Unitarily Invariant Coherence Rank

• The unitarily invariant coherence rank is defined as the number of non-zero eigenvalues in a trace-normalized 4×4 coherence matrix, remaining fixed under all energy-preserving unitary transformations.
• It refines entropy-based classifications by distinguishing unitary equivalence classes and obstructing reversible conversions between fields with different spectral ranks.
• Experimental protocols using filtering and randomization demonstrate fixed-entropy paths between ranks, enabling practical fringe-visibility tests for mode separation in optical fields.

Unitarily invariant coherence rank, in the optical setting of partially coherent fields with multiple degrees of freedom, is the rank of the trace-normalized coherence matrix $G$: the number of its non-zero eigenvalues. For fields with two binary degrees of freedom—polarization and two spatial modes—$G$ is a $4\times 4$ Hermitian, positive semidefinite, unit-trace matrix, and its rank is invariant under every deterministic, energy-preserving transformation of the form $G\mapsto UGU^\dagger$. In this framework, rank is not merely a bookkeeping device: it refines entropy-based classification, separates unitary equivalence classes that entropy alone cannot distinguish, and determines whether iso-entropy fields can or cannot be inter-converted unitarily (Harling et al., 2024).

1. Definition and mathematical setting

The relevant field has two binary degrees of freedom (DoFs): polarization with basis $\{\mathrm{H},\mathrm{V}\}$, and a spatial DoF with two points or modes $a,b$. The joint DoF is therefore four-dimensional, with natural basis

$$\{|a\mathrm{H}\rangle,\ |a\mathrm{V}\rangle,\ |b\mathrm{H}\rangle,\ |b\mathrm{V}\rangle\}.$$

Its first-order coherence is encoded in a $4\times 4$ coherence matrix $G$, whose entries are

$G_{kl}^{ij}=\langle E_k^i(E_l^j)^*\rangle,$

with $G$0, $G$1, and $G$2 denoting ensemble averaging. Diagonal entries are modal intensities; off-diagonal entries are mutual coherences between different polarization and spatial components.

After trace normalization, $G$3 is Hermitian, positive semidefinite, and satisfies $G$4. As with a density operator, it admits a spectral decomposition

$G$5

By a global unitary acting on the full four-dimensional space, every coherence matrix is unitarily equivalent to a diagonal form

$G$6

In this setting, the natural notion of unitarily invariant coherence rank is

$G$7

Physically, this counts the number of coherent modes with non-zero statistical weight. Because the rank is defined spectrally, it is basis-independent under arbitrary joint unitaries on the two DoFs.

2. Unitary invariance and the failure of entropy as a complete classifier

The entropy used for classification is the von Neumann entropy of the normalized coherence matrix,

$G$8

A deterministic, energy-preserving transformation is represented by a unitary $G$9 on the four-dimensional joint space, acting as

$4\times 4$0

Because this is a unitary similarity transformation, the eigenvalues of $4\times 4$1 are unchanged. Consequently, the spectrum, entropy, purity $4\times 4$2, and rank are all invariant under such transformations.

The special role of rank emerges by comparison with the single-binary-DoF case. For a $4\times 4$3 coherence matrix, the spectrum is $4\times 4$4, and the entropy

$4\times 4$5

is one-to-one in the relevant interval. There, entropy uniquely labels unitary equivalence classes: any two iso-entropy fields can be connected by a unitary.

For the $4\times 4$6 two-DoF problem, that equivalence fails. Iso-entropy fields can have different spectra and therefore belong to different unitary orbits. Rank is the coarsest additional spectral invariant that immediately blocks some conversions. In particular, inter-rank unitary conversion is impossible: if two fields have different numbers of zero eigenvalues, no unitary can relate them, even when their total entropy is the same (Harling et al., 2024).

3. Rank-resolved classification of iso-entropy fields

The space of normalized $4\times 4$7 coherence matrices is stratified by the possible ranks $4\times 4$8. Each rank sector has a distinct spectral geometry and a distinct relationship between entropy and unitary equivalence.

Rank of $4\times 4$9	Spectrum pattern	Entropy and unitary classification
$G\mapsto UGU^\dagger$0	$G\mapsto UGU^\dagger$1	$G\mapsto UGU^\dagger$2; all rank-1 fields are unitarily related
$G\mapsto UGU^\dagger$3	$G\mapsto UGU^\dagger$4	$G\mapsto UGU^\dagger$5; entropy uniquely determines the unitary orbit
$G\mapsto UGU^\dagger$6	$G\mapsto UGU^\dagger$7	$G\mapsto UGU^\dagger$8; iso-entropy sets are one-dimensional families of inequivalent spectra
$G\mapsto UGU^\dagger$9	$\{\mathrm{H},\mathrm{V}\}$0	$\{\mathrm{H},\mathrm{V}\}$1; iso-entropy sets are two-dimensional families of inequivalent spectra

For rank-1 fields, entropy and rank together are complete, but entropy alone is trivial because all such fields have $\{\mathrm{H},\mathrm{V}\}$2. For rank-2 fields, the non-zero spectrum is fully specified by one parameter $\{\mathrm{H},\mathrm{V}\}$3, and the entropy is again sufficient to determine the unitary class. This is why any pair of iso-entropy rank-2 fields can be converted into each other unitarily.

For rank-3 fields, the spectrum $\{\mathrm{H},\mathrm{V}\}$4 spans a two-dimensional simplex. Fixing entropy imposes one constraint, leaving a one-dimensional iso-entropy curve. Different points on that curve correspond to different spectra and therefore to different unitary equivalence classes. Rank-4 behaves analogously, except that the ambient simplex is three-dimensional, so fixing entropy yields a two-dimensional iso-entropy surface.

The complete unitary invariant is the full spectrum, including multiplicities. Entropy is therefore a coarse spectral invariant, while rank is the simplest discrete refinement. The unitarily invariant coherence rank is fundamental precisely because it separates reversible from non-reversible transformations at the most basic level: unitary dynamics preserve it, whereas non-unitary operations can change it (Harling et al., 2024).

4. Separability, classical entanglement, and locked entropy

The rank of $\{\mathrm{H},\mathrm{V}\}$5 is also tied to the separability structure of the two DoFs. Writing the joint coherence matrix as

$\{\mathrm{H},\mathrm{V}\}$6

one may trace over one DoF to obtain reduced coherence matrices for polarization and space. Their entropies quantify the degree of coherence in each subsystem, but unlike the total entropy $\{\mathrm{H},\mathrm{V}\}$7, these reduced entropies are not invariant under global unitary mixing of the DoFs.

A decisive condition for separability in the diagonal basis is

$\{\mathrm{H},\mathrm{V}\}$8

If and only if this holds can the diagonalized coherence matrix be separated into a direct product with respect to the two DoFs. Rank-2 fields automatically satisfy this condition, because their spectra have the form $\{\mathrm{H},\mathrm{V}\}$9. They are therefore always separable after a suitable global unitary, and all of their entropy can be concentrated into one DoF while the other becomes fully coherent. By contrast, rank-3 fields have spectra $a,b$0 with $a,b$1, so the separability condition is necessarily violated. Rank-3 fields are never separable with respect to their DoFs (Harling et al., 2023).

This non-separability produces what has been termed locked entropy. For rank-2 fields, one can always achieve

$a,b$2

depending on the chosen unitary. For rank-3 fields, neither DoF can be rid altogether of statistical fluctuations. The minimum entropy locked in one DoF is

$a,b$3

while the maximum entropy concentrated into the other is

$a,b$4

with

$a,b$5

When $a,b$6, one cannot concentrate $a,b$7 bit of entropy in a single DoF. This establishes rank-3 as an intrinsically non-separable coherence class, with rank functioning analogously to a Schmidt-structure constraint on entropy redistribution (Harling et al., 2023).

An experimentally useful consequence is interference-based rank discrimination. If there exists a polarization projection along which the field is spatially coherent, then the rank is at most $a,b$8. If there exist two orthogonal polarization projections along which the field is spatially coherent, then the rank is at most $a,b$9. These criteria convert the rank taxonomy into observable fringe-visibility tests.

5. Non-unitary rank change and experimental realization

Because rank is strictly preserved by unitaries, any rank-changing conversion at fixed entropy must be non-unitary. The relevant transformation is modeled either by a single linear operator,

$$\{|a\mathrm{H}\rangle,\ |a\mathrm{V}\rangle,\ |b\mathrm{H}\rangle,\ |b\mathrm{V}\rangle\}.$$0

or by a super-operator of the form

$$\{|a\mathrm{H}\rangle,\ |a\mathrm{V}\rangle,\ |b\mathrm{H}\rangle,\ |b\mathrm{V}\rangle\}.$$1

The first type describes filtering, such as differential attenuation, polarizers, or beam blocking. Filtering can reduce rank and typically reduces entropy after renormalization. The second type describes randomizing transformations, such as time-averaged unitary modulation; these tend to increase entropy and can increase rank by populating previously zero eigenmodes.

By combining filtering and randomization, one can preserve entropy while changing rank. Higher-to-lower rank conversion at fixed entropy proceeds by filtering chosen so that the target entropy is reached after renormalization. Lower-to-higher rank conversion requires a randomizer-plus-filter sequence: randomization first increases both rank and entropy, and filtering then lowers the entropy back to its original value while keeping the new rank. The paper explicitly constructs fixed-entropy paths such as rank-4 $$\{|a\mathrm{H}\rangle,\ |a\mathrm{V}\rangle,\ |b\mathrm{H}\rangle,\ |b\mathrm{V}\rangle\}.$$2 rank-3 $$\{|a\mathrm{H}\rangle,\ |a\mathrm{V}\rangle,\ |b\mathrm{H}\rangle,\ |b\mathrm{V}\rangle\}.$$3 rank-2 and the reverse at $$\{|a\mathrm{H}\rangle,\ |a\mathrm{V}\rangle,\ |b\mathrm{H}\rangle,\ |b\mathrm{V}\rangle\}.$$4 (Harling et al., 2024).

The experimental implementation begins from the source state

$$\{|a\mathrm{H}\rangle,\ |a\mathrm{V}\rangle,\ |b\mathrm{H}\rangle,\ |b\mathrm{V}\rangle\}.$$5

a maximum-entropy rank-4 field synthesized from unpolarized, spatially incoherent LED light passing through two slits. Static optical elements implement diagonal filters, while a rotating half-wave plate realizes randomizers. The coherence matrix is reconstructed by optical coherency matrix tomography. For diagonal $$\{|a\mathrm{H}\rangle,\ |a\mathrm{V}\rangle,\ |b\mathrm{H}\rangle,\ |b\mathrm{V}\rangle\}.$$6, only four intensity measurements are required, packaged into generalized Stokes parameters $$\{|a\mathrm{H}\rangle,\ |a\mathrm{V}\rangle,\ |b\mathrm{H}\rangle,\ |b\mathrm{V}\rangle\}.$$7, from which the eigenvalues $$\{|a\mathrm{H}\rangle,\ |a\mathrm{V}\rangle,\ |b\mathrm{H}\rangle,\ |b\mathrm{V}\rangle\}.$$8 are recovered.

Using this protocol, 114 distinct partially coherent fields with different ranks and entropies were synthesized and plotted in the “pyramid” representation of eigenvalue space. The experiments demonstrated both intra-rank steering along fixed-rank iso-entropy trajectories and inter-rank steering at fixed entropy. The observed behavior matched the theoretical prediction that every rank change is necessarily non-unitary, whereas unitary invariants—spectrum, entropy, and rank—remain unchanged under reversible transformations (Harling et al., 2024).

6. Related usages and conceptual boundaries

The optical notion of unitarily invariant coherence rank has already been used operationally in communications. In a highly scattering channel that mixes polarization and spatial modes, logical symbols were encoded in the rank of the field coherence matrix: rank-1, rank-2, rank-3, and rank-4 fields were chosen with maximal entropies $$\{|a\mathrm{H}\rangle,\ |a\mathrm{V}\rangle,\ |b\mathrm{H}\rangle,\ |b\mathrm{V}\rangle\}.$$9, $4\times 4$0, $4\times 4$1, and $4\times 4$2 bits, respectively. Because the channel acted unitarily on the full four-dimensional mode space, the rank was preserved even when conventional polarization or mode encodings were scrambled, and the reported communication protocol achieved $4\times 4$3 fidelity (Harling et al., 25 Jul 2025).

At the same time, the phrase should not be conflated with standard fixed-basis resource theory of coherence. In that setting, incoherent states are diagonal in a chosen reference basis, and coherence is inherently basis-dependent. A central negative result is that coherence measures cannot be induced by any unitary similarity invariant norm; the relevant symmetry is invariance under incoherent unitaries, not under arbitrary unitaries (Jing et al., 2020). A distinct channel-level usage appears in the resource theory of coherence for operations, where a maximally coherent state of rank $4\times 4$4 is the basic unit for simulating channels under incoherent operations, and cost and capacity are measured in $4\times 4$5 units rather than by the spectral rank of a field coherence matrix (Dana et al., 2017).

Other basis-independent developments use different rank objects. For qubit multi-states, coherence and imaginarity can be characterized by the rank of the Gram matrix of Bloch vectors: rank $4\times 4$6 corresponds to set incoherence, rank $4\times 4$7 to imaginarity-free families, and rank $4\times 4$8 to multi-state imaginarity (Li et al., 20 Jul 2025). For arbitrary finite families of density operators, low-order Bargmann invariants furnish a basis-independent hierarchy for deciding set coherence, with a universal fourth-order criterion based on pairwise commutators (Wang, 11 May 2026). These constructions are conceptually adjacent, but they are not the same quantity as the optical rank of $4\times 4$9.

In the optical two-DoF framework, unitarily invariant coherence rank therefore has a precise and narrow meaning: it is the rank of the normalized $G$0 coherence matrix. Its significance lies in three linked facts. It is invariant under arbitrary joint unitaries; it obstructs inter-rank reversible conversion even for iso-entropy fields; and it encodes structural information about separability, entropy localization, and coherent-mode content that entropy alone does not provide.