Information Convex Set in Quantum Topology

• Information convex set is defined as the convex collection of locally energy-minimizing reduced density matrices that characterizes local and global quantum topological features.
• Its convex structure and extremal points encode superselection sectors, anyon properties, and quantum dimensions through restriction maps and convex decompositions.
• Applications include analyzing 2D topological order, deriving topological entanglement entropy, and designing experimental protocols for probing quantum many-body systems.

The information convex is a set of reduced density matrices associated with a subsystem of a many-body quantum system, defined by local energy-minimization conditions rather than by a single global ground-state wavefunction. In the setting of 2D topological order governed by frustration-free local Hamiltonians, it provides a quantum-information characterization of bulk anyons, boundary superselection sectors, quantum dimensions, and bulk-to-boundary condensation rules through the convex geometry of locally consistent reduced states (Shi et al., 2018). In that literature, the object is denoted $\Sigma(\Omega)$ for a region $\Omega$. The same phrase also appears in unrelated convex-analysis and information-design contexts, where it refers respectively to characteristic-cone constructions for convex sets and to achievable expected-value sets under Bayesian information design (Mehdiloozad et al., 2015, Zhong, 2018).

1. Formal definition and basic structure

For a lattice model with a local, frustration-free Hamiltonian

$H=\sum_i h_i,$

each $h_i$ is Hermitian, supported on a finite-radius neighborhood, and has minimal eigenvalue $0$. Frustration-free means every ground state $|\psi\rangle$ satisfies $h_i|\psi\rangle=0$ for all $i$, so $\operatorname{tr}(h_i\rho)=0$ for the ground-state density matrix $\rho=|\psi\rangle\langle\psi|$ (Shi et al., 2018).

Given a region $\Omega$0, containing all local terms that overlap a smaller region $\Omega$1, one defines the restricted local Hamiltonian $\Omega$2 from those $\Omega$3 supported within $\Omega$4. A state $\Omega$5 is a local energy minimizer on $\Omega$6 when

$\Omega$7

The information convex is then

$\Omega$8

The notation $\Omega$9 denotes the minimal choice of $H=\sum_i h_i,$0 that contains all terms overlapping $H=\sum_i h_i,$1.

This construction filters reduced states on $H=\sum_i h_i,$2 to those compatible with a locally gapped environment. Its geometric role is immediate: $H=\sum_i h_i,$3 is convex, compact, and embedded in a finite-dimensional real vector space isomorphic to the Hermitian matrices on the Hilbert space of $H=\sum_i h_i,$4. If $H=\sum_i h_i,$5, then for any $H=\sum_i h_i,$6,

$H=\sum_i h_i,$7

Moreover, every extremal point of $H=\sum_i h_i,$8 has a purification in $H=\sum_i h_i,$9 by a pure state $h_i$0 that is itself a local minimizer.

A common misconception is that topological characterization from local reduced states must begin with ground-state wavefunctions alone. The information convex formulation explicitly works with density matrices and therefore organizes local sectors directly at the reduced-state level.

2. Convex geometry, extremal points, and restriction maps

The information convex is not merely a collection of locally admissible reduced states; its extremal structure is the carrier of superselection data. Extremal points correspond to pure local-minimizing sectors, and their labels and overlaps are determined by bulk or boundary superselection sectors and their quantum dimensions (Shi et al., 2018).

For nested regions $h_i$1, there is a natural restriction map

$h_i$2

This map is linear, completely positive and trace-preserving, and surjective. It preserves convex structure in the sense that images of line segments are line segments. Physically, it implements forgetting degrees of freedom while retaining the local minimizing condition. Faces map to faces, and extremal points in the larger region can map either to extremal points or to convex combinations in the smaller region, depending on whether the corresponding superselection data remain distinguishable inside $h_i$3.

This suggests a useful operational interpretation: inclusion maps encode the loss of topological distinguishability under coarse localization. In particular, when a region is deformed so that it touches a gapped boundary, the same CPTP structure constrains how bulk labels reduce to boundary labels. In the topological-order setting, the geometry is therefore not ancillary to the physics; it is a representation of physically allowed sector reduction under partial trace.

3. Bulk annuli, anyon sectors, and topological entanglement entropy

Let $h_i$4 be a bulk annulus. In a 2D topological order governed by a frustration-free commuting-projector Hamiltonian, $h_i$5 is a simplex whose extremal points are in one-to-one correspondence with bulk superselection sectors $h_i$6 (Shi et al., 2018). Its convex decomposition is

$h_i$7

The extremal points satisfy orthogonality and overlap relations

$h_i$8

and

$h_i$9

where $0$0 is the quantum dimension of anyon $0$1 and $0$2 denotes the vacuum. Their entropies obey

$0$3

These entropy shifts are topological invariants of the annular topology and are independent of microscopic details or deformations of $0$4.

The entropy-maximizing center element is

$0$5

Partitioning the annulus as in the Levin-Wen construction yields

$0$6

A general lower bound is

$0$7

and in the frustration-free commuting-projector setting, and more generally under mild assumptions, the bound is saturated:

$0$8

The significance is twofold. First, $0$9 is encoded by the geometry of $|\psi\rangle$0 through entropy maximization. Second, topological entanglement entropy appears not as an independent diagnostic but as a derived invariant of the information convex. This rules out the misconception that the formalism merely reproduces sector labels; it also captures global quantum-dimension data.

4. Boundary regions, boundary sectors, and condensation rules

When the region touches a gapped boundary, the information convex resolves boundary superselection sectors. Let $|\psi\rangle$1 be a strip annulus adjacent to an untwisted gapped boundary specified by a subgroup $|\psi\rangle$2. Then

$|\psi\rangle$3

where $|\psi\rangle$4 runs over deconfined boundary excitations (Shi et al., 2018).

The extremal boundary sectors satisfy

$|\psi\rangle$5

$|\psi\rangle$6

and

$|\psi\rangle$7

For $|\psi\rangle$8 with untwisted boundary $|\psi\rangle$9, boundary labels are

$h_i|\psi\rangle=0$0

where $h_i|\psi\rangle=0$1 is a double coset and $h_i|\psi\rangle=0$2 is an irrep of

$h_i|\psi\rangle=0$3

for a representative $h_i|\psi\rangle=0$4. The corresponding quantum dimension is

$h_i|\psi\rangle=0$5

with $h_i|\psi\rangle=0$6.

Bulk-to-boundary condensation is encoded by the restriction map. If a bulk anyon $h_i|\psi\rangle=0$7 decomposes at the boundary as

$h_i|\psi\rangle=0$8

with nonnegative integers $h_i|\psi\rangle=0$9, then the relation between extremal density matrices is

$i$0

In this way, condensation multiplicities, boundary sector content, and quantum dimensions are recovered directly from reduced density matrices.

A plausible implication is that the information convex provides a boundary-sensitive substitute for more operator-centric diagnostics. The data explicitly state that it recovers condensation rules without requiring string operators or modular matrices directly.

5. Quantum doubles and explicit solvable examples

For quantum doubles $i$1, bulk anyons are labeled by pairs

$i$2

where $i$3 is a conjugacy class in $i$4 with representative $i$5 and $i$6 is an irrep of the centralizer

$i$7

Their quantum dimension is

$i$8

and the total quantum dimension satisfies

$i$9

The paper constructs extremal elements explicitly by minimal diagram reductions: for a bulk annulus, $\operatorname{tr}(h_i\rho)=0$0 is obtained by applying appropriate ribbon operators and tracing out complements; for boundary-adjacent regions, $\operatorname{tr}(h_i\rho)=0$1 is obtained similarly using boundary ribbons (Shi et al., 2018).

Two examples organize the general structure.

Model	Key bulk structure	Key boundary structure
Toric code, $\operatorname{tr}(h_i\rho)=0$2	Anyons $\operatorname{tr}(h_i\rho)=0$3, all with $\operatorname{tr}(h_i\rho)=0$4 and $\operatorname{tr}(h_i\rho)=0$5	Rough boundary: $\operatorname{tr}(h_i\rho)=0$6 and $\operatorname{tr}(h_i\rho)=0$7 is confined; smooth boundary: $\operatorname{tr}(h_i\rho)=0$8 and $\operatorname{tr}(h_i\rho)=0$9 is confined
$\rho=|\psi\rangle\langle\psi|$0 quantum double, $\rho=|\psi\rangle\langle\psi|$1	Eight sectors with $\rho=|\psi\rangle\langle\psi|$2	Six boundary extremal points with $\rho=|\psi\rangle\langle\psi|$3; bulk processes reach only conjugacy-class averages

In the toric code, $\rho=|\psi\rangle\langle\psi|$4 has four extremal points $\rho=|\psi\rangle\langle\psi|$5, all with identical entropy because $\rho=|\psi\rangle\langle\psi|$6, and $\rho=|\psi\rangle\langle\psi|$7. For a rough boundary, boundary sectors satisfy $\rho=|\psi\rangle\langle\psi|$8 with $\rho=|\psi\rangle\langle\psi|$9, the charge $\Omega$00 condenses to vacuum, and $\Omega$01 is confined. For a smooth boundary, the roles of $\Omega$02 and $\Omega$03 are reversed.

The non-Abelian $\Omega$04 example is more intricate. The bulk sectors are

$\Omega$05

with quantum dimensions listed above. The boundary extremals are labeled by $\Omega$06 and all satisfy $\Omega$07. However, the subset accessible by bulk processes is only

$\Omega$08

with

$\Omega$09

Thus bulk processes do not reach all boundary extremals for non-Abelian $\Omega$10; boundary-string processes do.

The corresponding condensation rules include

$\Omega$11

$\Omega$12

A particularly distinctive feature appears for the boundary annulus $\Omega$13: $\Omega$14 has extremal points $\Omega$15, $\Omega$16, and a continuous family $\Omega$17 lying on a sphere $\Omega$18 due to the multiplicity-$\Omega$19 condensation $\Omega$20. Their overlap relation is

$\Omega$21

and the entropies satisfy

$\Omega$22

Boundary loop unitaries $\Omega$23 act by rotations on this $\Omega$24 family, realizing the $\Omega$25 group action.

6. Robustness, experimental access, and limitations

The convex structure of $\Omega$26 depends only on the topology of $\Omega$27 and on how $\Omega$28 meets the boundary. The labels, orthogonality relations, and entropy shifts are stable under continuous deformations of $\Omega$29 and under small local deformations of the Hamiltonian within the phase (Shi et al., 2018). For gapped phases with finite correlation length $\Omega$30, the construction can be generalized to allow a small local energy density $\Omega$31:

$\Omega$32

The deviation from the $\Omega$33 convex set is controlled by the fidelity bound

$\Omega$34

where $\Omega$35 is the local gap of $\Omega$36. Choosing $\Omega$37 thicker than $\Omega$38 by several $\Omega$39 keeps $\Omega$40 close in fidelity to $\Omega$41 and preserves the convex-topological structure.

The formalism also has a direct experimental interpretation through overlap measurements. Interference of two copies can probe

$\Omega$42

The protocol described in the data consists of preparing two identical systems and two states $\Omega$43, cooling subsystems $\Omega$44 while keeping excitations outside $\Omega$45, and measuring the overlap via swap/interference. Predicted signals include

• for a bulk annulus $\Omega$46 with anyon $\Omega$47 in one copy and vacuum in the other,

$\Omega$48

• for a boundary region $\Omega$49 with boundary excitation $\Omega$50 in one copy and vacuum in the other,

$\Omega$51

• for the $\Omega$52 family on $\Omega$53,

$\Omega$54

The data also delimit the scope of current explicit constructions. They assume frustration-free, commuting-projector Hamiltonians, specifically quantum doubles with untwisted gapped boundaries. Extensions to twisted boundaries, gapless edges, non-commuting local terms, and 3D topological orders remain open. While the framework is robust under small perturbations in gapped phases, precise bounds and generalizations to long-range correlated models are still under investigation.

7. Terminological disambiguation across research literatures

The phrase information convex set has distinct meanings in other arXiv literatures. In convex analysis, a related object is the characteristic cone of a set

$\Omega$55

used to study decomposition of closed convex sets containing no line, maximal elements, relative interior points of polyhedra, and strictly complementary solutions in linear programming (Mehdiloozad et al., 2015). There the central statements concern correspondences between extreme points and directions of $\Omega$56 and vertical or horizontal extreme rays of $\Omega$57, together with the representation

$\Omega$58

In information design, the phrase denotes the set of achievable expected objective vectors under Bayes-plausible signal structures. For finite state space $\Omega$59, prior $\Omega$60, and continuous objectives $\Omega$61, the set is

$\Omega$62

which is nonempty, convex, compact, and continuous as a correspondence in $\Omega$63 (Zhong, 2018). That usage is tied to Bayesian persuasion, finite-support implementation, and Lagrange concavification, not to topological order.

Accordingly, “information convex set” is not a universally fixed technical term. In the topological-order literature, it specifically denotes $\Omega$64, the convex set of locally energy-minimizing reduced density matrices. In optimization and information design, it refers to different convex-geometric constructions. The shared terminology reflects convexity and information constraints, but the mathematical objects, ambient spaces, and physical or economic interpretations are distinct.