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arXiv:2512.16777 · PRX Quantum

A magic criterion (almost) as nice as PPT, with applications in distillation and detection

Zhenhuan Liu, Tobias Haug, Qi Ye, Zi-Wen Liu, Ingo Roth

quant-phSubmitted Dec 18, 2025• Announced Dec 19, 2025• 2 citations
v1 · Dec 18, 2025v2 · Apr 20, 2026
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Abstract

We introduce a mixed-state magic criterion, the Triangle Criterion, which plays a role for magic analogous to the Positive Partial Transposition (PPT) criterion for entanglement: it combines strong detection capability, a clear geometric interpretation, and an operational link to magic distillation. Using this criterion, we uncover several new features of multi-qubit magic distillation and detection. We prove that genuinely multi-qubit magic distillation protocols are strictly more powerful than all single-qubit schemes by showing that the Triangle Criterion is not stable under tensor products, in sharp contrast to the PPT criterion. Moreover, we show that, with overwhelming probability, multi-qubit magic states with relatively low rank cannot be distilled by any single-qubit distillation protocol. We derive an upper bound on the minimal purity of magic states, which is conjectured to be tight with both numerical and constructive evidences. Using this minimal-purity result, we predict the existence of unfaithful magic states, namely states that cannot be detected by any fidelity-based magic witness, and reveal fundamental limitations of mixed-state magic detection in any single-copy scheme.

Journal reference
PRX Quantum 7, 033016 (2026)
DOI
10.1103/rcpf-8nh9
Comments
13 pages, 2 figures

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We develop a strong and computationally simple entanglement criterion. The criterion is based on an elementary positive map Phi which operates on state spaces with even dimension N > or = 4. It is shown that Phi detects many entangled states with a positive partial transposition (PPT) and that it leads to a class of optimal entanglement witnesses. This implies that there are no other witnesses which can detect more entangled PPT states. The map Phi yields a systematic method for the explicit construction of high-dimensional manifolds of bound entangled states.
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We express the positive-partial-transpose (PPT) separability criterion for symmetric states of multiqubit systems in terms of matrix inequalities based on the recently introduced tensor representation for spin states. We construct a matrix from the tensor representation of the state and show that it is similar to the partial transpose of the density matrix written in the computational basis. Furthermore, the positivity of this matrix is equivalent to the positivity of a correlation matrix constructed from tensor products of Pauli operators. This allows for a more transparent experimental interpretation of the PPT criteria for an arbitrary spin-jjj state. The unitary matrices connecting our matrix to the partial transpose of the state generalize the so-called magic basis that plays a central role in Wootters' explicit formula for the concurrence of a two-qubit system and the Bell bases used for the teleportation of a one- or two-qubit state.
Chapter 13 delves into the complex terrain of mixed-state entanglement, extending the discourse from pure-state entanglement to encompass the broader and more practical scenarios encountered in quantum systems. The chapter systematically explores the detection of entanglement in mixed states, introducing criteria and methods such as the Positive Partial Transpose (PPT) criterion and entanglement witnesses, which serve as diagnostic tools for identifying entanglement in a mixed quantum state. Furthermore, it addresses the quantification of entanglement in mixed states, discussing various measures like entanglement cost and distillable entanglement. These concepts highlight the operational aspects of entanglement, including its creation and extraction, within mixed-state frameworks. The chapter also introduces the notion of entanglement conversion distances, providing a quantitative approach to understanding the transformations between different entangled states.