On uniqueness of density matrix description as mixed state

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Discussion Overview

The discussion revolves around the uniqueness of the basis that diagonalizes a density matrix in quantum mechanics, particularly focusing on conditions that affect this uniqueness in both two-dimensional and higher-dimensional cases. Participants explore the mathematical properties of density matrices, including eigenvalues and the implications of changing basis vectors.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that a density matrix \(\rho\) can be diagonalized in a unique basis if it has distinct eigenvalues, while if the eigenvalues are equal, the basis is not unique.
  • Another participant challenges the notion of uniqueness, stating that the basis is never unique due to the ability to change the length or phase of basis vectors.
  • A further response questions the real-valued nature of the entries in the density matrix, suggesting that they do not contain phase information.
  • Participants clarify that any positive semidefinite Hermitian matrix of trace 1 qualifies as a density matrix, and discuss the implications of changing the length of basis vectors on linear independence.
  • There is a debate about whether changing the length of basis vectors preserves the condition that the trace of the density matrix remains fixed at 1.

Areas of Agreement / Disagreement

Participants express disagreement regarding the uniqueness of the basis for diagonalizing density matrices. While some argue that uniqueness is contingent on the eigenvalues, others maintain that the basis is inherently non-unique due to the flexibility in defining basis vectors.

Contextual Notes

Participants discuss the implications of mathematical properties of density matrices without resolving the conditions under which the trace remains fixed when changing basis vectors. There is also ambiguity regarding the interpretation of real-valued entries in the context of physical representations.

Who May Find This Useful

This discussion may be of interest to those studying quantum mechanics, particularly in the areas of quantum state representation and the mathematical properties of density matrices.

stevendaryl
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If you have a density matrix \rho, there is a basis |\psi_j\rangle such that
\rho is diagonal in that basis. What are the conditions on \rho such that the basis that diagonalizes it is unique?

It's easy enough to work out the answer in the simplest case, of a two-dimensional basis: Then \rho can be represented as a 2x2 matrix. It will have two eigenvalues, p and q. If p = q, then \rho is diagonal in every basis. If p \neq q, then there is a unique basis (up to permutations of elements and overall phase factor) that diagonalizes \rho.

Is there some result that is similar for bases of arbitrary dimension? What's the condition on \rho such that there is a unique way to diagonalize it?
 
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The basis is never unique as one can always change the length or the phase of the basis vectors. The necessary and sufficient condition for this being the only freedom is that the matrix has no multiple eigenvalues.
 
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A. Neumaier said:
The basis is never unique as one can always change the length or the phase of the basis vectors.
Aren't entries of density matrix real valued? Then there is no phase.
And what do you mean by changing length of basis vectors? If intensity in one output of PBS is twice the intensity of other output then it is so. You can not change that by mathematical manipulations.
 
zonde said:
Aren't entries of density matrix real valued?
No. Any positive semidefinite Hermitian matrix of trace 1 qualifies as a density matrix.

zonde said:
And what do you mean by changing length of basis vectors?
Changing the length of a basis vector preserves linear independence, hence we get another basis. This is a mathematical fact independent of physics. (Maybe stevendaryl intended to have an orthonormal basis - where lengths are fixed at 1 - but he didn't say so.)
 
A. Neumaier said:
No. Any positive semidefinite Hermitian matrix of trace 1 qualifies as a density matrix.
Okay. So let me restate my question:
Aren't entries of diagonal density matrix real valued?
A. Neumaier said:
This is a mathematical fact independent of physics.
If a mathematical fact disagrees with empirical fact then this particular mathematical fact does not describe particular physical situation and is irrelevant.
A. Neumaier said:
Changing the length of a basis vector preserves linear independence, hence we get another basis.
Does it preserve condition that the trace is fixed at 1?
 
zonde said:
Aren't entries of diagonal density matrix real valued?
Yes, but they do not contain the phase information.
zonde said:
Does it preserve condition that the trace is fixed at 1?
The trace is a basis-independent concept.
 

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