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Communications in Theoretical Physics

Quantum Hadamard Operators and Their Decomposition Derived by Virtue of IWOP Technique

Fan Hong-Yi, Qin Guo

Published Apr 1, 2008• 15 citations
Publisher

Abstract

We introduce the quantum Hadamard operator in continuum state vector space and find that it can be decomposed into a single-mode squeezing operator and a position-momentum mutual transform operator. The two-mode Hadamard operator in bipartite entangled state representation is also introduced, which involves the two-mode squeezing operator and |η⟩ ↔ ξ⟩ mutual transformation operator, where |η⟩ and ξ⟩ are mutual conjugate entangled states. All the discussions are proceeded by virtue of the IWOP technique.

DOI
10.1088/0253-6102/49/4/11

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We introduce the quantum Hadamard operator in continuum state vector space and find that it can be decomposed into a single-mode squeezing operator and a position-momentum mutual transform operator. The two-mode Hadamard operator in bipartite entangled state representation is also introduced, which involves the two-mode squeezing operator and |η⟩ ↔ ξ⟩ mutual transformation operator, where |η⟩ and ξ⟩ are mutual conjugate entangled states. All the discussions are proceeded by virtue of the IWOP technique.
According to the continuum bipartite entangled state representation |η> we derive the Schmidt decomposition of |η>, respectively in the coordinate representation, momentum representation and the particle number representation, and explain their physical significances. As the applications of the Schmidt decomposition, we directly derive the action of the single-mode squeeze operator on |η>, the two-mode squeeze operator's entangled state representation, as well as the matrix element of displacement operator in the Fock space. Generalization of our discussion in this paper to the multipartite entangled state cases is feasible.
We introduce the quantum fractional Hadamard transform with continuous variables. It is found that the corresponding quantum fractional Hadamard operator can be decomposed into a single-mode fractional operator and two single-mode squeezing operators. This is extended to the entangled case by using the bipartite entangled state representation. The new transformation presents more flexibility to represent signals in the fractional Hadamard domain with extra freedom provided by an angle and two-squeezing parameters.
We present the continuous state vector of the total coordinate of multi-particle and the state vector of their total momentum, respectively, which possess completeness relation in multi-mode Fock space by virtue of the integration within an order product (IWOP) technique. We also calculate the transition from classical transformation of variables in the states to quantum unitary operator, deduce a new multi-mode squeezing operator, and discuss its squeezing effect. In progress, it indicates that the IWOP technique provides a convenient way to construct new representation in quantum mechanics.