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I have the following two density operators, the paper I am reading says that these two operators have same eigenvalues

$$\rho^i = \frac{1}{3} ( |0\rangle \langle 0 | +|1\rangle \langle 1 |+|2\rangle \langle 2 |+a|0\rangle \langle 1 |+a|1\rangle \langle 0 |+c|1\rangle \langle 2 |+c|2\rangle \langle 1 |),$$

$$\rho^f = \frac{1}{3} ( |0\rangle \langle 0 | +|1\rangle \langle 1 |+|2\rangle \langle 2 |+ax|0\rangle \langle 1 |+ax^*|1\rangle \langle 0 |+cy|1\rangle \langle 2 |+cy^*|2\rangle \langle 1 |).$$

Here $a,c$ are real numbers and $x,y$ are unimodular complex numbers. Also $\{|0\rangle ,|1\rangle , |2\rangle \} $ are three orthonormal vectors of a 3-D Hilbert space.
My approach : its apparent that the operators are hermitian and thus have a spectral decomposition in terms of orthonormal vectors. Lets say one of the vectors in spectral decomposition of first operator is $$|\psi \rangle = A|0 \rangle + B|1\rangle,$$ then for the second operator the corresponding one would be $$|\psi \rangle = Ax^{\frac{1}{2}}|0 \rangle +(x^*)^{\frac{1}{2}} B|1\rangle$$ ( $A,B,C,D$ are some complex constants ). Similarly another vector in spectral decomposition of first operator will be of form $$|\psi \rangle = C|1 \rangle + D|2\rangle$$ and its counterpart in second operators spectral decomposition will be $$|\psi \rangle = y^{\frac{1}{2}}C|1 \rangle + (y^*)^{\frac{1}{2}}D|2\rangle.$$ Thus keeping eigen values same.
Doubt : But my approach seems to be an intuitive and vague proof, is there a way where it is evident that they have same eigenvalues in a simple manner?

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I have the following two density operators, the paper I am reading says that these two operators have same eigenvalues

$$\rho^i = \frac{1}{3} ( |0\rangle \langle 0 | +|1\rangle \langle 1 > |+|2\rangle \langle 2 |+a|0\rangle \langle 1 |+a|1\rangle \langle 0 > |+c|1\rangle \langle 2 |+c|2\rangle \langle 1 |),$$

$$\rho^f = \frac{1}{3} ( |0\rangle \langle 0 | +|1\rangle \langle 1 > |+|2\rangle \langle 2 |-ax|0\rangle \langle 1 |+ax^*|1\rangle \langle > 0 |+cy|1\rangle \langle 2 |+cy^*|2\rangle \langle 1 |).$$

They don't.

Did you mean to write "+ax" instead of "-ax" in the second matrix?

If so, then they do... in either case (if you meant to write "+ax") the eigenvalue equation is: $$ (1-\lambda)^3=(1-\lambda)(a^2+c^2) $$ giving $$ \lambda=1\;,\;1\pm\sqrt{a^2+c^2} $$

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  • $\begingroup$ yeah sorry corrected the mistake it was meant to be '+ax' $\endgroup$ – sashas Mar 9 '15 at 7:13
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Hints:

  1. What happens to the density operator and its eigenvalues under a change of the orthonormal basisvectors $|0\rangle $, $|1\rangle $, $|2\rangle$ by phase factors?

  2. More generally, what happens to the density operator and its eigenvalues under a unitary transformation $\rho\longrightarrow U^{\dagger}\rho U$?

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