Abstract
The Hilbert space for a quantum system contains states that behave in a manner that is similar to classical objects. There are, however, also states in Hilbert space that are enigmatic and nonclassical in the sense of being forbidden by classical physics.
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- 1.
Multiplying from the right by elements of \(\vert w\rangle\) is another, but distinct, way of defining the outer product. One needs to consistently use only the rule one has chosen.
- 2.
One can equivalently define the outer product by multiplying the entire row vector from the right into each element of the column vector on the left and obtain the same result.
- 3.
The density matrix should be termed the density operator since, in general, it is not a finite or infinite matrix; however, the term density matrix is so widely used that its proper definition is implicitly understood.
- 4.
One can take the complete basis states of the degrees of freedom that is of larger dimension N and use it for the other degree of freedom, with some left over unused basis vectors.
- 5.
A similar result holds for taking a partial trace over the x 1 degree of freedom.
- 6.
Namely, \(\sum \limits _{i=1}^{N}\vert \psi _{i}^{I}\rangle \langle \psi _{i}^{I}\vert = \mathbb{I}\).
- 7.
The maximally entangled state is the same whether the partial trace is performed over quantum system I or system II.
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Baaquie, B.E. (2013). Density Matrix: Entangled States. In: The Theoretical Foundations of Quantum Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6224-8_6
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