# On uniqueness of density matrix description as mixed state

• I
• stevendaryl
In summary: Changing the length of a basis vector does not affect the trace. In summary, the conditions for a unique basis that diagonalizes \rho are that \rho has no multiple eigenvalues, and that the entries of the density matrix are real-valued. Changing the length or phase of basis vectors does not affect the trace, and therefore does not affect the uniqueness of the diagonalization.
stevendaryl
Staff Emeritus
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?

zonde
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.

dextercioby
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.

## 1. What is a density matrix?

A density matrix is a mathematical tool used in quantum mechanics to describe the state of a quantum system. It is a matrix that contains all the necessary information about the system, such as its energy levels and probabilities of being in each state.

## 2. What is the uniqueness of density matrix description as a mixed state?

The uniqueness of density matrix description as a mixed state means that any mixed state can be uniquely represented by a density matrix. This is important because it allows us to use the same mathematical framework to describe both pure and mixed states.

## 3. How is a density matrix used to describe a mixed state?

A density matrix is used to describe a mixed state by representing the probabilities of the system being in different pure states. It does this by using a combination of diagonal and off-diagonal elements, where the diagonal elements represent the probabilities of the system being in a pure state, and the off-diagonal elements represent the correlations between different pure states.

## 4. Can a density matrix be used to describe a classical system?

No, a density matrix cannot be used to describe a classical system. Density matrices are only used in quantum mechanics to describe the behavior of quantum systems. Classical systems follow different rules and can be described using other mathematical tools, such as probability distributions.

## 5. What is the physical significance of the density matrix?

The physical significance of the density matrix lies in its ability to describe the behavior of quantum systems. It allows us to calculate the probabilities of different outcomes when measuring a system, and also to study the effects of decoherence, which is the loss of coherence in a quantum system due to interactions with its environment.

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