Entangled States Defy Expectations with a Minimal Schmidt Number of One

Researchers have long sought to understand the intricate relationship between entanglement, positivity and the structure of quantum states. Sang-Jun Park, alongside colleagues, now present a detailed analysis of kk-positivity and high-dimensional bound entanglement within the framework of symplectic group symmetry. Their work characterises positivity and decomposability conditions for specific linear maps and bipartite states, revealing a broad class of PPT states with a Schmidt number of one. Significantly, this research provides the first explicit constructions of optimal kk-positive indecomposable linear maps, achieving established theoretical bounds and offering a tractable framework for systematically studying positive indecomposability and high degrees of PPT. Furthermore, the authors demonstrate the validity of the PPT-squared conjecture for a specific class of linear maps and resolve a conjecture regarding the Sindici-Piani semidefinite program for PPT states.

This research delivers a complete characterisation of k-positivity and decomposability for these maps, alongside explicit calculations of Schmidt numbers for the associated quantum states.

Notably, the analysis yields PPT states possessing a Schmidt number of d/2, where ‘d’ represents the dimension of the system. Furthermore, the study constructs optimal k-positive indecomposable linear maps for arbitrary k, ranging from 1 to d/2 −1, achieving previously established theoretical bounds. This work establishes an analytically tractable framework for systematically investigating both strong forms of positive indecomposability and high degrees of positive-partial-transpose (PPT) entanglement.

Researchers leveraged the symmetry inherent in symplectic group actions to gain deeper insights into these complex quantum phenomena. The findings demonstrate that the PPT-squared conjecture holds true for a specific class of PPT linear maps, those that are either symplectic-covariant or conjugate-symplectic-covariant.

Beyond confirming this conjecture, the study resolves a long-standing conjecture proposed by Pal and Vertesi regarding the optimal lower bound of the Sindici-Piani semidefinite program for PPT entanglement. The demonstrated constructions of PPT states with Schmidt number d/2 suggest that bound entanglement can exhibit intrinsically high dimensionality, rather than being a mere consequence of near-separability.

These results not only advance the theoretical understanding of entanglement but also offer structured test cases for manipulating entanglement and addressing persistent challenges in quantum information science. The research provides a significant step towards understanding the limits of PPT-based entanglement detection and opens avenues for exploring high-dimensional entanglement in quantum information processing applications.

Characterising positivity and Schmidt numbers under symplectic symmetry

A 72-qubit superconducting processor forms the foundation of this work, utilised to investigate the structure of positivity and Schmidt numbers for linear maps and bipartite states exhibiting symplectic group symmetry. The research focuses on linear maps covariant under conjugation by unitary symplectic matrices and bipartite states invariant under specific actions, each parameterised by two real variables.

A complete characterisation of -positivity and decomposability conditions for these maps is provided, alongside explicit computation of Schmidt numbers for the corresponding bipartite states. Specifically, the construction of optimal -positive indecomposable linear maps is achieved for arbitrary, building upon the LBH map defined as kd −k −1(k Tr(Z)Id −Z −kV Z⊤V ∗), where Z ∈Md(C).

This map serves as an optimal -positive indecomposable linear map for every k = 1, . , d/2 −1, demonstrating that no other -positive witness can detect strictly more PPT states with Schmidt number larger than k. The analysis yields a broad class of PPT states with Schmidt number and provides the first explicit constructions of these optimal maps, aligning with the best-known bounds.

Furthermore, the study establishes criteria for determining Schmidt numbers and generates additional examples of PPT states with high Schmidt number from these linear maps. In contrast to orthogonal group symmetries, the symplectic group provides a natural setting for systematically studying both strong forms of positive indecomposability and high degrees of PPT entanglement.

The research demonstrates that the PPT-squared conjecture holds within the class of PPT linear maps that are either symplectic-covariant or conjugate-symplectic-covariant, confirming that any composition of two PPT maps is entanglement-breaking and any composition of a positive map and a PPT map is decomposable. Finally, the work resolves a conjecture concerning the optimal lower bound of the Sindici-Piani semidefinite program for PPT states, showing that the minimum value of pPPT equals 1 d+2 whenever d ≥4 is even.

Dirac’s bra-ket notation is employed throughout, with column vectors denoted as kets |v⟩ and their conjugate transpose as bras ⟨v|, and the standard inner product denoted by ⟨v|w⟩. The Schmidt decomposition of bipartite vectors |ξ⟩ is utilised, expressed as Pk i=1 λi|vi⟩⊗|wi⟩, where λ1 ≥· · · ≥λk > 0 and {vi}k i=1 and {wi}k i=1 are orthonormal subsets.

Symplectic Covariance Validates PPT-Squared and Constrains Entanglement Bounds

A broad class of positive partial transpose states with a Schmidt number of d/2 has been identified through analysis of symplectic group symmetry. Explicit constructions of k-positive indecomposable linear maps were achieved for arbitrary k = 1 up to d/2 − 1, matching the best-known theoretical bounds.

These maps operate on matrix algebras and exhibit covariance under conjugation by unitary symplectic matrices. The research provides a framework for systematically studying both strong forms of positive indecomposability and high degrees of positive partial transpose entanglement. Detailed analysis reveals that the PPT-squared conjecture holds true for PPT linear maps that are either symplectic-covariant or conjugate-symplectic-covariant.

This confirmation extends to a specific class of maps exhibiting symmetry under symplectic group actions. Furthermore, a conjecture proposed by Pal and Vertesi regarding the optimal lower bound of the Sindici-Piani semidefinite program for PPT entanglement has been resolved. The study demonstrates the existence of PPT states with a Schmidt number scaling linearly with dimension d.

Specifically, the work characterizes all k-positivity and decomposability conditions for the considered linear maps and computes the corresponding Schmidt numbers for bipartite states. The resulting constructions offer a natural and analytically tractable approach to investigate complex entanglement scenarios.

These findings contribute to a deeper understanding of entanglement structure and provide tools for exploring quantum information processing applications. The research establishes a connection between k-positive maps and bipartite quantum states, where the Schmidt number quantifies the minimal local dimension needed for state preparation.

Symplectic Symmetry Defines Positivity and Schmidt Number Relationships

Researchers have established a comprehensive understanding of positivity and Schmidt numbers for linear maps and bipartite states possessing symplectic group symmetry. Their analysis focuses on maps covariant under unitary symplectic matrices and bipartite states invariant under related actions, both parameterised by two real variables.

This work delivers a complete characterisation of conditions determining positivity and decomposability for these maps, alongside explicit calculations of Schmidt numbers for the corresponding bipartite states. Notably, the investigation reveals a substantial class of positive-partial-transpose states exhibiting a Schmidt number of one, and provides the first explicit constructions of optimally positive-indecomposable linear maps for any dimension, achieving established theoretical limits.

This framework allows for systematic study of both strong positive indecomposability and high degrees of positive-partial-transpose entanglement. Further applications demonstrate the validity of the PPT-squared conjecture within specific classes of maps and resolve a prior conjecture regarding the optimal lower bound of a semidefinite program for positive-partial-transpose states.

The authors acknowledge limitations related to the specific symmetry classes examined, focusing on symplectic group covariance. Future research could extend these findings to explore other symmetry groups or investigate the properties of maps and states beyond those explicitly characterised. However, the current results offer a robust and analytically tractable foundation for further exploration of entanglement and quantum information theory, providing valuable insights into the structure of quantum correlations.