Testing Product States, Quantum Merlin-Arthur Games and Tensor Optimization

A classical tensor product $${A \otimes B}$$ of complete lattices A and B, consisting of all down-sets in $${A \times B}$$ that are join-closed in either coordinate, is isomorphic to the complete lattice Gal(A,B) of Galois maps from A to B, turning arbitrary joins into meets. We introduce more general kinds of tensor products for closure spaces and for posets. They have the expected universal property for bimorphisms (separately continuous maps or maps preserving restricted joins in the two components) into complete lattices. The appropriate ingredient for quantale constructions is here distributivity at the bottom, a generalization of pseudocomplementedness. We show that the truncated tensor product of a complete lattice B with itself becomes a quantale with the closure of the relation product as multiplication iff B is pseudocomplemented, and that the tensor product has a unit element iff B is atomistic. The pseudocomplemented complete lattices form a semicategory in which the hom-set between two objects is their tensor product. The largest subcategory of that semicategory has as objects the atomic boolean complete lattices, which is equivalent to the category of sets and relations. More general results are obtained for closure spaces and posets.

In the domain of quantum cryptography, the implementation of quantum secret sharing stands as a pivotal element. In this paper, we propose a novel verifiable quantum secret sharing protocol using the d-dimensional product state and Lagrange interpolation techniques. This protocol is initiated by the dealer Alice, who initially prepares a quantum product state, selected from a predefined set of orthogonal product states within the ℂ d ⊗ ℂ d framework. Subsequently, the participants execute unitary operations on this product state to recover the underlying secret. Furthermore, we subject the protocol to a rigorous security analysis, considering both eavesdropping attacks and potential dishonesty from the participants. Finally, we conduct a comparative analysis of our protocol against existing schemes. Our scheme exhibits economies of scale by exclusively employing quantum product states, thereby realizing significant cost-efficiency advantages. In terms of access structure, we adopt a (t,n)-threshold architecture, a strategic choice that augments the protocol’s practicality and suitability for diverse applications. Furthermore, our protocol includes a rigorous integrity verification mechanism to ensure the honesty and reliability of the participants throughout the execution of the protocol.

Tight frames are extremely useful in applications. A scalable frame was recently introduced as a frame with the property of generating a tight frame by rescaling its frame vectors. In this paper, we consider piecewise scalable frames. We obtain some characterizations for them, and demonstrate that scalability is stable under unitary operators and isomorphisms between two Hilbert spaces. We further obtain a relation between the piecewise scalable frames in Hilbert spaces, and their tensor product

We show for any $d,m\ge 2$ with $(d,m)\neq (2,2)$, the matrix-valued generalization of the (tensor product) quantum correlation set of $d$ inputs and $m$ outputs is not closed. Our argument uses a reformulation of super-dense coding and teleportation in terms of $C^*$-algebra isomorphisms. Namely, we prove that for certain actions of cyclic group ${{\mathbb{Z}}}_d$, $$\begin{equation*}M_d(C^*({{\mathbb{F}}}_{d^2}))\cong{{\mathcal{B}}}_d\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d , M_d({{\mathcal{B}}}_d)\cong C^*({{\mathbb{F}}}_{d^2})\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d,\end{equation*}$$where ${{\mathcal{B}}}_d$ is the universal unital $C^*$-algebra generated by the elements $u_{jk}, \, 0 \le i, j \le d-1$, satisfying the relations that $[u_{j,k}]$ is a unitary operator, and $C^*({{\mathbb{F}}}_{d^2})$ is the universal $C^*$-algebra of $d^2$ unitaries. These isomorphisms provide a nice connection between the embezzlement of entanglement and the non-closedness of quantum correlation sets.

The objective of our study is to generalize the results on product states of the tensor product of two JC-algebras to infinite tensor product JC-algebras. Also, we characterize the tracial product state of the tensor product of two JC-algebras, and the tracial product state of infinite tensor products of JC-algebras.

This paper delves into the spectral theory of compact operators, focusing on eigenvalues that predominantly accumulate at zero and the implications for quantum information theory and machine learning. It highlights how the spectrum of compact, self-adjoint operators plays a crucial role in dimensionality reduction and quantum state decomposition. The research bridges abstract operator theory with practical computational applications, thereby enriching the theoretical underpinnings and practical applications of spectral theory. The study addresses how the eigenvalue decomposition of compact self-adjoint operators assists in reducing the dimensionality of large data sets and decomposing quantum states in tensor products. This connection between theory and application not only enhances the capabilities of Principal Component Analysis and machine learning methods but also proves critical in handling complex data structures and quantum systems. By exploring the structure and implications of compact operators within both classical and quantum domains, the findings offer a robust framework for future research. This includes potential expansions into more complex operator classes and their implications in other areas of functional analysis. Future research might explore the extension of these methods to tackle challenges posed by non-ideal, infinite-dimensional, or dynamically evolving systems.

Summary The rapid change of weather, change of climate and activity of wreckers, today negatively influence on the modern state of agricultural production. Efficiency of application of classic methods of monitoring is lost. In this connection for the increase of efficiency of production the newest technologies and instruments are used (for example, innovative decisions on the basis of the satellites systems and drones). Machine learning is actual modern instrument which used at the decision-making from the process control of vegetation. Importance of application of machine vegetation is conditioned by complication of task of analysis of the fields and prognostication of possible harvest, so as a lot of factors influence on a general results. For the analysis of the information got at monitoring of the state of environment, expert technologies are also successfully used. It is conditioned to those, that modern satellites pictures with distributive distance 3 meters do not give sufficient information for the high-quality control after the harvest of corn, that is why it follows to attract additional information. Therefore holds the search of new methods of receipt of additional specifying information, in particular with the use of technological decisions on the basis of drones.

The main topics of the second chapter is an introduction to module theory, aiming at application to the structure theorems about finitely generated modules over Dedekind domain (and general results about reductions of endomorphisms of vector spaces as a byproduct). On the way, the main tools of module theory (including tensor products) are introduced, as well as the rudiments of the language of homological algebra (exact sequences, left exactness of Hom, right exactness of tensor product).A particular emphasis is given to finitely generated modules, with detailed applications to rings of integers, Noetherian and Jacobson rings. Localization and projectivity of modules are carefully treated in order to be applied to Dedekind domains.

Matrix product states (MPS) are a class of physically-relevant quantum states which arise in the study of quantum many-body systems. A quantum state comprised of n qudits is said to be an MPS of bond dimension r if the reduced density matrix ψ1, …, k has rank r for each k ∊ {1, …, n}. When r = 1, this corresponds to the set of product states, i.e. states of the form |ψ1〉 ⊗ ⃛ ⊗ |ψn), which possess no entanglement. For larger values of r, this yields a more expressive class of quantum states, which are allowed to possess limited amounts of entanglement. Devising schemes for testing the amount of entanglement in quantum systems has played a crucial role in quantum computing and information theory. In this work, we study the problem of testing whether an unknown state |ψ〉 is an MPS in the property testing model. In this model, one is given m identical copies of |ψ〉, and the goal is to determine whether |ψ〉 is an MPS of bond dimension r or whether |ψ〉 is far from all such states. For the case of product states, we study the product test, a simple two-copy test previously analyzed by Harrow and Montanaro [17], and a key ingredient in their proof that QMA(2) = QMA(k) for k ≥ 2. We give a new and simpler analysis of the product test which achieves an optimal bound for a wide range of parameters, answering open problems in [17] and [23]. For the case of r ≥ 2, we give an efficient algorithm for testing whether |ψ〉 is an MPS of bond dimension r using m = O(nr2) copies, independent of the dimensions of the qudits, and we show that Ω(n1/2) copies are necessary for this task. This lower bound shows that a dependence on the number of qudits n is necessary, in sharp contrast to the case of product states where a constant number of copies suffices.

In recent years, friending recommendation has played a crucial role on the explosive growth of online social networks. Prior studies on this problem have the following two limitations: i) they assume that there is only one pair of source and target at one time; and ii) the intermediate user can indefinitely forward friending requests. These two assumptions are impractical for online social networks. To bridge this gap, we study how to maximize the acceptance probability for multi-pair active friending in online social networks. In this paper, we first establish a probability model for the Multi-pair Active Friending with Tolerance and then propose two efficient algorithms: Multi-pair Active Friending algorithm for Conflict Resolution (MAFCR) and Group-based Multi-pair Active Friending algorithm for Conflict Resolution (MAFCR-G). Through extensive experiments, we show that the algorithm MAFCR significantly outperforms the baseline algorithm in terms of average acceptance probability, and the algorithm MAFCR-G greatly reduces the running time while obtaining similar average acceptance probability.

Reliable Transportation of Humanitarian Supplies in Disaster Response: Model and Heuristic

We present examples of macroscopic systems entailing a quantum mechanical structure. One of our examples has a structure which is isomorphic to the spin structure for a spin 1/2 and another system entails a structure isomorphic to the structure of two spin 1/2 in the entangled singlet state. We elaborate this system by showing that an arbitrary tensor product state representing two entangled qubits can be described in a complete way by a specific internal constraint between the ray or density states of the two qubits, which describes the behavior of the state of one of the spins if measurements are executed on the other spin. Since any n‐qubit unitary operation can be decomposed into 2‐qubit gates and unary operations, we argue that our representation of 2‐qubit entanglement contributes to a better understanding of the role of n‐qubit entanglement in quantum computation. We illustrate our approach on two 2‐qubit algorithms proposed by Deutsch, respectively Arvind et al.

30 - Minimum entropy production

The representation of numbers by tensor product states of composite quantum systems is examined. Consideration is limitedto k — ary representations of length L and arithmetic mod K L An abstract representation on an L fold tensor product Hilbert space H arith of number states and operators for the basic arithmetic operation is described. Unitary maps onto a physical parameter based tensor product space H phy are defined and the relations between these two spaces and the dependence of algorithm dynamics on the unitary maps is discussed. The important condition of efficient implementation by physically realizable Hamiltonians of the basic arithmetic operations is also discussed.

This advanced textbook on linear algebra and geometry covers a wide range of classical and modern topics. Differing from existing textbooks in approach, the work illustrates the many-sided applications and connections of linear algebra with functional analysis, quantum mechanics and algebraic and differential geometry. The subjects covered in some