Two-Particle Quantum Holonomy Overview

• Two-particle quantum holonomy is defined by holonomic phenomena that emerge only in bipartite systems, including exotic eigenspace shifts in the Lieb–Liniger model and non-Abelian responses in interferometry.
• The framework covers distinct constructions such as correlation-induced non-Abelian transport, entanglement-dependent phase shifts, and purely geometric transformations within cyclic subspaces.
• These concepts have practical implications for quantum control, revealing design parameters and topological structures absent in single-particle systems.

Two-particle quantum holonomy denotes holonomic phenomena whose essential structure appears only in a two-particle or bipartite setting. In the literature, the term covers several distinct constructions: exotic adiabatic eigenspace permutation in the two-body Lieb–Liniger model, correlation-induced non-Abelian transport in bipartite interferometry, entanglement-dependent holonomic phase of two entangled photons under local evolution, and purely geometric transformations acting on cyclic subspaces of two-particle Hilbert spaces (Yonezawa et al., 2013, Johansson et al., 2010, Loredo et al., 2013, Neef et al., 16 Jun 2025). This suggests that “two-particle quantum holonomy” is best understood as a family of related notions rather than a single universally fixed definition.

1. Terminological scope and principal variants

The literature uses “holonomy” in at least four technically different senses when two particles are involved. In some cases the holonomic output is a scalar phase; in others it is a matrix-valued transport law or an outright permutation of eigenspaces.

Construction	Physical setting	Holonomic content
Exotic quantum holonomy	Two bosons in the Lieb–Liniger model	Eigenspace/eigenvalue permutation
Correlation-induced holonomy	Franson two-particle interferometer	Generally non-Abelian induced local unitary
Entanglement-dependent holonomic phase	Two entangled photons under local evolutions	Phase depending on tangle $\mathbb T$
Cyclic-subspace holonomy	Two-particle photonic subspaces	Purely geometric transformation with $\hat K=0$

This distinction is important because the familiar templates of Berry phase, Wilczek–Zee holonomy, and Aharonov–Anandan phase do not exhaust the two-particle cases. In the Lieb–Liniger setting, the final eigenspace may differ from the initial eigenspace even though the Hamiltonian returns to its original form (Yonezawa et al., 2013). In the Franson setting, the effective “phase” degree of freedom is promoted from $U(1)$ to a local unitary on the second subsystem, so the holonomy group is generally non-Abelian for correlated states (Johansson et al., 2010). In the entangled-photon experiment, the holonomic quantity is a measurable phase of the joint bipartite wavefunction, and its behavior changes from geometric to topological as the entanglement is tuned from zero to maximal (Loredo et al., 2013). In the integrated-photonics framework, the central object is a cyclic subspace whose evolution is purely geometric when the dynamical contribution vanishes, and such subspaces can exist for two particles even when the analogous one-particle sector is non-holonomic (Neef et al., 16 Jun 2025).

A recurrent source of confusion is the assumption that all two-particle holonomies are either Berry phases or Wilczek–Zee transports. The cited literature does not support that simplification. Some two-particle holonomies are scalar and Abelian, some are matrix-valued and non-Abelian, and some are best described as monodromies of eigenspace labels rather than as phases.

2. Exotic eigenspace holonomy in the two-body Lieb–Liniger model

In the Lieb–Liniger model, two identical bosons move on a one-dimensional ring with contact interaction. For $N=2$, the Hamiltonian is

$$H = -\frac{1}{2}\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right) +g\,\delta(x_1-x_2), \qquad x_j \sim x_j+2\pi,$$

with $L=2\pi$, $\hbar=1$, and particle mass $1$ (Yonezawa et al., 2013). The parameter cycle is

$g: 0 \to +\infty \to -\infty \to 0,$

with the first and third stages adiabatic and the $+\infty\to-\infty$ step implemented by identifying the finite Bethe roots at the two singular endpoints.

The general Bethe equations reduce for two particles to

$\hat K=0$0

with total momentum

$\hat K=0$1

and energy

$\hat K=0$2

At the free point $\hat K=0$3, the ordered quasimomenta are integers $\hat K=0$4 with $\hat K=0$5. After one full cycle, the exact holonomy rule is

$\hat K=0$6

Equivalently,

$\hat K=0$7

The mechanism is a two-step spectral flow through the Tonks–Girardeau and super-Tonks–Girardeau regimes. In the first adiabatic stage,

$\hat K=0$8

so the roots become half-integers at the Tonks–Girardeau endpoint. The second stage identifies

$\hat K=0$9

and the third stage repeats the same shift, yielding the full permutation of free-state labels. The final Hamiltonian is again the free Hamiltonian, but the system occupies a different free eigenstate. The paper therefore characterizes the effect as exotic quantum holonomy: an eigenspace or eigenvalue permutation rather than a phase (Yonezawa et al., 2013).

For a free state $U(1)$0, one cycle preserves total momentum,

$U(1)$1

but changes the energy to

$U(1)$2

which is strictly larger for ordered $U(1)$3. The two minimal free states are

$U(1)$4

and repeated cycles generate the families

$U(1)$5

$U(1)$6

By momentum translation these become the general minimal seeds $U(1)$7 and $U(1)$8.

The inverse map

$U(1)$9

fails on minimal states because the attractive continuation encounters clustering: for $N=2$0, the clustering sector is the formation of a bound pair as $N=2$1, and the eigenenergy diverges to $N=2$2. This gives the two-body classification a direct physical interpretation in terms of gas-like versus bound branches. The paper also notes that the two-body holonomy becomes especially transparent in center-of-mass and relative coordinates,

$N=2$3

for which the cycle preserves $N=2$4 and shifts the relative quantum number by one at the free endpoints: $N=2$5 The paper does not develop the center-of-mass/relative-coordinate reduction explicitly, but it is completely consistent with its formulas and helps interpret the two-body result (Yonezawa et al., 2013).

3. Correlation-induced non-Abelian holonomy in bipartite interferometry

A different notion of two-particle holonomy was formulated for bipartite interferometry in a Franson interferometer. Here the system is a general bipartite density operator on

$N=2$6

and one inserts local unitaries

$N=2$7

in the long arms of the two interferometers (Johansson et al., 2010). The operational quantity is the coincidence intensity

$N=2$8

The paper defines bipartite “parallelity” by maximizing this coincidence intensity with respect to $N=2$9 for a chosen $$H = -\frac{1}{2}\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right) +g\,\delta(x_1-x_2), \qquad x_j \sim x_j+2\pi,$$0.

In Stokes-tensor form, with

$$H = -\frac{1}{2}\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right) +g\,\delta(x_1-x_2), \qquad x_j \sim x_j+2\pi,$$1

the intensity becomes

$$H = -\frac{1}{2}\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right) +g\,\delta(x_1-x_2), \qquad x_j \sim x_j+2\pi,$$2

The central structural object is the correlation matrix $$H = -\frac{1}{2}\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right) +g\,\delta(x_1-x_2), \qquad x_j \sim x_j+2\pi,$$3 for $$H = -\frac{1}{2}\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right) +g\,\delta(x_1-x_2), \qquad x_j \sim x_j+2\pi,$$4. In the infinitesimal formulation, extremization yields

$$H = -\frac{1}{2}\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right) +g\,\delta(x_1-x_2), \qquad x_j \sim x_j+2\pi,$$5

and, provided $$H = -\frac{1}{2}\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right) +g\,\delta(x_1-x_2), \qquad x_j \sim x_j+2\pi,$$6 is invertible, the anti-Hermitian connection one-form is

$$H = -\frac{1}{2}\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right) +g\,\delta(x_1-x_2), \qquad x_j \sim x_j+2\pi,$$7

The transported local unitary on subsystem $$H = -\frac{1}{2}\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right) +g\,\delta(x_1-x_2), \qquad x_j \sim x_j+2\pi,$$8 is then

$$H = -\frac{1}{2}\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right) +g\,\delta(x_1-x_2), \qquad x_j \sim x_j+2\pi,$$9

The induced holonomy group for a state $L=2\pi$0 is defined as

$L=2\pi$1

generated by all sequences of local $L=2\pi$2-side operations satisfying $L=2\pi$3. The construction is state-dependent because $L=2\pi$4 depends on the full bipartite Stokes tensor, and it is generally non-Abelian because $L=2\pi$5 is matrix-valued in $L=2\pi$6, so successive infinitesimal steps need not commute.

The decisive result is that correlations induce the non-Abelianity. For product states,

$L=2\pi$7

and the maximizing operation is

$L=2\pi$8

so only a scalar $L=2\pi$9 compensation is required and the holonomy is Abelian. For correlated states, by contrast, the response depends on $\hbar=1$0 and generally spans noncommuting generators. The paper emphasizes that, for mixed states, $\hbar=1$1 captures general correlations rather than entanglement alone; separable but correlated mixed states may therefore already yield nontrivial non-Abelian behavior (Johansson et al., 2010).

For pure two-qubit states under local $\hbar=1$2 evolutions, the formalism contains Lévay’s geometric phase. In that specialization, the infinitesimal transport law reduces to

$\hbar=1$3

and the Franson intensity becomes identical to the quaternionic Mach–Zehnder intensity used in Lévay’s construction. The two-particle holonomy here is therefore not a consequence of spectral degeneracy, but of the tensor-product structure and the correlation matrix of the bipartite state (Johansson et al., 2010).

4. Entanglement-dependent holonomic phase of two entangled photons

A third line of work studies the holonomic phase of a genuinely entangled two-particle state under local unitary evolution. The theoretical starting point is a pure two-qubit state in Schmidt form,

$\hbar=1$4

with tangle

$\hbar=1$5

which for pure two-qubit states equals the square of the concurrence, $\hbar=1$6 (Loredo et al., 2013). The holonomic phase is defined kinematically as

$\hbar=1$7

where

$\hbar=1$8

For the “Schmidt evolution,” in which both qubits are locally rotated about their preferred Bloch directions, the entanglement-induced holonomic phase is

$\hbar=1$9

For a full $1$0 Schmidt cycle this becomes

$1$1

Hence the phase vanishes for separable states and is nonzero for entangled states, with a monotonic dependence on $1$2 for this class of evolutions.

The experiment implemented a zero-dynamical-phase evolution of polarization-entangled photons prepared in the family

$1$3

The local unitary applied to each photon was

$1$4

so the bipartite evolution was $1$5. Because the paths were concatenations of geodesics, the dynamical phase vanished identically. The measured phase was therefore the Pancharatnam phase of the bipartite state and thus directly the holonomic phase.

For this experimentally realized class of evolutions,

$1$6

or equivalently,

$1$7

with sign determined by which Schmidt component is more populated. At maximal entanglement, $1$8, and the phase reduces to the discrete values

$1$9

The phase therefore changes from continuously tunable geometric behavior to discrete topological behavior as the entanglement increases (Loredo et al., 2013).

The experiment used a displaced-Sagnac interferometer. The coincidence probability was

$g: 0 \to +\infty \to -\infty \to 0,$0

and was rewritten as

$g: 0 \to +\infty \to -\infty \to 0,$1

with visibility $g: 0 \to +\infty \to -\infty \to 0,$2. The holonomic phase was extracted from the shift of the coincidence fringe as $g: 0 \to +\infty \to -\infty \to 0,$3 was scanned. In the low-entanglement regime, with fitted tangle

$g: 0 \to +\infty \to -\infty \to 0,$4

the phase behaved as the sum of two local geometric phases. In the high-entanglement regime, with fitted tangle

$g: 0 \to +\infty \to -\infty \to 0,$5

the phase clustered near $g: 0 \to +\infty \to -\infty \to 0,$6 and $g: 0 \to +\infty \to -\infty \to 0,$7; independent tomography gave

$g: 0 \to +\infty \to -\infty \to 0,$8

The observed flattening of the phase versus path parameter $g: 0 \to +\infty \to -\infty \to 0,$9 was interpreted as increased resilience to evolution changes with increasing entanglement (Loredo et al., 2013).

5. Emergent purely geometric two-particle subspace holonomies

A more recent framework defines quantum holonomies as purely geometric transformations within cyclic subspaces. If a time-dependent subspace returns to itself after an evolution time $+\infty\to-\infty$0, the subspace evolution is

$+\infty\to-\infty$1

where $+\infty\to-\infty$2 is the geometric connection and $+\infty\to-\infty$3 is the dynamical contribution (Neef et al., 16 Jun 2025). When $+\infty\to-\infty$4, the transformation is a holonomy in the strict sense used in that work.

The paper’s central claim is that increasing the particle number can generate new holonomies even when the corresponding single-particle system has no holonomy in the relevant sector. For a single particle in $+\infty\to-\infty$5 modes, the Hilbert space is $+\infty\to-\infty$6-dimensional. For two particles, the dimension becomes $+\infty\to-\infty$7 for indistinguishable bosons, $+\infty\to-\infty$8 for indistinguishable fermions, and $+\infty\to-\infty$9 for distinguishable particles. For the experimentally studied case $\hat K=0$00, the dimensions are $\hat K=0$01, $\hat K=0$02, and $\hat K=0$03, respectively (Neef et al., 16 Jun 2025). The enlargement of Hilbert space creates many more candidate cyclic subspaces.

For a linear Hermitian coupled-mode Hamiltonian,

$\hat K=0$04

the single-particle holonomic condition requires the effective couplings $\hat K=0$05 inside the cyclic subspace to vanish. For two particles, however, the dynamical matrix elements are statistics-dependent combinations. In the notation of the paper,

$\hat K=0$06

with upper and lower signs for bosons and fermions. The two-particle geometric connection likewise lifts from the one-particle one according to

$\hat K=0$07

Because the dynamical contribution is filtered through Kronecker deltas, occupation patterns, and exchange symmetry, a carefully chosen two-particle subspace may satisfy $\hat K=0$08 even when some one-particle couplings remain nonzero (Neef et al., 16 Jun 2025).

The experimental platform was an integrated photonic four-waveguide structure fabricated by femtosecond laser direct writing in fused silica. The Hamiltonians were shaped so that

$\hat K=0$09

and the ideal cycle occurred at $\hat K=0$10, where $\hat K=0$11. At this length the single-particle propagator implemented a double flip between modes $\hat K=0$12 and $\hat K=0$13. Because off-ideal device lengths break exact cyclicity, the experiment enclosed each subspace operationally by input-state selection, output post-selection, and renormalization to success and failure inside the intended subspace (Neef et al., 16 Jun 2025).

Several realized examples illustrate the emergent two-particle character. The single-photon outer-mode subspace $\hat K=0$14 was holonomic, with a plateau width of about $\hat K=0$15 mm. The outer two-photon subspace $\hat K=0$16 was also holonomic, again with a plateau width about $\hat K=0$17 mm. Restricting further to the bunched outer states $\hat K=0$18 yielded an even more stable holonomy, with plateau width about $\hat K=0$19 mm. By contrast, the inner single-photon subspace $\hat K=0$20 was cyclic but non-holonomic, with plateau width about $\hat K=0$21 mm, and the inner two-photon subspace $\hat K=0$22 remained non-holonomic, with plateau width only about $\hat K=0$23 mm. However, restricting to the inner bunched two-photon subspace $\hat K=0$24 removed the problematic dynamical contribution and produced a holonomy with plateau width about $\hat K=0$25 mm. The paper also realized a distinguishable-particle holonomic sub-subspace,

$\hat K=0$26

with measured plateau width about $\hat K=0$27 mm (Neef et al., 16 Jun 2025).

These examples establish that the two-particle phenomenon is not merely a tensor-product lift of a one-particle holonomy. The paper states that the four-waveguide structure supports $\hat K=0$28 distinct non-Abelian holonomies for two indistinguishable photons, with dimensions ranging from $\hat K=0$29 to $\hat K=0$30 basis states. Particle number and particle type thereby become explicit design parameters for holonomic control (Neef et al., 16 Jun 2025).

6. Topological formulations, mixed-state extensions, and conceptual boundaries

Several broader frameworks help organize the preceding results while also clarifying their limits. A topological formulation of exotic quantum holonomy identifies the essential object not as a phase but as an ordered set of eigenprojectors. In the two-level case, the ordered pair lives on $\hat K=0$31, the unordered pair lives on $\hat K=0$32, and the nontrivial holonomy is the $\hat K=0$33 monodromy that exchanges the two eigenspaces (Tanaka et al., 2014). The paper interprets the effect as a disclination of a director field and emphasizes that exotic quantum holonomy is a permutation action on eigenspaces rather than an ordinary Berry phase. It does not primarily study two-particle systems, but its covering-space and projector-based language provides a template for interpreting eigenspace permutations in composite settings (Tanaka et al., 2014).

This topological viewpoint is directly relevant to the two-body Lieb–Liniger problem. After complexifying the coupling $\hat K=0$34, the two-body energy sheets form a Riemann surface joined at exceptional points, and the authors argue that the real $\hat K=0$35 cycle can be viewed as the real remnant of a more general complex deformation whose encirclement of many exceptional points approximates the shift

$\hat K=0$36

In that sense, the real-axis spectral permutation is tied to hidden branch-point topology in the complexified coupling plane (Yonezawa et al., 2013).

A different extension concerns mixed-state holonomy. A general bundle-and-connection framework for mixed states treats the standard purification bundle

$\hat K=0$37

with structure group $\hat K=0$38, and yields both Uhlmann holonomy and a reduced-bundle holonomy reproducing the Sjöqvist et al. interferometric phase (Andersson, 2019). The thesis does not develop a dedicated two-particle theory, but it states that the formalism can be specialized directly to a bipartite Hilbert space $\hat K=0$39. This gives a general holonomy theory for the joint state of two particles, while leaving open the relation between joint-state holonomy, subsystem holonomies, and entanglement-specific structure (Andersson, 2019).

Taken together, these frameworks delimit the concept of two-particle quantum holonomy. Not every such holonomy is topological, not every one is non-Abelian, and not every one is a phase. In the existing literature, the defining two-particle feature may be an interaction-driven permutation of Bethe branches, a correlation-induced local-unitary response, an entanglement-dependent bipartite phase, or an emergent purely geometric transformation within a two-particle cyclic subspace. The common theme is that the holonomic object is controlled by structure that is absent, trivial, or incomplete at the one-particle level.