Questions tagged [purification]
For questions about purification of quantum states.
12
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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 ...
- 151
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2
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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?
...
- 43
2
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0
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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 $$
\rho^x = \...
- 39
1
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1
answer
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Simple proof that entangled pure states are not separable
I am trying to understand more about the notion of separable states. For clarity, I will only use the word entangled for pure states, even if a non-separable state is sometimes called entangled too.
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- 101
1
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1
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Implementing circuits with post-selection in Stim
I want to use Stim to calculate logical error rates with circuits that involve post-selection on measurement outcomes. Is it possible to do so with Stim and if yes, how do I go about it?
To provide ...
- 111
2
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0
answers
62
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List of inequalities for purity of a traced out bipartite system
I would like to know if there are inequalities related to the purity of the partial trace of a bipartite system.
The purity $P$ of a density matrix $\rho$ is given by
$$P(\rho) = Tr(\rho^2).$$
The ...
- 2,778
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0
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81
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Equivalence between entanglement purification and quantum error correction
I've recently been reading the seminal paper recently (BDSW 1996), where they established the equivalence between 1-EPP (one-way classical communication) and error correction through explicit ...
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368
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Prove the equality conditions in the triangle inequality $S(A,B)\ge |S(A)-S(B)|$ for the von Neumann entropy
The triangle inequality or Araki-Lieb inequality of the von Neumann entropy is
$$
S(A,B)\ge|S(A)-S(B)|
$$
this is proven by introducing a system $R$ which purifies systems $A$ and $B$. Applying ...
- 831
2
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0
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70
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Fidelity and Uhlmann's theorem in the context of source coding
In quantum source coding, we have an encoder $\mathcal{E}$ and a decoder $\mathcal{D}$ which are some quantum channels. Given a state $\rho_A$ on Hilbert space $\mathcal{H}_A$, we wish to encode and ...
- 3,085
2
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2
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433
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Does the unitary freedom in choice of Kraus operators come from the freedom in the choice of purifications?
Does the unitary freedom in the choice of Kraus operators for a given quantum channel just come from the unitary freedom in choice of purification of a quantum state?
Here's what I'm thinking. If I ...
- 559
4
votes
1
answer
138
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Closeness of unitary dilations of CPTP maps
Let $\Phi_1,\Phi_2 \colon S(\mathcal{H}) \to S(\mathcal{H})$ be CPTP maps on the same Hilbert space $\mathcal{H}$ which are $\varepsilon$-close in diamond norm, and let $U_1,U_2$ be respective unitary ...
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Uhlmann's theorem analogue for channels
Let the stabilized channel fidelity between two channels $M_{A\rightarrow B}$ and $N_{A\rightarrow B}$ be defined as
$$F(M,N) = \min\limits_{\vert\psi\rangle_{AR}} F\left((M\otimes I_R)\vert\psi\...
- 3,085