Minimal Tensor Research Articles

The Schmidt Rank for the Commuting Operator Framework

In quantum information theory, the Schmidt rank is a fundamental measure for the entanglement dimension of a pure bipartite state. Its natural definition uses the Schmidt decomposition of vectors on bipartite Hilbert spaces, which does not exist (or at least is not canonically given) if the observable algebras of the local systems are allowed to be general C*-algebras. In this work, we generalize the Schmidt rank to the commuting operator framework where the joint system is not necessarily described by the minimal tensor product but by a general bipartite algebra. We give algebraic and operational definitions for the Schmidt rank and show their equivalence. We analyze bipartite states and compute the Schmidt rank in several examples: the vacuum in quantum field theory, Araki–Woods-Powers states, as well as ground states and translation invariant states on spin chains which are viewed as bipartite systems for the left and right half chains. We conclude with a list of open problems for the commuting operator framework.

Hypersurfaces of the Space form for Which the Covariant Derivative of Their Ricci Tensors has Vanishing Minimal Norm Tensor

Hypersurfaces of the Space form for Which the Covariant Derivative of Their Ricci Tensors has Vanishing Minimal Norm Tensor

Coupling capacity in C*-algebras

Given two unital C*-algebras equipped with states and a positive operator in the enveloping von Neumann algebra of their minimal tensor product, we define three parameters that measure the capacity of the operator to align with a coupling of the two given states. Further, we establish a duality formula that shows the equality of two of the parameters for operators in the minimal tensor product of the relevant C*-algebras. In the context of abelian C*-algebras, our parameters are related to quantitative versions of Arveson's null set theorem and to dualities considered in the theory of optimal transport. On the other hand, restricting to matrix algebras we recover and generalize quantum versions of Strassen's theorem. We show that in the latter case our parameters can detect maximal entanglement and separability.

On Tsirelson pairs of C*-algebras

We introduce the notion of a Tsirelson pair of C*-algebras, which is a pair of C*-algebras for which the space of quantum strategies obtained by using states on the minimal tensor product of the pair is dense in the space of quantum strategies obtained by using states on the maximal tensor product. We exhibit a number of examples of such pairs that are “nontrivial” in the sense that the minimal tensor product and the maximal tensor product of the pair are not isomorphic. For example, we prove that any pair containing a C*-algebra with Kirchberg’s QWEP property is a Tsirelson pair. We then introduce the notion of a C*-algebra with the Tsirelson property (TP) and establish a number of closure properties for this class. We also show that the class of C*-algebras with the TP forms an elementary class (in the sense of model theory), but that this class does not admit an effective axiomatization.

A Convergent Inflation Hierarchy for Quantum Causal Structures

A causal structure is a description of the functional dependencies between random variables. A distribution is compatible with a given causal structure if it can be realized by a process respecting these dependencies. Deciding whether a distribution is compatible with a structure is a practically and fundamentally relevant, yet very difficult problem. Only recently has a general class of algorithms been proposed: These so-called inflation techniques associate to any causal structure a hierarchy of increasingly strict compatibility tests, where each test can be formulated as a computationally efficient convex optimization problem. Remarkably, it has been shown that in the classical case, this hierarchy is complete in the sense that each non-compatible distribution will be detected at some level of the hierarchy. An inflation hierarchy has also been formulated for causal structures that allow for the observed classical random variables to arise from measurements on quantum states—however, no proof of completeness of this quantum inflation hierarchy has been supplied. In this paper, we construct a first version of the quantum inflation hierarchy that is provably convergent. It takes an additional parameter, r, which can be interpreted as an upper bound on the Schmidt rank of the observables involved. For each r, it provides a family of increasingly strict and ultimately complete compatibility tests for correlations that are compatible with a given causal structure under this Schmidt rank constraint. From a technical point of view, convergence proofs are built on de Finetti theorems, which show that certain symmetries (which can be imposed in convex optimization problems) imply independence of random variables (which is not directly a convex constraint). A main technical ingredient to our proof is a Quantum de Finetti Theorem that holds for general tensor products of C^*-algebras, generalizing previous work that was restricted to minimal tensor products.

Operational Independence, Faithful Maps and Minimal Tensor Products

Operational Independence, Faithful Maps and Minimal Tensor Products

Principle of Information Causality Rationalizes Quantum Composition.

The principle of information causality, proposed as a generalization of no signaling principle, has efficiently been applied to outcast beyond quantum correlations as unphysical. In this Letter, we show that this principle, when utilized properly, can provide physical rationale toward structural derivation of multipartite quantum systems. In accordance with the no signaling condition, the state and effect spaces of a composite system can allow different possible mathematical descriptions, even when descriptions for the individual systems are assumed to be quantum. While in one extreme, namely, the maximal tensor product composition, the state space becomes quite exotic and permits composite states that are not allowed in quantum theory, the other extreme-minimal tensor product composition-contains only separable states, and the resulting theory allows only Bell local correlation. As we show, none of these compositions is commensurate with information causality, and hence cannot be the bona-fide description of nature. Information causality therefore promises an information-theoretical derivation of self duality of the state and effect cones for composite quantum systems.

Annihilating Entanglement Between Cones

Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform the k-fold maximal tensor product of a cone {textsf{C}}_1 into the k-fold minimal tensor product of a cone {textsf{C}}_2, and the pair ({textsf{C}}_1,{textsf{C}}_2) is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that ({textsf{C}}_1,{textsf{C}}_2) is resilient if either {textsf{C}}_1 or {textsf{C}}_2 is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.

Factorization properties for unbounded local positive maps

In this paper we present some factorization properties for unbounded local positive maps. We show that an unbounded local positive map ϕ on the minimal tensor product of the locally C∗-algebras A and C∗(DE), where DE is a Fréchet quantized domain, that is dominated by φ⊗id is of the forma ψ⊗id, where ψ is an unbounded local positive map dominated by φ. As an application of this result, we show that given a local positive map φ:A→B, the local positive map φ⊗idMnℂ is local decomposable for some n≥2 if and only if φ is a local CP-map. Also, we show that an unbounded local CCP-map ϕ on the minimal tensor product of the unital locally C∗ -algebras A and B, that is dominated by φ⊗ψ is of the forma φ⊗ψ˜, where ψ˜ is an unbounded local CCP- map dominated by ψ, whenever φ is pure.

Entangleability of cones

We solve a long-standing conjecture by Barker, proving that the minimal and maximal tensor products of two finite-dimensional proper cones coincide if and only if one of the two cones is generated by a linearly independent set. Here, given two proper cones \({\mathcal {C}}_1\), \({\mathcal {C}}_2\), their minimal tensor product is the cone generated by products of the form \(x_1\otimes x_2\), where \(x_1\in {\mathcal {C}}_1\) and \(x_2\in {\mathcal {C}}_2\), while their maximal tensor product is the set of tensors that are positive under all product functionals \(\varphi _1\otimes \varphi _2\), where \(\varphi _1|_{{\mathcal {C}}_1}\geqslant 0\) and \(\varphi _2|_{{\mathcal {C}}_2}\geqslant 0\). Our proof techniques involve a mix of convex geometry, elementary algebraic topology, and computations inspired by quantum information theory. Our motivation comes from the foundations of physics: as an application, we show that any two non-classical systems modelled by general probabilistic theories can be entangled.

The Weak Tracial Rokhlin Property for Finite Group Actions on Simple C*-Algebras

We develop the concept of weak tracial Rokhlin property for finite group actions on simple (not necessarily unital) C*-algebras and study its properties systematically. In particular, we show that this property is stable under restriction to invariant hereditary C*-algebras, minimal tensor products, and direct limits of actions. Some of these results are new even in the unital case and answer open questions asked by N. C. Phillips in full generality. We present several examples of finite group actions with the weak tracial Rokhlin property on simple stably projectionless C*-algebras. We prove that if \alpha \colon G \rightarrow \text{Aut}(A) is an action of a finite group G on a simple C*-algebra A with tracial rank zero and \alpha has the weak tracial Rokhlin property, then the crossed product A \rtimes_{\alpha} G and the fixed point algebra A^{\alpha} are simple with tracial rank zero. This extends a result of N. C. Phillips to the nonunital case. We use the machinery of Cuntz subequivalence to work in this nonunital setting.

Exact correlators from conformal Ward identities in momentum space and the perturbative TJJ vertex

We present a general study of 3-point functions of conformal field theory in momentum space, following a reconstruction method for tensor correlators, based on the solution of the conformal Ward identities (CWI's), introduced in recent works by Bzowski, McFadden and Skenderis (BMS). We investigate and detail the structure of the CWI's, their non-perturbative solutions and the transition to momentum space, comparing them to perturbation theory by taking QED as an example. We then proceed with an analysis of the TJJ correlator, presenting independent and detailed re-derivations of the conformal equations in the reconstruction method of BMS, originally formulated using a minimal tensor basis in the transverse traceless sector. A careful comparison with a second basis introduced in previous studies shows that this correlator is affected by one anomaly pole in the graviton (T) line, induced by renormalization. The result shows that the origin of the anomaly, in this correlator, should be necessarily attributed to the exchange of a massless effective degree of freedom. Our results are then exemplified in massless QED at one-loop in d-dimensions, expressed in terms of perturbative master integrals. An independent analysis of the Fuchsian character of the solutions, which bypasses the 3K integrals, is also presented. We show that the combination of field theories at one-loop – with a specific field content of degenerate massless scalar and fermions – is sufficient to generate the complete non-perturbative solution, in agreement with a previous study in coordinate space. The result shows that free conformal field theories, in specific dimensions, arrested at one-loop, reproduce the general result for the TJJ. Analytical checks of this correspondence are presented in d=3,4 and 5 spacetime dimensions. This implies that the generalized 3K integrals of the BMS solution can be expressed in terms of the two single master integrals B0 and C0 of 2- and 3-point functions, with significant simplifications.

Faithfulness of bifree product states

Given a nontrivial family of pairs of faces of unital C ∗ \mathrm {C}^* -algebras where each pair has a faithful state, it is proved that if the bifree product state is faithful on the reduced bifree product of this family of pairs of faces, then each pair of faces arises as a minimal tensor product. A partial converse is also obtained.

$K$-Continuity Is Equivalent To $K$-Exactness

Let $A$ be a $C^{*}$-algebra. It is well known that the functor $B \mapsto A \otimes B$ of taking the minimal tensor product with $A$ preserves inductive limits if and only if it is exact. $C^{*}$-algebras with this property play an important role in the structure and finite-dimensional approximation theory of $C^{*}$-algebras. We consider a $K$-theoretic analogue of this result and show that the functor $B \mapsto K_{0}(A \otimes B)$ preserves inductive limits if and only if it is half-exact.

Examining the effects of altering hip orientation on gluteus medius and tensor fascae latae interplay during common non-weight-bearing hip rehabilitation exercises

BackgroundImproving activity and strength of the gluteus medius muscle is a common goal among clinicians aiming to rehabilitate lower extremity and low back injuries. The functional anatomy of the hip is complex, particularly how position-dependent the activity and strength of many muscles surrounding the hip are, and the optimal exercise technique to isolate gluteus medius remains controversial. The objective of this study was to quantify the effect of altering hip orientation during side-lying clamshell and hip abduction exercises on the relative muscle activation profiles of gluteus medius and tensor fascae latae. MethodsThe ratio of gluteus-medius-to-tensor-fascae-latae peak electromyography signal amplitude of 13 healthy, male participants was compared across variations of the clamshell and abduction exercises. The hip flexion angle was varied from 30°, 45°, and 60° for the clamshell, while hip rotation orientation was varied from internal, neutral, and external rotation for the abduction exercise. FindingsVarying hip angle – flexion in the clamshell exercise and internal/external rotation in the abduction exercise – did not significantly affect the interplay between gluteus medius and tensor fascae latae activation levels. Both exercises remained gluteus medius-dominant across all variations, but the gluteus-medius-to-tensor-fascae-latae ratio was far greater for the clamshell than for the abduction exercise; the clamshell may be the preferred rehabilitative exercise to prescribe when minimal tensor fascae latae muscle activation is desired by the clinician. InterpretationThese findings provide information for clinical decision-making pertaining to effective gluteus medius activation in lower extremity and low back exercise rehabilitation programs.

The locality axiom in quantum field theory and tensor products of C*-algebras

The prototypes of mutually independent systems are systems which are localized in spacelike separated regions. In the framework of locally covariant quantum field theory, we show that the commutativity of observables in spacelike separated regions can be encoded in the tensorial structure of the functor which associates unital C*-algebras (the local observable algebras) to globally hyperbolic spacetimes. This holds under the assumption that the local algebras satisfy the split property and involves the minimal tensor product of C*-algebras.

On Nuclearity of $C*$-algebras of Fell Bundles over Étale Groupoids

We show the nuclearity of the reduced C^* -algebra of a Fell bundle over an étale amenable groupoid, whose fibers over the unit space are all nuclear. We introduce the (minimal) tensor product of a Fell bundle and a C^* -algebra in order to show the uniqueness of tensor product norms.

Operator system structures on the unital direct sum of $C^*$-algebras

This work is motivated by Radulescu's result on the comparison of C*-tensor norms on C*(F_n) x C*(F_n). For unital C*-algebras A and B, there are natural inclusions of A and B into their unital free product, their maximal tensor product and their minimal tensor product. These inclusions define three operator system structures on the internal sum A+B, the first of which we identify as the coproduct of A and B in the category of operator systems. Partly using ideas from quantum entanglement theory, we prove various interrelations between these three operator systems. As an application, the present results yield a significant improvement over Radulescu's bound on C*(F_n) x C*(F_n). At the same time, this tight comparison is so general that it cannot be regarded as evidence for a positive answer to the QWEP conjecture.

The Glimm space of the minimal tensor product of C*-algebras

We show that for C*-algebras A and B, there is a natural open bijection from Glimm (A) × Glimm (B) to Glimm (A ⊗α B) (where A ⊗α B denotes the minimal C*-tensor product), and identify a large class of C*-algebras A for which the map is continuous for arbitrary B. As a consequence, we determine the structure space of the centre of the multiplier algebra ZM(A ⊗α B) in terms of Glimm (A) and Glimm (B), and give necessary and sufficient conditions for the inclusion ZM(A) ⊗ ZM(B) ⊆ ZM(A ⊗α B) to be surjective. Further we show that when the Glimm spaces are considered as sets of ideals, the map (G, H) ↦ G ⊗α B+A ⊗α H implements the above bijection, extending a result of Kaniuth by eliminating the assumption of property (F).

A CANONICAL RANK-THREE TENSOR MODEL WITH A SCALING CONSTRAINT

A rank-three tensor model in canonical formalism has recently been proposed. The model describes consistent local-time evolutions of fuzzy spaces through a set of first-class constraints which form an on-shell closed algebra with structure functions. In fact, the algebra provides an algebraically consistent discretization of the Dirac–DeWitt constraint algebra in the canonical formalism of general relativity. However, the configuration space of this model contains obvious degeneracies of representing identical fuzzy spaces. In this paper, to delete the degeneracies, another first-class constraint representing a scaling symmetry is added to propose a new canonical rank-three tensor model. A consequence is that, while classical solutions of the previous model have typically runaway or vanishing behaviors, the new model has a compact configuration space and its classical solutions asymptotically approach either fixed points or cyclic orbits in time evolution. Among others, fixed points contain configurations with group symmetries, and may represent stationary symmetric fuzzy spaces. Another consequence on the uniqueness of the local Hamiltonian constraint is also discussed, and a minimal canonical tensor model, which is unique, is given.