Representing Mixed States in Hilbert Space

In summary, mixed states cannot be represented as rays in a Hilbert space like pure states because they correspond to a statistical mixture of pure states and do not have a unique direction where a measurement outcome is certain. This can be proven algebraically by showing that not every matrix can be written as a product of two vectors, and the concept of pure and mixed states is defined in terms of positive operators and convex sums. Therefore, it is not possible to represent all quantum states as rays in a Hilbert space, as some states are operators.
  • #1
Muthumanimaran
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Why cannot we represent mixed states with a ray in a Hilbert space like a Pure state.
I know Mixed states corresponds to statistical mixture of pure states, If we are able to represent Pure state as a ray in Hilbert space, why we can't represent mixed states as ray or superposition of rays in Hilbert space.
 
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  • #2
Because the statistics don't work out.

If you have a spin-1/2 particle in a given pure state, the outcome of a spin measurement depends on the direction you measure. For example, there's always a direction where you get one of the outcomes with certainty. For mixed states, such a direction doesn't exist.
 
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  • #3
The best way to understand why this cannot be done is to try to do it on an explicit simple example. Have you tried?
 
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  • #4
Muthumanimaran said:
why we can't represent mixed states as ray or superposition of rays in Hilbert space.
Algebraically, it is closely related to the following claim. Not every matrix ##C_{ij}## can be written as ##A_iB_j##.

How to prove it? By counterexample. Assume ##C_{ij}=A_iB_j##. As an example, consider the case ##C_{12}=0##. Then either ##A_1=0## or ##B_2=0##. But if ##A_1=0## then ##C_{11}=0##, and if ##B_2=0## then ##C_{22}=0##. Therefore the assumption cannot be satisfied when ##C_{12}=0##, ##C_{11}\neq 0##, and ##C_{22}\neq 0##. Q.E.D.
 
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How to prove it? By counterexample. Assume ##C_{ij}=A_iB_j##. As an example, consider the case ##C_{12}=0##. Then either ##A_1=0## or ##B_2=0##. But if ##A_1=0## then ##C_{11}=0##, and if ##B_2=0## then ##C_{22}=0##. Therefore the assumption cannot be satisfied when ##C_{12}=0##, ##C_{11}\neq 0##, and ##C_{22}\neq 0##. Q.E.D.[/QUOTE]

Thank you, your answer is simple and elegant.
 
  • #6
No need to prove anything. Its simple.

By definition quantum states are positive operators of unit trace. Forget this Hilbert space stuff - that's the kiddy version. By definition pure states are operators of the form |u><u|. Mixed states, again by definition, are convex sums of pure states. It can be shown all states are mixed or pure (start a new thread if you want to discuss it). Obviously pure states can be mapped to rays in a Hilbert space - but keep in mind in general states are operators.

Thanks
Bill
 

1. What is a mixed state in Hilbert Space?

A mixed state in Hilbert Space refers to a quantum state that cannot be described by a single pure state, but instead is a statistical combination of multiple pure states. This means that the state does not have a definite value for a particular observable, but instead has a probability distribution for the possible values.

2. How do we represent mixed states in Hilbert Space?

Mixed states are represented by density matrices in Hilbert Space. The density matrix is a mathematical tool that describes the statistical mixture of pure states that make up the mixed state. It is a Hermitian matrix with unit trace, and its eigenvalues represent the probabilities of the pure states in the mixture.

3. What are the differences between pure states and mixed states in Hilbert Space?

Pure states in Hilbert Space are described by a single vector, while mixed states are described by density matrices. Pure states have a definite value for a particular observable, while mixed states have a probability distribution for the possible values. Pure states also have a purity of 1, while mixed states have a purity less than 1.

4. How do we measure the purity of a mixed state in Hilbert Space?

The purity of a mixed state can be calculated by taking the trace of the square of the density matrix. This value will be between 0 and 1, with a purity of 1 indicating a pure state and a purity of 0 indicating a completely mixed state. The closer the purity is to 1, the more pure the mixed state is.

5. Can mixed states be entangled in Hilbert Space?

Yes, mixed states can be entangled in Hilbert Space. Entanglement refers to the correlation between two or more quantum systems, and this correlation can exist even in mixed states. The entanglement between the pure states that make up a mixed state can affect the overall properties and behavior of the mixed state.

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