Kubo-Ando Means and Rigidity of Quantum Positivity Cones

Abstract: We investigate the stability of quantum positivity cones under nonlinear operator means. Specifically, we examine how Kubo--Ando means interact with the separable, positive partial transpose (PPT), and Schmidt-number cones. By analyzing the curvature of operator monotone functions at the identity, we give a strict rigidity phenomenon: weighted arithmetic means are the only Kubo--Ando means that preserve the separable cone in all dimensions. We show that the strictly positive curvature of any non-arithmetic mean explicitly forces a violation of the PPT condition, even in the foundational two-qubit setting, and can strictly increase the Schmidt number of the resulting operator. Finally, using the Choi--Jamiołkowski correspondence, we translate these geometric obstructions to the map-theoretic setting, concluding that convex mixing is the uniquely permissible Kubo--Ando operation for preserving entanglement-breaking quantum channels.

• The paper establishes that arithmetic Kubo-Ando means, with zero curvature, are the only operations preserving the separability, PPT, and Schmidt-number cones in quantum systems.
• It uses spectral analysis and Taylor expansions to quantify curvature effects, demonstrating that non-arithmetic means yield negative eigenvalues and increased Schmidt numbers.
• The findings significantly impact quantum information theory, restricting entanglement-preserving channel averaging to convex mixing methods.

Rigidity of Kubo-Ando Means for Quantum Positivity Cones

Background and Motivation

Kubo-Ando means are the canonical nonlinear binary operations on the cone of positive semidefinite operators, characterized by operator monotonicity, order, and congruence invariance. These encompass arithmetic, geometric, harmonic, and logarithmic means, and are pivotal in operator theory and quantum information. The study presented in "Kubo-Ando Means and Rigidity of Quantum Positivity Cones" (2605.26272) systematically analyzes the interaction of these means with quantum positivity cones beyond the full positive cone—specifically, the cones central to entanglement theory: the separable cone, PPT cone, and Schmidt-number cones.

Quantum positivity cones encapsulate entanglement structure in bipartite systems; preservation under operator means is highly nontrivial and fundamental for the study of quantum channels and states.

Curvature and the Rigidity Phenomenon

A central analytic tool in the paper is the curvature $\kappa_\sigma = -f''(1)$ of the representing function $f$ associated with a Kubo-Ando mean $\sigma$. Weighted arithmetic means correspond to affine $f$, yielding zero curvature, while non-arithmetic means manifest strictly positive curvature.

The principal rigidity result is as follows: the separable cone $\mathcal{S}_1^{m,n}$ is preserved under $\sigma$ if and only if $\sigma$ is a weighted arithmetic mean. Non-arithmetic Kubo-Ando means are strictly incompatible with separability-preserving structure, and this obstruction is quantified by curvature. This result is proven first in the two-qubit $(2 \otimes 2)$ case, then extended to arbitrary dimensions via local isometric embeddings.

Violations for PPT and Schmidt-number Cones

The rigidity extends to PPT cones $\mathcal{PPT}^{m,n}$, as well as to all intermediate cones between separability and PPT. Non-arithmetic means yield mean operators outside PPT even when inputs belong to the interior of the separable cone. Theoretical analysis is grounded in explicit spectral calculations for block-diagonal forms under partial transpose, leveraging Taylor expansions in the representing function and quantifying non-preservation in terms proportional to curvature.

For Schmidt-number cones, the paper furnishes constructions that strictly increase Schmidt number, showing that non-arithmetic Kubo-Ando means applied to two elements of $\mathcal{S}_r^{m,n}$ can generate an operator in $f$0 for $f$1, with the lower bound attained precisely in the construction. Lemma 4 provides multiplicativity of Schmidt number in tensor products, facilitating the explicit calculation.

Choi–Jamiołkowski Correspondence and Quantum Channels

Quantum channels correspond to positive maps whose structure is captured via the Choi–Jamiołkowski isomorphism. The aforementioned rigidity properties are translated to the map-theoretic setting: convex mixing is the sole Kubo-Ando operation preserving entanglement-breaking channels, i.e., channels whose Choi matrices are separable.

Non-arithmetic Kubo-Ando means may map two entanglement-breaking channels to a channel whose Choi matrix fails the PPT test or exhibits increased Schmidt number, violating entanglement-breaking properties. Normalization of Choi matrices to restore trace-preservation is shown to preserve cone memberships, rendering the negative result robust under operational constraints.

Numerical and Structural Claims

Strong numerical claims are substantiated in the construction:

• For any non-arithmetic mean with curvature $f$2, the smallest eigenvalue of the partial transpose of the mean operator is strictly negative for sufficiently small perturbations, even when input operators are drawn from the interior of the separable cone.
• Schmidt-number is increased from $f$3 to at least $f$4 via tensorial constructions.

No exceptions are allowed by the class of cones interpolating between separability and PPT, including rank-constrained $f$5-convex hulls. Only arithmetic means maintain closure.

Implications and Prospects

Theoretical implications are multifold: the findings delimit the applicability of operator means in quantum information theory, imposing a sharp restriction on nonlinear combination schemes for states and channels that respect the entanglement structure. The convexity inherent to arithmetic mixing is the unique compatible nonlinear operation for all positivity cones associated with entanglement. This result unearths a fundamental geometric incompatibility between non-arithmetic nonlinear means and quantum entanglement structure.

Practically, these results constrain schemes for channel averaging, entanglement distillation, and quantum resource theories built upon operator means, indicating potential failure of entanglement-preserving operations except for convex mixing. The constructions generalize to arbitrary dimensions and all cones between separability and PPT, rendering the restrictions pervasive.

Future directions may probe the modification or relaxation of operator mean properties to achieve partial compatibility, explore alternative invariance under congruence or partial trace, and extend the analysis to multipartite or non-bipartite entanglement cones. Investigations may also address the quantum operational meaning of geometric curvature in generalized averaging of quantum channels and states.

Conclusion

Weighted arithmetic means are uniquely rigid among Kubo-Ando means in preserving quantum positivity cones such as separable, PPT, and Schmidt-number cones. Non-arithmetic means are strictly excluded due to positive curvature-induced violations. These findings rigorously delimit compatible nonlinear operations in quantum information, as articulated both for states and quantum channels via their Choi representations. The restriction deepens understanding of the geometry of quantum positivity and operator means, setting boundaries for quantum entanglement theory and the algebraic structure of quantum maps.