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Let $\rho$ be the density matrix for a system and let the POVMs be $\{E_m\}$ such that $\sum_i {E_m} = I$. The probability of getting the outcome $m$ is $\operatorname{Tr}(E_m \rho)$.

The source I am reading from ( Semi definite programming to characterize quantum set , section 2.2 second paragraph ) says that the same probabilities of getting the outcomes can be obtained by performing projective measurements on a pure state (there is no restriction given on what the pure state is). That is there exists a pure state $|\phi\rangle$ ( of some arbitary dimension ) and projective opererators $\{P_m\}$ such that for every $m$ , $\langle \phi | P_m |\phi \rangle = \operatorname{Tr}(E_m\rho)$. To state it more clearly I want to prove that general POVMs can be done as projective measurements on a pure state of a larger dimension Hilbert space.

For more clarity the exact paragraph from the source I mentioned above , which I am trying to prove is

Any any general measurement on a given Hilbert space can be viewed as a projective measurement on a larger Hilbert space, and any mixed state ρ can be viewed as a subsystem of a larger system in a pure state.


I tried but can't find a way to prove the statement . I was able to prove that for POVMs for a mixed state $\rho$ can be seen as POVMs for a pure state $|\psi\rangle$ of a larger dimension ( using purification ). But I have no idea oh how to prove that POVMs on a pure state can be seen as projective measurements on some other pure state ( of some other dimension ). I am not even sure if this is the correct way to proceed.

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  • $\begingroup$ Projective measurement sutisfies $P=P^{\dagger}$. You want to prove that for any $E_m$ and pure state exist $P=E_m\otimes A$ such that $P=P^{\dagger}$ i.e. $E_m\otimes A=E_m^{\dagger}\otimes A^{\dagger}$ .Thus you need to prove that for any $E_m$ you can find $A$ that sutisfies this relation. $\endgroup$ – Alexander Jul 19 '15 at 8:02
  • $\begingroup$ @Alexander How would this answer the question? -- Since the comments have been deleted, let me say again that this is answered in Preskill's lecture notes, chapter 3.1.4, "Neumark's theorem". $\endgroup$ – Norbert Schuch Jul 19 '15 at 9:48
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    $\begingroup$ Read also my answer here physics.stackexchange.com/questions/184524/… - if I remember correctly, I also more or less sketched a proof of Naimark's dilation theorem that Norbert Schuch referred to. $\endgroup$ – Martin Jul 23 '15 at 10:27

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