An HD space is a generalized sample space each of whose subspaces has, as its logic, an orthomodular poset.We study HD spaces which allow full sets of dispersion free weights: in a sense, these sample spaces can be said to admit "hidden variables".For such a space, we show that any proposition in the logic contains a minimal proposition, and that any maximal refinement ideal of operations is generated by a single operation.While the former ensures that the logic is atomic, the latter provides a generalized analog of the classical "grand canonical operation".In fact, we obtain a stronger result concerning atomicity, namely, that the logic of any HD space in which all maximal refinement ideals are principal is atomic.An HD space is a generalized sample space each of whose subspaces has, as its logic, an orthomodular poset.We study HD spaces which allow full sets of dispersion free weights: in a sense, these sample spaces can be said to admit "hidden variables".For such a space, we show that any proposition in the logic contains a minimal proposition, and that any maximal refinement ideal of operations is generated by a single operation.While the former ensures that the logic is atomic, the latter provides a generalized analog of the classical "grand canonical operation".In fact, we obtain a stronger result concerning atomicity, namely, that the logic of any HD space in which all maximal refinement ideals are principal is atomic.

Rocky Mountain Journal of MathematicsDec 1, 19790 citations

Entanglement in C $^*$ -algebras: tensor products of state spacesEntanglement in C$^*$-algebras: tensor products of state spaces

Magdalena Musat, Mikael Rørdam

We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C $^*$ -algebras. Relatedly, we study entanglement in (infinite dimensional) C $^*$ -algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C $^*$ -algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C $^*$ -algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C $^*$ -algebras. We show that the minimal and maximal tensor products of state spaces of C $^*$ -algebras agree precisely when one of the two C $^*$ -algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C $^*$ -algebras. Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C $^*$ -algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C $^*$ -algebras. This enables a systematic way of determining the trace simplex of a tensor product of C $^*$ -algebras.We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C$^*$-algebras. Relatedly, we study entanglement in (infinite dimensional) C$^*$-algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C$^*$-algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C$^*$-algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C$^*$-algebras. We show that the minimal and maximal tensor products of state spaces of C$^*$-algebras agree precisely when one of the two C$^*$-algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C$^*$-algebras. Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C$^*$-algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C$^*$-algebras. This enables a systematic way of determining the trace simplex of a tensor product of C$^*$-algebras.

math.OACross-listedarXiv:2512.10410Dec 12, 2025

Tensor products and probability weights

Matthias P. Kl�y, Charles H. Randall, David J. Foulis

International Journal of Theoretical PhysicsMar 1, 198766 citations

Observables and States in Tensor Products of Hilbert Spaces

Kalyanapuram Rangachari Parthasarathy

Suppose (Ω i , F i ), 1 ≤ i ≤ n are sample spaces describing the elementary outcomes and events concerning n different statistical systems in classical probability. To integrate them into a unified picture under the umbrella of a single sample space one takes their cartesian product (Ω, F) where Ω = Ω1 x … x Ω n , F = F1 x … x F n , the smallest σ-algebra containing all rectangles of the form F1 x F1 x … F n , F j ∈ F j for each j. Now we wish to search for an analogue of this description in quantum probability when we have n systems where the events concerning the j-th system are described by the set P(ℋ j ) of all projections in a Hubert space ℋ j j = 1, 2,…, n. Such an attempt leads us to consider tensor products of Hilbert spaces. We shall present a somewhat statistically oriented approach to the definition of tensor products which is at the same time coordinate free in character. To this end we introduce the notion of a positive definite kernel.Suppose (Ω i , F i ), 1 ≤ i ≤ n are sample spaces describing the elementary outcomes and events concerning n different statistical systems in classical probability. To integrate them into a unified picture under the umbrella of a single sample space one takes their cartesian product (Ω, F) where Ω = Ω1 x … x Ω n , F = F1 x … x F n , the smallest σ-algebra containing all rectangles of the form F1 x F1 x … F n , F j ∈ F j for each j. Now we wish to search for an analogue of this description in quantum probability when we have n systems where the events concerning the j-th system are described by the set P(ℋ j ) of all projections in a Hubert space ℋ j j = 1, 2,…, n. Such an attempt leads us to consider tensor products of Hilbert spaces. We shall present a somewhat statistically oriented approach to the definition of tensor products which is at the same time coordinate free in character. To this end we introduce the notion of a positive definite kernel.

Birkhäuser Basel eBooksJan 1, 19920 citations

Asymptotically Good Quantum Codes with Addressable Transversal T Gates

Tongyin Lin, Bujiao Wu, Bin Cheng

Designing quantum codes with both transversal non-Clifford gates and good error-correcting parameters is an important goal in fault-tolerant quantum computation. Here, we construct an explicit family of binary CSS codes with constant rate and linear distance that admit fully addressable transversal $T$ gates. Specifically, any prescribed tensor product of logical powers of $T$ is implemented by a tensor product of physical powers of $T$ , without any subsequent correction. Our construction combines algebraic-geometry codes with suitable binary embedding to obtain generalized divisibility of binary codes. This divisibility then enables fully addressable transversal $T$ gates. Furthermore, we formulate the minimum binary embedding length problem as a minimum-weight problem over an affine space and numerically improve the constants in the asymptotic rate and relative-distance bounds.Designing quantum codes with both transversal non-Clifford gates and good error-correcting parameters is an important goal in fault-tolerant quantum computation. Here, we construct an explicit family of binary CSS codes with constant rate and linear distance that admit fully addressable transversal $T$ gates. Specifically, any prescribed tensor product of logical powers of $T$ is implemented by a tensor product of physical powers of $T$, without any subsequent correction. Our construction combines algebraic-geometry codes with suitable binary embedding to obtain generalized divisibility of binary codes. This divisibility then enables fully addressable transversal $T$ gates. Furthermore, we formulate the minimum binary embedding length problem as a minimum-weight problem over an affine space and numerically improve the constants in the asymptotic rate and relative-distance bounds.

quant-pharXiv:2609.35129Sep 29, 20260 citations

Entanglement in C

^*

-algebras: tensor products of state spaces

We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C

^*

-algebras. Relatedly, we study entanglement in (infinite dimensional) C

^*

-algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C

^*

-algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C

^*

-algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C

^*

-algebras. We show that the minimal and maximal tensor products of state spaces of C

^*

-algebras agree precisely when one of the two C

^*

-algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C

^*

-algebras. Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C

^*

-algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C

^*

-algebras. This enables a systematic way of determining the trace simplex of a tensor product of C

^*

-algebras.

T

gates. Specifically, any prescribed tensor product of logical powers of

T

is implemented by a tensor product of physical powers of

T

, without any subsequent correction. Our construction combines algebraic-geometry codes with suitable binary embedding to obtain generalized divisibility of binary codes. This divisibility then enables fully addressable transversal

T

gates. Furthermore, we formulate the minimum binary embedding length problem as a minimum-weight problem over an affine space and numerically improve the constants in the asymptotic rate and relative-distance bounds.