Categorical quantum mechanics

Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably . The primitive objects of study are physical processes, and the different ways that these can be composed. It was pioneered in 2004 by Samson Abramsky and Bob Coecke. Categorical quantum mechanics is entry 18M40 in MSC2020. Mathematically, the basic setup is captured by a : composition of morphisms models sequential composition of processes, and the tensor product describes parallel composition of processes. The role of the dagger is to assign to each state a corresponding test. These can then be adorned with more structure to study various aspects. For instance: A allows one to distinguish between an "input" and "output" of a process. In the diagrammatic calculus, it allows wires to be bent, allowing for a less restricted transfer of information. In particular, it allows entangled states and measurements, and gives elegant descriptions of protocols such as quantum teleportation. In quantum theory, it being compact closed is related to the Choi-Jamiołkowski isomorphism (also known as process-state duality), while the dagger structure captures the ability to take adjoints of linear maps. Considering only the morphisms that are completely positive maps, one can also handle mixed states, allowing the study of quantum channels categorically. Wires are always two-ended (and can never be split into a Y), reflecting the no-cloning and no-deleting theorems of quantum mechanics. Special commutative dagger Frobenius algebras model the fact that certain processes yield classical information, that can be cloned or deleted, thus capturing classical communication. In early works, dagger biproducts were used to study both classical communication and the superposition principle. Later, these two features have been separated. Complementary Frobenius algebras embody the principle of complementarity, which is used to great effect in quantum computation, as in the ZX-calculus.

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No-deleting theorem

In physics, the no-deleting theorem of quantum information theory is a no-go theorem which states that, in general, given two copies of some arbitrary quantum state, it is impossible to delete one of the copies. It is a time-reversed to the no-cloning theorem, which states that arbitrary states cannot be copied. This theorem seems remarkable, because, in many senses, quantum states are fragile; the theorem asserts that, in a particular case, they are also robust. Physicist Arun K. Pati along with Samuel L.

Dagger compact category

In , a branch of mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from their category of finite-dimensional continuous unitary representations (that is, ). They also appeared in the work of John Baez and James Dolan as an instance of semistrict k-tuply , which describe general topological quantum field theories, for n = 1 and k = 3.

No-cloning theorem

In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James Park, in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist (the same result would be independently derived in 1982 by Wootters and Zurek as well as Dieks the same year).

Empirical Sample Complexity of Neural Network Mixed State Reconstruction

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Quantum state reconstruction using Neural Quantum States has been proposed as a viable tool to reduce quantum shot complexity in practical applications, and its advantage over competing techniques has been shown in numerical experiments focusing mainly on ...

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In this paper we consider the compilation of quantum state permutations into quantum gates for physical quantum computers. A sequence of generic single-target gates, which realize the input permutation, are extracted using a decomposition based reversible ...

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Ultra-coherent nano-mechanical resonators for quantum optomechanics at room temperature

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