Odd Entropy: Multiple Meanings in Research

• Odd Entropy is a polysemous concept that denotes distinct entropy measures defined by odd structures, such as odd partial quotients, time-reversal parity, and odd replica indices.
• In continued fractions, odd entropy arises from algorithms with odd-digit constraints yielding constant entropy plateaus, while in stochastic thermodynamics odd-parity variables generate unique fluctuation contributions.
• In quantum information, odd entropy is obtained via odd replica continuations of the partial transpose and connects to holographic entanglement measures, with similar parity effects seen in combinatorial models.

Odd Entropy is not a single universally standardized quantity. In the arXiv literature, the expression denotes several technically distinct constructions whose common feature is an “odd” structure: odd partial quotients in continued fractions, odd-parity variables under time reversal in stochastic thermodynamics, odd replica limits of partially transposed density matrices in quantum information, and odd/even sector decompositions in combinatorial and lattice models. In one line of work it is the entropy of $\alpha$-continued fraction expansions with odd partial quotients and is exactly constant on a large parameter interval (Hartono et al., 2021). In another, it refers to parity-sensitive contributions to entropy production that arise only when odd variables such as velocities are present (Spinney et al., 2012, Lee et al., 2012, Yeo et al., 2015). In quantum many-body theory and holography, it denotes the odd entanglement entropy obtained by analytic continuation from odd Rényi moments of a partial transpose (Mollabashi et al., 2020, Basu et al., 2023, Biswas et al., 2023). The term therefore functions as a family resemblance rather than a single definition.

1. Terminological scope and shared structural motifs

The most important clarification is that the word “odd” does not mean the same thing across these literatures. In the theory of $\alpha$-continued fractions, “odd” refers to the digit set: the partial quotients are constrained to $\{1,3,5,\dots\}$ (Hartono et al., 2021). In stochastic thermodynamics, “odd” refers to variables that reverse sign under time reversal, such as velocity or momentum, with parity map $x^\dagger=\epsilon x$ or $Eq=(\epsilon_1 q_1,\dots,\epsilon_N q_N)$ (Spinney et al., 2012, Lee et al., 2012). In entanglement theory, “odd” refers to the replica branch: one studies $\mathrm{Tr}[(\rho_{AB}^{T_B})^{n_o}]$ for odd integers $n_o$ and analytically continues $n_o\to1$ (Mollabashi et al., 2020, Basu et al., 2023). In lattice-coloring and dimer models, “odd” labels a sector distinguished by chirality parity or width parity, and entropy is then compared between odd and even sectors (Cepas, 2016, Si, 2012).

A common misconception is that these uses define one transferable entropy notion. They do not. The continued-fraction entropy is a Kolmogorov–Sinai entropy for a dynamical system (Hartono et al., 2021). The stochastic-thermodynamic quantities are trajectory-level entropy productions and their decompositions (Spinney et al., 2012, Yeo et al., 2015). The quantum-information quantity is a mixed-state entanglement functional built from a partial transpose (Mollabashi et al., 2020). This suggests that “Odd Entropy” is best treated as a polysemous research term whose meaning is fixed by domain-specific constructions.

Another shared motif is that “oddness” often produces an additional invariant, sector, or branch that is invisible in the corresponding even-only formulation. Examples include the parity-sensitive term $\Delta S_3$ in stochastic thermodynamics (Spinney et al., 2012), the odd-replica continuation distinct from negativity’s even branch (Basu et al., 2023), and the odd/even chirality sectors on the hexagonal lattice that standard loop dynamics cannot connect (Cepas, 2016).

2. Odd entropy in $\alpha$-continued fractions with odd partial quotients

In the continued-fraction setting, Boca and Merriman introduced an “odd” $\alpha$0-continued fraction algorithm on the interval $\alpha$1 for $\alpha$2, where $\alpha$3 and $\alpha$4 (Hartono et al., 2021). For $\alpha$5, the transformation is

$\alpha$6

with $\alpha$7 and

$\alpha$8

Iteration yields a signed continued fraction

$\alpha$9

where the digits $\{1,3,5,\dots\}$0 are odd positive integers (Hartono et al., 2021).

The natural extension has universal form

$\{1,3,5,\dots\}$1

with invariant probability density

$\{1,3,5,\dots\}$2

For $\{1,3,5,\dots\}$3, and in fact for all $\{1,3,5,\dots\}$4, the natural extensions are metrically isomorphic (Hartono et al., 2021). Because Kolmogorov–Sinai entropy is invariant under metric isomorphism, the entropy is constant on that entire interval:

$\{1,3,5,\dots\}$5

The mechanism producing this plateau is explicit digit surgery. The isomorphisms are realized by singularizations and insertions that preserve the odd-digit constraint after suitable insertions. In the range $\{1,3,5,\dots\}$6 one uses identities such as

$\{1,3,5,\dots\}$7

together with an insertion restoring oddness; analogous adapted formulas apply for $\{1,3,5,\dots\}$8 (Hartono et al., 2021). These surgeries preserve the invariant density and induce measure-preserving bijections between the natural extension domains.

Outside the entropy plateau, the function $\{1,3,5,\dots\}$9 is non-monotone near $x^\dagger=\epsilon x$0. Matching is defined by the existence of $x^\dagger=\epsilon x$1 such that

$x^\dagger=\epsilon x$2

On intervals where the matching exponents are constant, the sign of $x^\dagger=\epsilon x$3 determines local monotonicity: if $x^\dagger=\epsilon x$4, then $x^\dagger=\epsilon x$5 is increasing; if $x^\dagger=\epsilon x$6, then $x^\dagger=\epsilon x$7 is constant; if $x^\dagger=\epsilon x$8, then $x^\dagger=\epsilon x$9 is decreasing (Hartono et al., 2021). Accordingly, in any neighborhood of $Eq=(\epsilon_1 q_1,\dots,\epsilon_N q_N)$0 there are intervals on which the entropy is increasing, decreasing, and constant.

This continued-fraction use of “Odd Entropy” is therefore a literal entropy theory for a one-parameter family of piecewise Möbius interval maps. It is unrelated to partial transposition or time-reversal parity, except at the level of terminology.

3. Odd entropy in stochastic thermodynamics with odd-parity variables

In stochastic thermodynamics, the phrase refers to entropy-production contributions that arise when the state space contains variables odd under time reversal, such as velocities (Spinney et al., 2012, Lee et al., 2012). For a Markovian trajectory $Eq=(\epsilon_1 q_1,\dots,\epsilon_N q_N)$1, time reversal must include parity reversal:

$Eq=(\epsilon_1 q_1,\dots,\epsilon_N q_N)$2

The total entropy production is defined by the path-probability ratio

$Eq=(\epsilon_1 q_1,\dots,\epsilon_N q_N)$3

and obeys the integral fluctuation theorem

$Eq=(\epsilon_1 q_1,\dots,\epsilon_N q_N)$4

(Spinney et al., 2012).

A central result is the three-way decomposition

$Eq=(\epsilon_1 q_1,\dots,\epsilon_N q_N)$5

Here $Eq=(\epsilon_1 q_1,\dots,\epsilon_N q_N)$6 is the excess or Hatano–Sasa part and satisfies

$Eq=(\epsilon_1 q_1,\dots,\epsilon_N q_N)$7

The generalized housekeeping part $Eq=(\epsilon_1 q_1,\dots,\epsilon_N q_N)$8 also satisfies an IFT for all Markovian stochastic dynamics,

$Eq=(\epsilon_1 q_1,\dots,\epsilon_N q_N)$9

while the transient housekeeping term

$\mathrm{Tr}[(\rho_{AB}^{T_B})^{n_o}]$0

does not obey an IFT in general and vanishes on average only in the stationary state: $\mathrm{Tr}[(\rho_{AB}^{T_B})^{n_o}]$1 (Spinney et al., 2012).

This decomposition identifies a specifically odd-parity contribution. When the stationary distribution is symmetric under parity, $\mathrm{Tr}[(\rho_{AB}^{T_B})^{n_o}]$2, one has $\mathrm{Tr}[(\rho_{AB}^{T_B})^{n_o}]$3 and the standard adiabatic/non-adiabatic split is recovered (Spinney et al., 2012). When that symmetry fails, $\mathrm{Tr}[(\rho_{AB}^{T_B})^{n_o}]$4 is the additional boundary-like term generated by odd variables. A related discrete-state analysis separates

$\mathrm{Tr}[(\rho_{AB}^{T_B})^{n_o}]$5

where $\mathrm{Tr}[(\rho_{AB}^{T_B})^{n_o}]$6 is activated whenever $\mathrm{Tr}[(\rho_{AB}^{T_B})^{n_o}]$7 and does not satisfy an IFT in general (Lee et al., 2012).

For continuous stochastic dynamics, the housekeeping entropy can be decomposed in infinitely many ways characterized by a parameter $\mathrm{Tr}[(\rho_{AB}^{T_B})^{n_o}]$8 (Yeo et al., 2015). One part, denoted in the paper as the detailed-balance-breaking housekeeping entropy, satisfies an IFT for arbitrary $\mathrm{Tr}[(\rho_{AB}^{T_B})^{n_o}]$9, while the remaining contribution is tied to parity asymmetry of the stationary state distribution and generally does not (Yeo et al., 2015). This suggests a robust structural distinction: entropy production from detailed-balance breaking can retain FT structure, whereas entropy production from odd-parity asymmetry typically cannot.

Active-matter models make these odd-parity effects concrete. In one-dimensional active Brownian dynamics,

$n_o$0

with $n_o$1 odd and $n_o$2 even under time reversal, the reservoir entropy production contains a Jacobian-generated term

$n_o$3

and a cross-coupling term

$n_o$4

These are explicitly identified as excess odd contributions beyond the Clausius form (Chaudhuri, 2016). Numerical simulations confirm the detailed fluctuation theorem for the total entropy production in this setting (Chaudhuri, 2016).

A data-driven extension estimates entropy production with odd-parity variables from trajectory data using multiple neural networks (Kim et al., 2021). For underdamped and odd-parity Markov jump systems, the estimator must learn parity-sensitive pieces such as mirror-state asymmetry and waiting-time distribution asymmetry; even-only estimators are biased because the reversed dataset is mis-specified (Kim et al., 2021).

4. Odd entanglement entropy in quantum information, CFT, and holography

In quantum-information and field-theoretic usage, Odd Entropy usually means odd entanglement entropy, abbreviated OEE. For a bipartite state with reduced density matrix $n_o$5, one defines the partial transpose by

$n_o$6

The odd Rényi entropy is

$n_o$7

and the odd entanglement entropy is its odd-branch continuation,

$n_o$8

(Basu et al., 2023, Mollabashi et al., 2020). This differs from logarithmic negativity, which uses an even replica sequence and continuation $n_o$9 (Basu et al., 2023).

For Gaussian free scalar theories, OEE has a closed covariance-matrix form. If $n_o\to1$0 are the partially transposed symplectic eigenvalues of $n_o\to1$1, then

$n_o\to1$2

with

$n_o\to1$3

For pure states, $n_o\to1$4 (Mollabashi et al., 2020). In the free-boson study, $n_o\to1$5 and $n_o\to1$6 were observed numerically, while $n_o\to1$7 was found to be non-monotonic in interval size and separation and could become negative when classical correlations dominate (Mollabashi et al., 2020). This is an important qualification: outside holographic regimes, $n_o\to1$8 is not a universal positive correlation measure.

In large-$n_o\to1$9 holographic CFT$\Delta S_3$0, the proposal

$\Delta S_3$1

relates OEE to the entanglement wedge cross section (Basu et al., 2023, Biswas et al., 2023). Multiple works extend this relation. In $\Delta S_3$2-deformed CFT$\Delta S_3$3s, a replica technique computes leading $\Delta S_3$4-corrections to OEE for disjoint intervals, adjacent intervals, and single intervals in thermal states, and the high-temperature holographic calculation in finite-cutoff BTZ reproduces the field-theory result (Basu et al., 2023). In BCFT$\Delta S_3$5, replica computations for static and moving-mirror configurations give formulas for $\Delta S_3$6 that match the bulk EWCS in AdS$\Delta S_3$7/BCFT$\Delta S_3$8 (Kumari et al., 2023). A fully covariant AdS$\Delta S_3$9/CFT$\alpha$0 construction extends this to time-dependent states in pure AdS$\alpha$1, BTZ, and rotating BTZ geometries, with large-$\alpha$2 agreement up to additive constants (Biswas et al., 2023).

The construction also extends beyond relativistic CFT$\alpha$3. In Galilean conformal field theories, the odd replica correlator yields OEE formulas involving GCFT cross ratios $\alpha$4, and the difference $\alpha$5 matches the entanglement wedge cross section in asymptotically flat holography (Basak et al., 2022). In thermofield double states of free scalar QFTs, OEE exhibits linear growth followed by saturation, with finite-size oscillations on the circle and a zero-mode-induced logarithmic growth regime for a single degree of freedom on each side (Ghasemi et al., 2021).

Topological theories provide a further refinement. In $\alpha$6-dimensional Chern–Simons theory, the regulated odd entropy

$\alpha$7

obeys

$\alpha$8

with $\alpha$9 a classical Shannon term from fluctuating anyon sectors (Berthiere et al., 2020). In the cases studied, reflected entropy coincides with mutual information, while the odd entropy and reflected entropy both admit simple EWCS interpretations up to that classical piece (Berthiere et al., 2020).

5. Other uses: topological sectors, parity effects, and odd-dimensional entropy

In several combinatorial and statistical-mechanical settings, odd entropy denotes an entropy restricted to an odd sector rather than a new entropy functional. On the hexagonal lattice with Baxter’s three-coloring constraint, local chiralities $\alpha$00 define total chirality

$\alpha$01

and a $\alpha$02 invariant

$\alpha$03

distinguishes even and odd sectors (Cepas, 2016). Standard two-color loop flips preserve $\alpha$04, making standard Monte Carlo nonergodic across parity sectors. By adding triple-stranded loop moves, one can change parity and restore ergodicity (Cepas, 2016). The principal thermodynamic conclusion is that odd and even sectors have the same entropy density in the thermodynamic limit,

$\alpha$05

even though finite-size state counts generally differ and depend strongly on geometry and aspect ratio (Cepas, 2016).

A related parity-sensitive entropy effect appears in the finite square-lattice dimer film model. The melting temperature is determined by

$\alpha$06

and in the thermodynamic limit

$\alpha$07

with $\alpha$08 (Si, 2012). Because the finite-size entropy per site differs between even and odd widths, the paper reports $\alpha$09 at fixed $\alpha$10 (Si, 2012). Fusion of two toroidal films shows a parity-dependent entropy change: fusing two small toruses with even number of length reduces entropy, while fusing two small toruses with odd number of length increases entropy (Si, 2012).

Odd-dimensional entanglement entropy with boundaries provides a different use of the word. For odd-dimensional CFTs without boundary, there is no local Weyl anomaly and no universal logarithmic term in entanglement entropy. With boundaries, however, an integrated boundary anomaly exists, and when the entangling surface crosses the boundary there is a logarithmic contribution (Fursaev et al., 2016). In $\alpha$11, the integrated anomaly is

$\alpha$12

and the anomaly-derived logarithmic coefficient is

$\alpha$13

where $\alpha$14 is the number of intersection points and $\alpha$15 are intersection angles (Fursaev et al., 2016). This is not an “odd entropy” in the partial-transpose sense, but it is an entropy phenomenon specific to odd spacetime dimension.

A non-equilibrium continuum-mechanics use appears in passive odd viscoelasticity. There, an entropy-production analysis shows that odd viscosity is nondissipative and that passive chiral viscoelastic fluids can exhibit odd viscoelastic responses, whereas passive linear chiral solids cannot host odd elastic moduli (Lier et al., 2021). The paper explicitly characterizes which parity-odd coefficients contribute to entropy production and which do not, again linking “odd” structure to a refined entropy balance (Lier et al., 2021).

6. Comparative perspective and unresolved directions

Across these fields, Odd Entropy usually signals that a standard construction has been split by an odd branch, odd sector, or odd-parity symmetry. In dynamical systems, odd partial quotients produce an isomorphism class with a large constant-entropy plateau and matching-induced non-monotonicity near $\alpha$16 (Hartono et al., 2021). In stochastic thermodynamics, odd variables force a refinement of entropy production into pieces with different fluctuation-theorem status (Spinney et al., 2012, Yeo et al., 2015). In entanglement theory, odd replica continuation yields a mixed-state quantity closely related to EWCS in holographic regimes but not universally positive after subtracting $\alpha$17 in generic free theories (Mollabashi et al., 2020, Biswas et al., 2023). In combinatorial models, odd and even sectors can have different finite-size counts yet identical entropy densities asymptotically (Cepas, 2016).

Several open directions are explicit in the literature. For odd $\alpha$18-continued fractions, it is natural to conjecture that matching holds for Lebesgue almost every $\alpha$19, but proving this would require a finer description of the exceptional parameter set where matching fails (Hartono et al., 2021). In stochastic thermodynamics, the role of parity asymmetry in continuous dynamics leads to infinitely many housekeeping decompositions characterized by a single parameter, and the precise operational status of those decompositions remains subtle (Yeo et al., 2015). In holography and mixed-state entanglement, the large-$\alpha$20 relation $\alpha$21 is well supported in several settings, but additive constants, non-universal functions, and the behavior away from the holographic limit remain active issues (Basu et al., 2023, Biswas et al., 2023).

The strongest general conclusion is therefore negative in the best encyclopedic sense: Odd Entropy is not a single theory. It is a recurring label for entropy-like quantities whose technical content depends on what is odd—digits, parity under time reversal, replica index, chirality sector, or spatial dimension. The term acquires precision only within the formalism that defines it.