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Questions tagged [purification]

For questions about purification of quantum states.

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4 votes
2 answers
101 views

Show that all extensions of $\rho$ can be obtained as a channel applied to its purification

I am struggling with this exercise here: Let $H:A, H_E$ and $H_{E′}$ denote complex Euclidean spaces. Consider a purification $|ψ_{AE}⟩⟨ψ_{AE}| ∈ D(H_A ⊗ H_E)$ of a quantum state $ρ_A ∈ D(H_A)$ and a ...
Pink Elephants's user avatar
1 vote
2 answers
67 views

Why is it impossible to make a mixed-state qubit into a pure-state qubit?

How is it impossible to make a mixed-state qubit (having Bloch vector of length $l < 1$) into a pure-state qubit (having Bloch vector of length $l' = 1$) via a quantum operation that is invertible? ...
Sudhir Kumar's user avatar
2 votes
0 answers
73 views

Purification of classical-quantum state

For a classical-quantum state in $\mathcal{H}_{X \otimes A}$ $$ \rho = \sum_x p(x) \vert x\rangle \langle x \vert \otimes \rho^x, $$ I can use spectral decomposition for $ \rho^x $ to get $$ \rho^x = \...
Fireond's user avatar
  • 39
1 vote
1 answer
198 views

Simple proof that entangled pure states are not separable

I am trying to understand more about the notion of separable states. For clarity, I will only use the word entangled for pure states, even if a non-separable state is sometimes called entangled too. ...
user8622655's user avatar
1 vote
1 answer
137 views

Implementing circuits with post-selection in Stim

I want to use Stim to calculate logical error rates with circuits that involve post-selection on measurement outcomes. Is it possible to do so with Stim and if yes, how do I go about it? To provide ...
transistor's user avatar
1 vote
0 answers
47 views

List of inequalities for purity of a traced out bipartite system

I would like to know if there are inequalities related to the purity of the partial trace of a bipartite system. The purity $P$ of a density matrix $\rho$ is given by $$P(\rho) = Tr(\rho^2).$$ The ...
MonteNero's user avatar
  • 2,646
1 vote
0 answers
76 views

Equivalence between entanglement purification and quantum error correction

I've recently been reading the seminal paper recently (BDSW 1996), where they established the equivalence between 1-EPP (one-way classical communication) and error correction through explicit ...
AndyLiuin's user avatar
  • 325
4 votes
0 answers
337 views

Prove the equality conditions in the triangle inequality $S(A,B)\ge |S(A)-S(B)|$ for the von Neumann entropy

The triangle inequality or Araki-Lieb inequality of the von Neumann entropy is $$ S(A,B)\ge|S(A)-S(B)| $$ this is proven by introducing a system $R$ which purifies systems $A$ and $B$. Applying ...
Sooraj S's user avatar
  • 821
2 votes
0 answers
69 views

Fidelity and Uhlmann's theorem in the context of source coding

In quantum source coding, we have an encoder $\mathcal{E}$ and a decoder $\mathcal{D}$ which are some quantum channels. Given a state $\rho_A$ on Hilbert space $\mathcal{H}_A$, we wish to encode and ...
user1936752's user avatar
  • 2,997
2 votes
2 answers
376 views

Does the unitary freedom in choice of Kraus operators come from the freedom in the choice of purifications?

Does the unitary freedom in the choice of Kraus operators for a given quantum channel just come from the unitary freedom in choice of purification of a quantum state? Here's what I'm thinking. If I ...
SescoMath's user avatar
  • 507
4 votes
1 answer
131 views

Closeness of unitary dilations of CPTP maps

Let $\Phi_1,\Phi_2 \colon S(\mathcal{H}) \to S(\mathcal{H})$ be CPTP maps on the same Hilbert space $\mathcal{H}$ which are $\varepsilon$-close in diamond norm, and let $U_1,U_2$ be respective unitary ...
nickspoon's user avatar
4 votes
0 answers
89 views

Uhlmann's theorem analogue for channels

Let the stabilized channel fidelity between two channels $M_{A\rightarrow B}$ and $N_{A\rightarrow B}$ be defined as $$F(M,N) = \min\limits_{\vert\psi\rangle_{AR}} F\left((M\otimes I_R)\vert\psi\...
user1936752's user avatar
  • 2,997