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.
