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Optical and Quantum Electronics

On the probability amplitude of quantum entanglement and the Pauli matrices

F.J. Duarte, T S Taylor, J. C. Slaten

Published Feb 1, 2020• 11 citations
Publisher

Abstract

Abstract The probability amplitudes for quantum entanglement, also known as Bell sates, are utilized to arrive explicitly at the identity matrix I and the $$\sigma_{x}$$ σx , $$\sigma_{y}$$ σy , and $$\sigma_{z}$$ σz Pauli matrices, via a straight-forward $$2 \times 2$$ 2×2 matrix representation that utilizes the vector direct product. It is also indicated that this approach is completely equivalent to the utilization of the Kronecker product $$\otimes$$ ⊗ to multiply the relevant ket vectors. Furthermore, it is shown that the polarization rotation matrix R, operating on the various versions of the probability amplitude for quantum entanglement, yields four identities elegantly interconnecting these probability amplitudes.

DOI
10.1007/s11082-020-2205-1

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Abstract The probability amplitudes for quantum entanglement, also known as Bell sates, are utilized to arrive explicitly at the identity matrix I and the σx\sigma_{x}σx​ σx , σy\sigma_{y}σy​ σy , and σz\sigma_{z}σz​ σz Pauli matrices, via a straight-forward 2×22 \times 22×2 2×2 matrix representation that utilizes the vector direct product. It is also indicated that this approach is completely equivalent to the utilization of the Kronecker product ⊗\otimes⊗ ⊗ to multiply the relevant ket vectors. Furthermore, it is shown that the polarization rotation matrix R, operating on the various versions of the probability amplitude for quantum entanglement, yields four identities elegantly interconnecting these probability amplitudes.
Using the vector direct product , the probability amplitudes for quantum entanglement, ψ + , ψ − , ψ + , and ψ − are expressed in a matrix form and shown to be represented by the Pauli matrices plus the identity matrix. The Hadamard matrix is derived from a beam splitter Hong-Ou-Mandel (HOM) configuration. Numerous mathematical operations with the polarization rotator matrix ( R = σ x ) and the Hadamard matrix operating on quantum entanglement states are demonstrated. This chapter concludes with a discussion of reversibility in quantum mechanics between quantum interference and quantum entanglement.
Matrices are widely, and extensively, used in quantum mechanics. Indeed, one of the avenues to quantum mechanics, that of Born, Heisenberg, and Jordan, is via matrices. One additional asset of matrices is that they are the natural mathematical language to describe quantum computing. Furthermore, matrices identify themselves in the physical tangible world via optical elements such as mirrors, beam splitters, polarization rotators, and interferometers. And these are the elements that comprise the hardware of photonic quantum computers. Here, we introduce the Pauli matrices https://www.w3.org/1998/Math/MathML" display="inline"> σ x , σ y , σ z , plus the identity matrix I , via the probability amplitudes of quantum entanglement. Once this is accomplished, we show how these matrices can be used to perform mathematical transformations, on quantum entanglement states, that form the bases of quantum computing as treated in the next chapter.
Following an introduction on interferometric physical computers, the discussion turns to Feynman’s contribution to quantum computing and qbits. The ψ + , ψ − , ψ + , and ψ − quantum entanglement states are expressed in terms of the σ x , σ y , σ z , and I matrices and the discussion continues, on quantitative terms, Pauli gates, the Hadamard gate, and the CNOT gate. The equivalence between σ x and the polarization rotational matrix R is noted. Interaction between the ψ + , ψ − , ψ + , and ψ − states, R and the Hadamard gate give rise to whole families of quantum mathematical operations. Physical representations for R and the Hadamard gate are explicitly identified.
In quantum mechanics, classical matrix bases such as the Pauli matrices are often generalized to higher dimensions. So, it is useful to express their corresponding operators using the Dirac Bra and Ket. In this paper, to express the corresponding operators we review the Kronecker-Pauli matrices and how to construct them for an N-dimensional system, with N a prime integer, N>2. Then, we give the expression of the Kronecker-Pauli operators and show that their matrices with respect to the standard basis fulfill the conditions to form a set of Kronecker-Pauli matrices.
In quantum mechanics, classical matrix bases such as the Pauli matrices are often generalized to higher dimensions. So, it is useful to express their corresponding operators using the Dirac Bra and Ket. In this paper, to express the corresponding operators we review the Kronecker-Pauli matrices and how to construct them for an N-dimensional system, with N a prime integer, N>2. Then, we give the expression of the Kronecker-Pauli operators and show that their matrices with respect to the standard basis fulfill the conditions to form a set of Kronecker-Pauli matrices. Relationship between the Kronecker-Pauli operators and the Weyl operators has been studied.