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Pure and Mixed States

K. Kong Wan

Published Jul 9, 2019• 0 citations

Abstract

So far we have only considered states in Postulate 25.1(PS) which correspond to a maximal amount of information of the system. These are pure states φ s describable by unit vectors φ → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6289.tif"/> . Often we may only have a partial knowledge of a system. As an example, consider a less than ideal state preparation process which is unable to prepare the system in a desired pure state φ s . Instead the state preparation process can only determine the system to within a set of possible pure states φ l s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6290.tif"/> , ℓ = 1, 2,…, namely the system may end up to be in a pure state φ 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6291.tif"/> , or φ 2 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6292.tif"/> and so on. Suppose the state preparation process can also tell us the probabilities w ℓ of the system ending up in these pure states. In other words, we do not know for certain which pure state the system is actually in. We only know that the system has a probability w ℓ to be in pure state φ l s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6293.tif"/> . The system is then said to be in a classical mixture of states . In more abstract terms, the concept can be stated as A classical mixture of states is a characterisation of a given quantum system in terms of a set of possible pure states φ → l https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6294.tif"/> together with a corresponding set of probabilities w ℓ of their occurrence .

DOI: 10.1201/b22504-31

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So far we have only considered states in Postulate 25.1(PS) which correspond to a maximal amount of information of the system. These are pure states φ s describable by unit vectors φ → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6289.tif"/> . Often we may only have a partial knowledge of a system. As an example, consider a less than ideal state preparation process which is unable to prepare the system in a desired pure state φ s . Instead the state preparation process can only determine the system to within a set of possible pure states φ l s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6290.tif"/> , ℓ = 1, 2,…, namely the system may end up to be in a pure state φ 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6291.tif"/> , or φ 2 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6292.tif"/> and so on. Suppose the state preparation process can also tell us the probabilities w ℓ of the system ending up in these pure states. In other words, we do not know for certain which pure state the system is actually in. We only know that the system has a probability w ℓ to be in pure state φ l s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6293.tif"/> . The system is then said to be in a classical mixture of states . In more abstract terms, the concept can be stated as A classical mixture of states is a characterisation of a given quantum system in terms of a set of possible pure states φ → l https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq6294.tif"/> together with a corresponding set of probabilities w ℓ of their occurrence .

In the previous chapter, we did not examine the change of a state with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . It has been assumed that we have been examining a state at a given moment https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . In the general case, the state of a system changes with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . For example, say the motion of an electron in a potential field, its wave function depends on time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_4.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> : https://www.w3.org/1998/Math/MathML"> Ψ = Ψ q , t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_5.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . Such a dependence has to be described by a differential equation in which we may have https://www.w3.org/1998/Math/MathML"> ∂ Ψ ∂ t , ∂ 2 Ψ ∂ t 2 , ⋯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_6.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> .

The quantised angular momentum operators are obtained in §27.4 . These operators do not commute. They satisfy the commutation relations in Eqs. (27.111) to (27.113) . It follows that L ^ c x , L ^ c y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7135.tif"/> and L ^ c z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7136.tif"/> are essentially incompatible. They do not possess a complete orthonormal set of common eigenvectors. However, there is a state, the state of zero orbital angular momentum values, in which they do possess a definite value, i.e., the value 0. 1 L ^ c z , L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7137.tif"/> are compatible and they possess common eigenvectors and are simultaneously measurable. This is also true for L ^ c x , L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7138.tif"/> and for L ^ c y , L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7139.tif"/> . However, the eigenvectors of L ^ c z , L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7140.tif"/> are different from that of L ^ c x , L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7141.tif"/> or that of L ^ c y , L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq7142.tif"/> .

Bounded operators on a Hilbert space H → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq3665.tif"/> have many of the properties of operators defined on V → N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq3666.tif"/> . For example, Theorem 13.1(1) and Corollary 13.1(1) apply, e.g., for bounded operators we have ( 18.1 ) 〈 ϕ → | A ^ ϕ → 〉 = 〈 ϕ → | B ^ ϕ → 〉 ∀ ϕ → ∈ H → ⇒ A ^ = B ^ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq3667.tif"/>