Quantum Teleportation and Super-Dense Coding in Operator Algebras

Quantum entanglement is the most representative feature of quantum mechanics to classical mechanics. It is the delocalization, non-classical strong correlation among multiple quantum system. It has been developed for nearly one hundred years and a lot of methods of production and applications have been proposed. Aiming at the generation mode of quantum entanglement, this study presents two general preparation methods of quantum entanglement sources, including spontaneous parametric down-conversion (SPDC) and the preparation of quantum entanglement sources based on the four-wave mixing (FWM) effect. The principle, advantages and disadvantages of quantum entanglement sources are analyzed respectively. Finally, the research and application direction of the preparation method of quantum entanglement source are prospected. With the ability of preparation of quantum entanglement source, the application of quantum entanglement developed quickly, (e.g., quantum teleportation, entanglement swapping, quantum key distribution and quantum dense coding in the area of communication), and other area such as quantum computer, quantum imaging, quantum ranging and quantum clock synchronization. These results shed light on guiding further exploration of entanglement fabrication.

Teaching Quantum Computing with the QuIDE Simulator

This introduction to quantum computing from a classical programmer's perspective is meant for students and practitioners alike. More than 50 quantum techniques and algorithms are explained with mathematical derivations and code for simulation, using an open-source code base in Python and C++. New material throughout this fully revised and expanded second edition includes new chapters on Quantum Machine Learning, State Preparation, and Similarity Tests. Coverage includes algorithms exploiting entanglement, black-box algorithms, the quantum Fourier transform, phase estimation, quantum walks, and foundational QML algorithms. Readers will find detailed, easy-to-follow derivations and implementations of Shor's algorithm, Grover's algorithm, SAT3, graph coloring, the Solovay-Kitaev algorithm, Moettoenen's algorithm, quantum mean, median, and minimum finding, Deutsch's algorithm, Bernstein-Vazirani, quantum teleportation and superdense coding, the CHSH game, and, from QML, the HHL algorithm, Euclidean distance, and PCA. The book also discusses productivity issues like quantum noise, error correction, quantum programming languages, compilers, and techniques for transpilation.

We investigate the usefulness of a recently introduced five-qubit state by Brown et al. [I. D. K. Brown, S. Stepney, A. Sudbery, and S. L. Braunstein, J. Phys. A 38, 1119 (2005)] for quantum teleportation, quantum-state sharing, and superdense coding. It is shown that this state can be utilized for perfect teleportation of arbitrary single and two-qubit systems. We devise various schemes for quantum-state sharing of an arbitrary single- and two-particle state via cooperative teleportation. We later show that this state can be used for superdense coding as well. It is found that five classical bits can be sent by sending only three quantum bits.

In this chapter we review the employment of quantum entanglement as a resource for information processing and transmission. In particular we introduce and discuss the notion of completely positive maps operating on observable algebras of physical systems in order to have a model to construct communication channels based on quantum processes. Then we discuss advantages and limitations of entanglement-assisted quantum communication schemes like quantum teleportation and dense coding.

Quantum correlations and their significance in quantum teleportation under [formula omitted]-symmetric operation

Richard Feynman's observation that certain quantum mechanical effects cannot be simulated efficiently on a computer led to speculation that computation in general could be done more efficiently if it used these quantum effects. This speculation proved justified when Peter Shor described a polynomial time quantum algorithm for factoring intergers. In quantum systems, the computational space increases exponentially with the size of the system, which enables exponential parallelism. This parallelism could lead to exponentially faster quantum algorithms than possible classically. The catch is that accessing the results, which requires measurement, proves tricky and requires new nontraditional programming techniques. The aim of this paper is to guide computer scientists through the barriers that separate quantum computing from conventional computing. We introduce basic principles of quantum mechanics to explain where the power of quantum computers comes from and why it is difficult to harness. We describe quantum cryptography, teleportation, and dense coding. Various approaches to exploiting the power of quantum parallelism are explained. We conclude with a discussion of quantum error correction.

Steerability is a characteristic nonlocal trait of quantum states lying in between entanglement and Bell nonlocality. A given quantum state is considered to be steerable if it violates a suitably chosen steering inequality. A quantum state which otherwise satisfies a certain inequality can violate the inequality under a global change of basis i.e, if the state is transformed by a nonlocal unitary operation. Intriguingly there are states which preserve their non-violation(pertaining to the said inequality) under any global unitary operation. The present work explores the effect of global unitary operations on the steering ability of a quantum state which live in two qubits. We characterize such states in terms of a necessary and sufficient condition on their spectrum. Such states are also characterized in terms of some analytic characteristics of the set to which they belong. Looking back at steerability the present work also provides a relation between steerability and quantum teleportation together with the derivation of a result related to the optimal violation of steering inequality . An analytic estimation of the size of such non-violating states in terms of purity is also obtained. Interestingly the estimation in terms of purity also gives a necessary and sufficient condition in terms of bloch parameters of the state. Illustrations from some signature class of quantum states further underscore our observations.

Quantum teleportation is an important method for transmitting quantum states between two quantum communication nodes. In this paper, we propose a low-latency quantum communication scheme based on Greenberger-Horne-Zeilinger (GHZ) state. To implement two-qubit transmission, the source and all intermediate nodes perform quantum measurements simultaneously and the measurement results are transmitted to the destination node through classical channel independently. The destination node performs appropriate unitary operation on its particles to recover the original two-qubit state based on received measurement results. Our scheme effectively reduces the end-to-end quantum communication delay.

Geometry of C⁎-algebras, and the bidual of their projective tensor product

The utility of a five-qubit entangled state for quantum teleportation, quantum state sharing and superdense coding is investigated. The state can be utilized for perfect teleportation and quantum state sharing of an arbitrary single- and two-qubit state. The capacity of superdense coding of the state reaches the "Holevo bound", which means that five classical bits can be transmitted by sending three qubits. The preparation of the five-qubit state and detection of the multipartite states in cavity QED are discussed. The distinct advantage of the feasible cavity QED technology that we use is insensitive to the thermal field and the cavity decay.

We give a test that can distinguish efficiently between product states of n quantum systems and states that are far from product. If applied to a state | ψ 〉 whose maximum overlap with a product state is 1 − ε , the test passes with probability 1 − Θ ( ε ), regardless of n or the local dimensions of the individual systems. The test uses two copies of | ψ 〉. We prove correctness of this test as a special case of a more general result regarding stability of maximum output purity of the depolarizing channel. A key application of the test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain several structural results that had been previously conjectured, including the fact that efficient soundness amplification is possible and that two Merlins can simulate many Merlins: QMA( k ) = QMA(2) for k ≥ 2. Building on a previous result of Aaronson et al., this implies that there is an efficient quantum algorithm to verify 3-SAT with constant soundness, given two unentangled proofs of Õ (√ n ) qubits. We also show how QMA(2) with log-sized proofs is equivalent to a large number of problems, some related to quantum information (such as testing separability of mixed states) as well as problems without any apparent connection to quantum mechanics (such as computing injective tensor norms of 3-index tensors). As a consequence, we obtain many hardness-of-approximation results, as well as potential algorithmic applications of methods for approximating QMA(2) acceptance probabilities. Finally, our test can also be used to construct an efficient test for determining whether a unitary operator is a tensor product, which is a generalization of classical linearity testing.

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

This article delves into algebraic topology, specifically (co)homology theory, which is essential in various mathematical fields. It explores different types of (co)homology groups such as Hochschild, cyclic, reflexive, and dihedral, focusing on dihedral cohomology generated by the dihedral group operating on simplicial complexes. The text discusses when dihedral and reflexive cohomology vanishes and provides examples, along with proving a homomorphism between dihedral cohomology groups. Mathematical concepts like Banach spaces, C*-algebras, and tensor products are elucidated to understand the algebraic structures involved. Theorems and proofs establish the relationships and properties of these cohomology groups, culminating in a canonical isomorphism theorem for the product of two algebras. Overall, the article aims to provide a comprehensive exploration of dihedral cohomology and its interplay with other algebraic structures, offering insights into its applications and theoretical foundations. Additionally, it studies the (co)homology theory of C*-algebras, focusing on the triviality of cohomology groups of operator algebras and discovering the canonical isomorphism between any two unital K -algebras A and A′: HDn(A×A′)≅HDn(A)⊕HDn(A′).

As is known, there exists an alternative, way to present basic notions and results of quantum functional analysis (= operator space theory). This approach is based on considering, instead of matrix spaces, a single space, consisting, roughly speaking, of vectors from the initial linear space equipped with coefficients taken from some good operator algebra. It seems that so far there was no systematical exposition of the theory in the framework of the non-matricial approach. We believe, however, that in a number of topics the non-matricial approach gives a more elegant and transparent theory. In this paper we introduce, using only the non-matricial language, both quantum versions of the classical (Grothendieck) projective tensor product of normed spaces. These versions correspond to the matricial Haagerup and operator-projective tensor products. We define them in terms of the universal property with respect to some classes of bilinear operators, corresponding to matricial multiplicatively bounded and completely bounded, and then produce their explicit constructions. Among the relevant results, we shall show that both tensor products are actually quotient spaces of some genuine projective tensor products. Moreover, the Haagerup tensor product is itself a genuine projective tensor product, however not of just normed spaces but of some normed modules.