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Questions tagged [purification]

For questions about purification of quantum states.

Purification of a mixed state $\rho$ on system A is a pure state $|\psi\rangle$ on a larger composite system AB such that $\mathrm{tr}_B(|\psi\rangle\langle\psi|)=\rho$.

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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 dilations on some larger space $\mathcal{H}…
nickspoon
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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 quantum state $σ_{AE′} ∈ D(H_A ⊗ H_{E′})$ such…
Pink Elephants
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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 subadditivity obtains $S(R)+S(A)\ge S(A,R)$. $ABR$ is…
SOORAJ SOMAN
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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\rangle\langle\psi\vert, (N\otimes…
user1936752
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Can a qubit pair have a high fidelity thanks to purification yet be near its life time and have a high probability to die?

I have found that a qubit has two life times: $T_1$ during which it stays excited and $T_2$ during which it stays "coherent", meaning (from my understanding) that it is in its initial state. Now I am interested in knowing how long I can keep a pair…
Rome
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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 have two representations of the same quantum…
Sergio Escobar
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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 = \sum_{i_x} \lambda_{i,x} | i_x \rangle \langle…
Fireond
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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 context, I want to calculate thresholds for codes…
transistor
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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 most trivial inequality is $1/n \leq P \leq 1$,…
MonteNero
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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 then decode it to…
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Regarding which is more efficient at improving the fidelity of entangled pairs, error correction or purification

To enhance the fidelity of entangled pairs, common methods include error correction and purification. For different channel error rates (corresponding to different initial fidelities), these methods might have their respective advantages, especially…
QIAN
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About an inequality derived from Uhlmann's theorem

Consider two density operators $\rho, \xi \in \mathrm{D}(\mathcal{H} \otimes \mathcal{L})$ where $\mathrm{dim}(\mathcal{H}) \le \mathrm{dim}(\mathcal{L})$. Define \begin{align*} \epsilon := \mathrm{min}\{\|\,|\phi\rangle - |\psi\rangle\,\| :…
mazyloron
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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? Can anyone give a proof of it?
Sudhir Kumar
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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. Where could I find a simple proof that pure,…
user8622655
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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 construction. Is there are any similar results that hold…
AndyLiuin
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