Bipartite Stabilizer Mutual Information

• Bipartite stabilizer mutual information is defined as I(A:B)=S(A)+S(B)-S(AB) for stabilizer states, leveraging flat entanglement spectra and algebraic reductions such as binary ranks.
• The decomposition into Bell pair and GHZ components clarifies how quantum entanglement and classical correlations contribute distinctly to mutual information in these systems.
• In tensor network and graph state frameworks, geometric minimal cuts and code-theoretic methods enable practical, operational computations of this entropic quantity.

Searching arXiv for recent and foundational papers on bipartite stabilizer mutual information, stabilizer tensor networks, noisy stabilizer states, and graph-state monogamy. arXiv search query: "bipartite stabilizer mutual information stabilizer tensor networks graph states noisy stabilizer states" Bipartite stabilizer mutual information is the mutual information

$I(A:B)=S(A)+S(B)-S(AB)$

evaluated on stabilizer states, graph states, stabilizer tensor networks, or noisy stabilizer marginals. The subject is distinctive because stabilizer structure renders the relevant entropies unusually rigid: flat entanglement spectra reduce von Neumann entropies to second moments, graph-state entropies reduce to binary ranks, and noisy spectra reduce to syndrome distributions of associated stabilizer codes. In random stabilizer tensor networks, subsystem entropies are governed by Ryu–Takayanagi minimal cuts and the genuinely tripartite GHZ contribution is only $O(1)$, so $I(A:B)/2$ is, up to $O(1)$ corrections, approximately the number of extractable Bell pairs between $A$ and $B$ (Nezami et al., 2016). Under Pauli or dephasing noise, the same quantity is computable from orthogonal conjugate-code sectors indexed by syndromes or cosets (Goodenough et al., 2024), and for graph states it reduces to ranks of adjacency submatrices over $\mathbb Z_2$ (Fuentes et al., 24 Nov 2025).

1. Definitions and stabilizer entropy identities

In the stabilizer tensor-network setting, the entropy is the usual von Neumann entropy, measured in units of $\log_p$ bits,

$S(A)=-\operatorname{tr}\rho_A\log_p\rho_A,$

and the bipartite mutual information is

$I(A:B)=S(A)+S(B)-S(AB).$

The associated tripartite information is

$O(1)$0

or, for a global pure state,

$O(1)$1

A stabilizer-specific simplification is that stabilizer states have flat entanglement spectrum, so the von Neumann entropy equals the Rényi-2 entropy: $O(1)$2 For the unnormalized tensor-network state $O(1)$3, with normalized state $O(1)$4, the entropy identity

$O(1)$5

underlies the second-moment derivation of the entropy formula in random stabilizer tensor networks (Nezami et al., 2016).

These identities already indicate why bipartite mutual information is especially tractable in the stabilizer setting. The general quantity $O(1)$6 is defined in the standard entropic way, but stabilizer structure converts the required subsystem entropies into algebraic objects that can be computed from second moments, binary ranks, or code spectra rather than from generic spectral decompositions.

2. Exact decomposition in tripartite stabilizer states

A central structural fact is that any pure tripartite stabilizer state $O(1)$7 is locally equivalent, up to local product factors, to

$O(1)$8

Here $O(1)$9 count maximally entangled $I(A:B)/2$0-level Bell pairs shared by the corresponding pairs, and $I(A:B)/2$1 counts $I(A:B)/2$2-dimensional GHZ triples. For this normal form,

$I(A:B)/2$3

This formula isolates two distinct contributions to bipartite mutual information: a quantum-entanglement contribution $I(A:B)/2$4 from Bell pairs, and a GHZ contribution $I(A:B)/2$5, which is classical-correlation content because tracing out one party of GHZ gives a classically correlated separable state on $I(A:B)/2$6. The same normal form obeys the entropy sum rule

$I(A:B)/2$7

An exact GHZ extraction formula is

$I(A:B)/2$8

Equivalently,

$I(A:B)/2$9

These relations provide the precise sense in which bipartite mutual information in a stabilizer state need not be purely bipartite entanglement (Nezami et al., 2016).

The conceptual significance is twofold. First, the formula $O(1)$0 rules out the naive identification of mutual information with distillable Bell content for arbitrary stabilizer states. Second, it makes the obstruction explicit: the contamination comes from inherited correlations of GHZ type, not from an undifferentiated many-body remainder. The tripartite stabilizer normal form therefore supplies an exact benchmark against which random-network and noisy-state results can be interpreted.

3. Geometric control in random stabilizer tensor networks

A random stabilizer tensor network is defined on a graph with bulk vertices $O(1)$1 and boundary vertices $O(1)$2. One places random stabilizer states $O(1)$3 on bulk vertices and contracts them with maximally entangled edge states $O(1)$4: $O(1)$5 The normalized boundary state is $O(1)$6. For a boundary region $O(1)$7, the entropy satisfies the universal upper bound

$O(1)$8

where $O(1)$9 is a cut separating $A$0 from $A$1, and each edge carries bond dimension $A$2. In the large-$A$3 limit, this bound is saturated on average,

$A$4

with $A$5 meaning equality up to $A$6, independent of $A$7, and the result can be strengthened to

$A$8

Since mutual information is built from subsystem entropies, the leading behavior is therefore geometric: $A$9 The appendix derives the entropy bound using the fact that random stabilizer states form a projective $B$0-design and the estimate

$B$1

which leads to a lower bound matching the universal upper bound up to $B$2 terms (Nezami et al., 2016).

This gives bipartite stabilizer mutual information a direct geometric interpretation in the tensor-network regime. The quantity is not merely an abstract correlation functional on a boundary state; to leading order it is fixed by minimal cuts of the underlying graph. This is the stabilizer-tensor-network analogue of the Ryu–Takayanagi relation and is the basic reason mutual information can be analyzed operationally in these ensembles.

4. Scarce GHZ content, monogamy, and higher-partite residual structure

The major multipartite result for random stabilizer tensor networks is that GHZ-type tripartite entanglement is scarce. For a tripartition $B$3 of the boundary, the expected number of extractable GHZ states is $B$4 in the large-$B$5 limit. More explicitly, for $B$6,

$B$7

with $B$8, $B$9 the numbers of minimal cuts, and $\mathbb Z_2$0 the maximal number of residual bulk connected components after removing minimal cuts. In typical situations with unique minimal cuts and one residual component,

$\mathbb Z_2$1

Because entropies and mutual informations generically scale like $\mathbb Z_2$2, while $\mathbb Z_2$3 does not, the GHZ contribution is subleading. Combining this with

$\mathbb Z_2$4

yields

$\mathbb Z_2$5

Operationally, $\mathbb Z_2$6 approximately counts extractable maximally entangled pairs, so mutual information becomes, up to $\mathbb Z_2$7 corrections, an entanglement measure in this ensemble. The technical route is a spin-model evaluation of

$\mathbb Z_2$8

using the identity

$\mathbb Z_2$9

and a third-moment formula for random stabilizer states that maps the average to a classical ferromagnetic spin model on the bulk vertices (Nezami et al., 2016).

The same result supplies an operational interpretation of the monogamy of mutual information,

$\log_p$0

equivalently

$\log_p$1

Because one half of the mutual information is, up to $\log_p$2 corrections, an entanglement measure and is approximately equal to quantities such as squashed entanglement $\log_p$3, monogamy of mutual information is interpreted as arising from monogamy of quantum entanglement rather than from a purely formal entropy inequality. The higher-partite extension is equally sharp. For a partition into $\log_p$4, the average number of maximally entangled pairs extractable between $\log_p$5 and $\log_p$6 is

$\log_p$7

For four parties, after extracting these Bell pairs pair by pair, one reaches a residual state $\log_p$8 whose pairwise mutual informations are all only $\log_p$9, and whose entropies satisfy

$S(A)=-\operatorname{tr}\rho_A\log_p\rho_A,$0

Thus pairwise mutual information captures the Bell-pair component, whereas $S(A)=-\operatorname{tr}\rho_A\log_p\rho_A,$1 diagnoses the genuinely four-partite remainder.

5. Noisy stabilizer states and code-theoretic computation

The code-theoretic treatment of noisy stabilizer states studies pure stabilizer states shared across a bipartition $S(A)=-\operatorname{tr}\rho_A\log_p\rho_A,$2, with Alice holding $S(A)=-\operatorname{tr}\rho_A\log_p\rho_A,$3 qubits and Bob $S(A)=-\operatorname{tr}\rho_A\log_p\rho_A,$4 qubits, under independent Pauli noise, specialized most explicitly to uniform depolarizing noise and uniform dephasing. The key representation uses code projectors

$S(A)=-\operatorname{tr}\rho_A\log_p\rho_A,$5

Under Pauli noise, a code projector is mapped to a convex mixture of orthogonal conjugate code projectors, with

$S(A)=-\operatorname{tr}\rho_A\log_p\rho_A,$6

and this equals $S(A)=-\operatorname{tr}\rho_A\log_p\rho_A,$7 iff $S(A)=-\operatorname{tr}\rho_A\log_p\rho_A,$8. Hence the noisy state is diagonal in orthogonal code sectors indexed by cosets of $S(A)=-\operatorname{tr}\rho_A\log_p\rho_A,$9. For a pure stabilizer state with stabilizer group $I(A:B)=S(A)+S(B)-S(AB).$0, the reduced state on Bob is

$I(A:B)=S(A)+S(B)-S(AB).$1

where contraction by $I(A:B)=S(A)+S(B)-S(AB).$2 means keeping only stabilizer elements with no support on $I(A:B)=S(A)+S(B)-S(AB).$3 and then removing the $I(A:B)=S(A)+S(B)-S(AB).$4-coordinates. Under uniform depolarizing noise, the probability of a conjugate code projector $I(A:B)=S(A)+S(B)-S(AB).$5 is

$I(A:B)=S(A)+S(B)-S(AB).$6

and the entropy formula is

$I(A:B)=S(A)+S(B)-S(AB).$7

where $I(A:B)=S(A)+S(B)-S(AB).$8 is the syndrome entropy. The paper’s primary focus is coherent information, but this spectral description is stronger: once the spectra of the noisy global state and noisy reduced states are known, the subsystem entropies and therefore the bipartite mutual information follow directly (Goodenough et al., 2024).

This makes noisy bipartite stabilizer mutual information computable from code data rather than by diagonalizing $I(A:B)=S(A)+S(B)-S(AB).$9 density matrices. The same mechanism applies to graph states under dephasing. In that case the relevant object is the biadjacency matrix $O(1)$00, and the paper states that $O(1)$01 can be interpreted as the generator matrix of a classical code. The resulting syndrome entropy controls the reduced-state entropy. A direct implication is that noisy bipartite mutual information in this regime reduces to classical linear-code data plus rank terms determined by the contraction or deletion codes. The paper also reports numerically that “better codes lead to more robust states” and that the robustness of stabilizer states is correlated with how good the associated codes are, for both dephasing and depolarizing noise.

6. Graph-state formulas, rank expressions, and pairwise-overlap structure

For qubit stabilizer states, entropy can be computed from the binary stabilizer tableau. An $O(1)$02-qubit stabilizer state is encoded by an $O(1)$03 binary tableau, and for a subsystem $O(1)$04 the entropy is

$O(1)$05

where $O(1)$06 is the rank of the projected tableau. A direct consequence is the bipartite mutual-information formula

$O(1)$07

for disjoint $O(1)$08 and $O(1)$09. For graph states the expression is even more transparent. If $O(1)$10 is the adjacency matrix and $O(1)$11 is the cut matrix, then

$O(1)$12

over $O(1)$13, so

$O(1)$14

Because every stabilizer state is local-Clifford equivalent to a graph state and local Clifford operations preserve all bipartite entropies, these graph-state formulas characterize stabilizer bipartite mutual information after passing to an LC-equivalent representative. In generalized-star geometries, the paper derives a sharper structural identity: $O(1)$15 and similarly for $O(1)$16 and $O(1)$17. Pairwise mutual information is therefore the dimension of the overlap of connectivity patterns into the center (Fuentes et al., 24 Nov 2025).

The graph-state analysis also isolates a limitation of pairwise mutual information. In the four-star $O(1)$18, which is local-Clifford equivalent to $O(1)$19, one has

$O(1)$20

and

$O(1)$21

so monogamy of mutual information fails. The same pattern occurs in star-like generalizations. By contrast, the state $O(1)$22 discussed in the same work has the same relevant pairwise mutual informations,

$O(1)$23

but satisfies

$O(1)$24

Thus equal pairwise mutual informations do not determine tripartite information or MMI behavior. This corrects a common overreading of pairwise rank formulas: they determine the bipartite contribution, but they do not exhaust the multipartite entropy structure.

7. Related operational frameworks: Choi-state scrambling and correlation-only measurement frames

Two adjacent lines of work place bipartite stabilizer mutual information in broader operational context. The first is the Choi-state analysis of bipartite unitaries. For a bipartite unitary $O(1)$25, the pure four-party Choi state satisfies

$O(1)$26

because $O(1)$27 and $O(1)$28. Exact vanishing of the conditional mutual information implies an exact Bell-pair product structure: $O(1)$29 equivalently a criss-cross normal form for the unitary. At the opposite extreme, maximal negativity of $O(1)$30 in the equal-dimension case means every bipartite reduction is maximally mixed, so the Choi state is a four-party perfect tensor or absolutely maximally entangled state. The paper notes that such maximally $O(1)$31-scrambling unitaries exist in sufficiently large prime dimension because a stabilizer state chosen at random will be a perfect tensor with high probability (Ding et al., 2016). This does not give a stabilizer-specific mutual-information formula, but it connects stabilizer entropic structure to routing, delocalization, and scrambling.

The second line is the construction of bipartite entangled stabilizer mutually unbiased bases. That work does not define mutual information in the Shannon or von Neumann sense; its central notion is mutual unbiasedness. Nevertheless, it is directly relevant to correlation-only measurement of bipartite stabilizer structure. The basis states are maximally entangled stabilizer states of the form

$O(1)$32

or more generally

$O(1)$33

Because these vectors are maximally entangled, each subsystem is maximally mixed, and the construction “provide[s] no local information” while being “sufficient and minimal” for decomposing locally maximally mixed operators. Mutual unbiasedness is reduced to the Clifford trace criterion

$O(1)$34

and complete BES MUBs correspond to maximum cliques of a Cayley graph on $O(1)$35 (Dam et al., 2011). The significance for bipartite stabilizer mutual information is terminological and methodological: entropic mutual information measures total correlations through subsystem entropies, whereas BES MUBs furnish a stabilizer-compatible measurement frame that isolates the nonlocal operator sector after local marginals have been projected out.

Together, these frameworks delimit the scope of bipartite stabilizer mutual information. In stabilizer tensor networks and graph states, $O(1)$36 is often computable by minimal cuts or binary ranks; in noisy stabilizer states it is computable from code spectra; in Choi-state settings it diagnoses routing versus delocalization; and in BES-MUB constructions it is not the quantity being measured, but the underlying correlation-only sector is isolated by maximally entangled stabilizer probes. The common theme is that stabilizer structure turns mutual-information questions into finite-field, code-theoretic, or graph-theoretic ones without eliminating the distinction between bipartite correlation and genuinely multipartite organization.