Mutual Robustness of Magic in Quantum Ising Chain

• Mutual robustness of magic is a bipartite measure derived from log-free robustness that quantifies non-stabilizer correlations beyond product state marginals.
• The method combines quantum Monte Carlo tomography with convex optimization to reconstruct reduced density matrices and compute robustness across phase transitions.
• This metric reveals distinct finite-temperature and dissipative behaviors compared to entanglement, highlighting regimes of magic death and rebirth in quantum systems.

to=arxiv_search уйғурлар.search ุ้นบาท content: {"query":"\"mutual robustness of magic\" robustness of magic Ising chain quantum monte carlo tomography", "max_results": 10} to=arxiv_search 手机版天天中彩票.search ՞նչ content: {"query":"mutual robustness of magic quantum Ising chain", "max_results": 10} to=arxiv_search 人人中彩票.search 手机天天中彩票 content: {"query":"mutual robustness of magic quantum Ising chain", "max_results": 10} Mutual robustness of magic is a bipartite quantity in the resource theory of non-stabilizer states, defined from the log-free robustness of magic of a joint state and its marginals. For two disjoint partitions $A$ and $B$, it is introduced as

$$MR(A:B)\equiv LR(\rho_{AB})-LR(\rho_A\otimes\rho_B),$$

where $LR(\rho)=\log_2 R(\rho)$ and $R(\rho)$ is the robustness of magic. By construction, $MR(A:B)\ge 0$, and $MR=0$ for product states. In the quantum Ising chain, this quantity has been used to study magic as a bipartite correlation across the quantum phase transition and at finite temperature, using a hybrid protocol that combines stochastic sampling of reduced density matrices via quantum Monte Carlo with estimators for the robustness of magic of mixed states (Timsina et al., 17 Jul 2025). Related work on dissipative dynamics places mutual robustness within a broader family of robustness-based magic diagnostics that behave differently from entanglement under noise, including regimes of magic death and rebirth under local amplitude damping (Cao, 21 May 2026).

1. Resource-theoretic setting

The underlying free set is the stabilizer polytope. On $n$ qubits, the set of free, or non-magical, states is

$$S=\operatorname{conv}\{\,|\phi\rangle\langle\phi|:\ |\phi\rangle\ \text{a pure stabilizer state}\,\}.$$

For an $n$-qubit mixed state $B$0, the robustness of magic is defined by expressing $B$1 as an affine, or pseudo-, mixture of pure stabilizer states $B$2 and minimizing the $B$3-norm of the coefficients: $B$4 An equivalent matrix formulation is

$B$5

with $B$6, $B$7, and $B$8 the Pauli basis on $B$9 qubits. The same resource admits a dual formulation,

$$MR(A:B)\equiv LR(\rho_{AB})-LR(\rho_A\otimes\rho_B),$$0

and, in the matrix notation used for convex optimization,

$$MR(A:B)\equiv LR(\rho_{AB})-LR(\rho_A\otimes\rho_B),$$1

The normalization is exact: $$MR(A:B)\equiv LR(\rho_{AB})-LR(\rho_A\otimes\rho_B),$$2 if and only if $$MR(A:B)\equiv LR(\rho_{AB})-LR(\rho_A\otimes\rho_B),$$3, while $$MR(A:B)\equiv LR(\rho_{AB})-LR(\rho_A\otimes\rho_B),$$4 quantifies how far $$MR(A:B)\equiv LR(\rho_{AB})-LR(\rho_A\otimes\rho_B),$$5 lies outside the stabilizer polytope. The log-free version,

$$MR(A:B)\equiv LR(\rho_{AB})-LR(\rho_A\otimes\rho_B),$$6

is subadditive under tensor products (Timsina et al., 17 Jul 2025).

Within this framework, mutual robustness of magic is the bipartite construction built from $$MR(A:B)\equiv LR(\rho_{AB})-LR(\rho_A\otimes\rho_B),$$7. It is not an independent primitive of the resource theory; rather, it is a derived quantity that isolates the excess log-free robustness in the joint state beyond that of the product of its marginals. This suggests an interpretation as a magic-correlation diagnostic, although the formal statement supplied in the literature is the nonnegativity condition and its vanishing on product states.

2. Definition of the mutual quantity

For two disjoint partitions $$MR(A:B)\equiv LR(\rho_{AB})-LR(\rho_A\otimes\rho_B),$$8 and $$MR(A:B)\equiv LR(\rho_{AB})-LR(\rho_A\otimes\rho_B),$$9, with joint state $LR(\rho)=\log_2 R(\rho)$0 and marginals $LR(\rho)=\log_2 R(\rho)$1 and $LR(\rho)=\log_2 R(\rho)$2, the mutual log-free robustness is defined as

$LR(\rho)=\log_2 R(\rho)$3

The construction is explicitly introduced for the study of magic as a bipartite correlation in the quantum Ising chain. The quantity is rigorous in the sense that it is built directly from a bona fide measure of magic for mixed states, namely the robustness of magic. The available study computes it for partitions up to $LR(\rho)=\log_2 R(\rho)$4 sites, embedded into a much larger system (Timsina et al., 17 Jul 2025).

Two structural properties are immediate from the definition and are stated explicitly. First, $LR(\rho)=\log_2 R(\rho)$5. Second, $LR(\rho)=\log_2 R(\rho)$6 for product states. These properties make $LR(\rho)=\log_2 R(\rho)$7 analogous in form to mutual-information-type constructions, but the resource being isolated is non-stabilizerness rather than entropy or entanglement. A plausible implication is that $LR(\rho)=\log_2 R(\rho)$8 is most naturally viewed as a correlation measure internal to the stabilizer-resource theory.

The formulation used in current work is specifically the mutual log-free robustness rather than a direct difference of $LR(\rho)=\log_2 R(\rho)$9 values. That choice matters because $R(\rho)$0 is the quantity stated to be subadditive under tensor products. The subtraction $R(\rho)$1 therefore has a direct algebraic fit with bipartite decomposition.

3. Quantum Monte Carlo tomography and numerical estimation

The principal computational setting to date is the quantum Ising chain, where mutual robustness is extracted by combining finite-temperature stochastic sampling with explicit reduced-state reconstruction. The protocol is described as follows.

First, the full spin chain of length $R(\rho)$2 with PBC is partitioned into three regions: $R(\rho)$3, $R(\rho)$4, and the environment $R(\rho)$5, where $R(\rho)$6 is the subsystem of interest. A standard finite-$R(\rho)$7 Stochastic Series Expansion QMC simulation of $R(\rho)$8 is then performed, except that in the imaginary-time propagation one imposes open, rather than periodic, boundary conditions on the spins in $R(\rho)$9. Each SSE configuration carries an initial and final basis-state string $MR(A:B)\ge 0$0 and $MR(A:B)\ge 0$1 on $MR(A:B)\ge 0$2, while $MR(A:B)\ge 0$3 remains periodic. The matrix element of the reduced density matrix is sampled by counting how often the bra-ket on $MR(A:B)\ge 0$4 are $MR(A:B)\ge 0$5: $MR(A:B)\ge 0$6 where $MR(A:B)\ge 0$7 is the total number of QMC samples and $MR(A:B)\ge 0$8 is the number of samples with $MR(A:B)\ge 0$9. Up to a common normalization, this reconstructs the full $MR=0$0 matrix $MR=0$1 (Timsina et al., 17 Jul 2025).

Once $MR=0$2 is obtained, the robustness optimization is solved by two different methods depending on subsystem size. “Naïve” linear programming is used for $MR=0$3 qubits. For $MR=0$4 up to $MR=0$5 qubits, the calculation uses a Column-Generation method. In that method, one starts from a small column set of stabilizers chosen by largest overlaps, solves the dual problem, identifies violated constraints $MR=0$6, adds those stabilizers, and reiterates until all constraints are satisfied.

This workflow is technically important because direct stabilizer enumeration becomes impractical as subsystem size grows. The hybrid QMC-tomography-plus-convex-optimization scheme is therefore the mechanism that makes the reported mutual-robustness calculations feasible for nontrivial embedded partitions.

4. Critical behavior in the quantum Ising chain

At the critical point, specified as $MR=0$7, the reported study analyzes how mutual robustness scales in the ground-state limit $MR=0$8. The detailed summary states that the calculation does not study $MR=0$9 as a function of the geometric distance $n$0 between $n$1 and $n$2. Instead, it investigates how $n$3 scales with the total system size $n$4 for two choices of partition sizes: $n$5, denoted “2+2”, and $n$6, denoted “4+4” (Timsina et al., 17 Jul 2025).

The reported scaling is a power law,

$n$7

The fitted exponents are

$n$8

The exponent increases with partition size: $n$9 for total $$S=\operatorname{conv}\{\,|\phi\rangle\langle\phi|:\ |\phi\rangle\ \text{a pure stabilizer state}\,\}.$$0, and $$S=\operatorname{conv}\{\,|\phi\rangle\langle\phi|:\ |\phi\rangle\ \text{a pure stabilizer state}\,\}.$$1 for total $$S=\operatorname{conv}\{\,|\phi\rangle\langle\phi|:\ |\phi\rangle\ \text{a pure stabilizer state}\,\}.$$2. No closed-form scaling relation in $$S=\operatorname{conv}\{\,|\phi\rangle\langle\phi|:\ |\phi\rangle\ \text{a pure stabilizer state}\,\}.$$3 is given beyond these two data points.

Partition	Critical scaling form	Fitted exponent
$$S=\operatorname{conv}\{\,|\phi\rangle\langle\phi|:\ |\phi\rangle\ \text{a pure stabilizer state}\,\}.$$4	$$S=\operatorname{conv}\{\,|\phi\rangle\langle\phi|:\ |\phi\rangle\ \text{a pure stabilizer state}\,\}.$$5	$$S=\operatorname{conv}\{\,|\phi\rangle\langle\phi|:\ |\phi\rangle\ \text{a pure stabilizer state}\,\}.$$6
$$S=\operatorname{conv}\{\,|\phi\rangle\langle\phi|:\ |\phi\rangle\ \text{a pure stabilizer state}\,\}.$$7	$$S=\operatorname{conv}\{\,|\phi\rangle\langle\phi|:\ |\phi\rangle\ \text{a pure stabilizer state}\,\}.$$8	$$S=\operatorname{conv}\{\,|\phi\rangle\langle\phi|:\ |\phi\rangle\ \text{a pure stabilizer state}\,\}.$$9

The significance of these data is that mutual robustness is directly tied to critical behavior in the model. A plausible implication is that the size dependence of $n$0 carries information about the scaling structure of non-stabilizer correlations at criticality, although no explicit field-theoretic identification of the exponent is provided in the supplied material.

5. Finite-temperature behavior and comparison with entanglement

The same study introduces an inverse-temperature scale $n$1 beyond which mutual robustness and ordinary log-free robustness saturate to their low-temperature plateau. The operational definition is precise: $n$2 is the value of $n$3 at which $n$4 differs by more than $n$5 from its low-$n$6 plateau value. Equivalently, one may define an effective critical temperature $n$7 (Timsina et al., 17 Jul 2025).

Both $n$8 and $n$9 grow from zero at high $B$00 to a plateau at low $B$01 without ever vanishing at any finite $B$02. The dependence of $B$03 on system size is reported to be algebraic: $B$04 The fitted exponents are

$B$05

and

$B$06

In terms of $B$07, this implies $B$08.

The comparison with mixed-state entanglement is explicit. In contrast to measures such as negativity, which exhibit “sudden death” and vanish above a threshold temperature $B$09, the robustness of magic and its mutual version remain strictly positive at all finite temperatures. The supplied interpretation is that magic correlations survive thermal noise far beyond the point where entanglement becomes PPT-positive and undistillable. This is presented as a distinction between non-Clifford resources and entanglement rather than a claim that magic universally dominates entanglement under all noise models.

6. Relation to dissipative robustness, magic rebirth, and state classification

Broader robustness-of-magic results sharpen the contrast with entanglement under local noise. For the $B$10-qubit GHZ family

$B$11

subjected to local amplitude damping $B$12, the evolved state takes the form

$B$13

with

$B$14

and

$B$15

For these real-GHZ–$B$16 states, the stabilizer criterion is

$B$17

This yields a magic-death threshold $B$18, a magic-rebirth threshold

$B$19

and an entanglement-death threshold

$B$20

The relation

$B$21

holds for every $B$22 (Cao, 21 May 2026).

For small $B$23, the reborn branch $B$24 lies entirely after the state has become fully separable, yet $B$25. In that regime the reborn magic is described as purely nonlocal, hidden from all proper marginals, which are diagonal. A parity-syndrome measurement of $B$26, followed by postselection of the all-$B$27 outcome and Clifford decoding via a CNOT cascade, concentrates the resource onto a single qubit with success probability

$B$28

and the extraction is stated to be “lossless” in expectation: $B$29

The same dissipative analysis also divides pure stabilizer states into two classes under homogeneous amplitude damping. A pure stabilizer state $B$30 is a magic-insulator if $B$31 for all $B$32, and a magic-generator if $B$33 for every $B$34. The criterion is stated in affine-support form: if

$B$35

with $B$36 an affine subspace, then $B$37 is an insulator if and only if all $B$38 have the same Hamming weight; otherwise it is a generator. At two qubits,

$B$39

is an insulator with $B$40 for all $B$41, whereas

$B$42

is a generator with

$B$43

These results are not themselves a theory of mutual robustness of magic, but they are directly relevant to its interpretation. They show that robustness-based magic diagnostics can remain informative when entanglement has already vanished, and that non-stabilizer structure can re-emerge under local dissipation. This suggests that mutual robustness of magic belongs to a broader class of quantities whose qualitative behavior is not constrained by the same monotonicity patterns that govern entanglement under local Markovian noise.