $$\mathbf R\mapsto \mathbf U\mathbf R\mathbf U^\dagger$$7, the unnormalized first-order coherence

$$\mathbf R\mapsto \mathbf U\mathbf R\mathbf U^\dagger$$8

the second-order statistics, and anomalous correlators relevant to squeezing. For a Kerr nonlinear oscillator, the formalism finds a reduction in reflection that is not due to photon leakage but rather associated to the squeezing of the output light (Daniel et al., 9 Sep 2025). In this setting “absolute” means unnormalized output moments rather than normalized ratios such as $$\mathbf R\mapsto \mathbf U\mathbf R\mathbf U^\dagger$$9.

Continuous-basis resource theory replaces ideal diagonalization by a physical dephasing channel. For position coherence, the channel

$\mu_{ij}$0

acts in position space as

$\mu_{ij}$1

Because position eigenstates are generalized and the usual diagonal-state picture fails, the paper defines coherence by dephasing disturbance rather than distance to a nonempty set of diagonal states. Two resulting quantifiers are

$\mu_{ij}$2

and

$\mu_{ij}$3

For physically relevant kernels with $\mu_{ij}$4 for $\mu_{ij}$5, the fixed-point set contains no normal states, so every normal output state has strictly positive continuous-basis coherence in this sense (Sajnok et al., 9 May 2026).

Non-equilibrium open-system studies introduce another output-oriented notion: residual steady-state coherence. For two coupled two-level atoms interacting with two heat baths at different temperatures, the secular master equation yields a diagonal steady state in the energy basis, while the non-secular master equation gives a stationary density matrix with off-diagonal element

$\mu_{ij}$6

The paper studies the absolute value $\mu_{ij}$7 as the residual steady coherence, finds that it vanishes at equilibrium, is largest near resonance, is strongest in the configuration where each atom couples to its own bath, and disappears completely when both atoms couple to both baths (Huangfu et al., 2017). Here the “output” is the long-time reduced state of the open system, and the coherence quantity is the absolute value of a surviving off-diagonal element.

6. Monitoring, thermodynamic control, and conceptual boundaries

Finite-time quantum thermodynamics makes the dependence on monitoring especially explicit. In a steady-state quantum Otto engine, finite-time unitary work strokes generate off-diagonal terms in the instantaneous energy basis, while incomplete thermalization allows residual coherence to persist from stroke to stroke. Pointer-based monitoring schemes modify that coherence through Gaussian suppression factors of the form

$\mu_{ij}$8

or analogous expressions involving work and heat differences. Three schemes are analyzed: S1 with four energy measurements, S2 with three delayed measurements of stroke-wise work and heat, and S3 with two delayed measurements of total work and hot-bath heat (Shastri et al., 2023).

For a two-level working substance, all three schemes recover the unmonitored average work in the limit of infinitely weak measurement. In the strong-measurement limit, only S1 and S2 reproduce the two-point projective measurement benchmark, while S3 generally does not because measuring only total work and heat leaves degenerate histories unresolved and therefore allows some off-diagonal contributions to survive (Shastri et al., 2023). In that literature, output coherence is not a separate thermodynamic observable; it is the controllable residual coherence of the working substance, inferred through its imprint on average work and work fluctuations.

Feedback-controlled open quantum systems show a related but sharper effect. Projective measurement can create zero-probability regions in forward trajectory space, producing absolute irreversibility. Quantum coherent driving, implemented by Hamiltonian terms off-diagonal in the measurement basis, can immediately spread the post-measurement state over the Hilbert space and thereby remove those zero-probability regions. The resulting suppression of absolute irreversibility is presented as a thermodynamic advantage of coherent driving (Murashita et al., 2017). This is not output coherence in a resource-theoretic sense, but it does show that coherence can alter the support of observable output trajectories themselves.

Across these literatures, the principal boundary condition is that absolute output coherence is seldom an intrinsic scalar of the underlying system alone. It is usually tied to a chosen output representation. In three-dimensional polarization it is frame dependent rather than rotationally invariant (Gil, 2020). In basis-dependent quantum resource theory it depends on the incoherent basis (Baumgratz et al., 2013). In theory-independent formulations it depends on the preferred observable used to define decoherence (Sun et al., 2017). In continuous position space it depends on the physically fixed dephasing kernel $\mu_{ij}$9 and its monitoring scale (Sajnok et al., 9 May 2026). In finite-time engines it depends on the monitoring protocol and pointer resolution (Shastri et al., 2023).

A plausible implication is that “absolute output coherence” is not a single cross-disciplinary invariant but a recurrent structural theme: coherence assessed at the level of outputs, with the relevant absoluteness supplied by a fixed channel decomposition, basis choice, frame choice, or measurement model. The mathematically precise object is therefore domain specific, but the unifying lesson is consistent: output coherence is meaningful only after the output space itself has been specified.