One-Shot distillation in a General Resource Theory

arxiv:1906.04959

Madhav Krishnan V

UTS 

Collaborators : Min-Hsiu Hsieh (UTS)

                                    Eric Chitambar (U. Illinois)

What is a resouce theory?

A resource allows you to overcome a restriction to perform a task

Operational definition

(Physically motivated)

\mathcal{O}

Free operations :

\mathcal{F}

Free states:

  • Easy to implement

  • Generated by Free operations

Resource states:

\rho
\notin
\mathcal{F}

Axiomatic definition

The pair 

\{ \mathcal F , \mathcal O \}

defines a resource theory

It must hold that

\forall \gamma \in \mathcal F, \Lambda \in \mathcal{O} \\ \Lambda (\gamma) \in \mathcal{F}

Examples

\mathcal F
\mathcal O
R.T.
Entanglement LOCC, SEP, PPT,...
Coherence MIO, IO, DIO,...
\sum_i p_i \rho_i \otimes \sigma_i
\sum_i p_i | i \rangle \langle i |_{R}

One-shot distillation

\rho
\Phi_M
\Lambda \in \mathcal O
\epsilon

One-shot distillable resource

R(\rho, \epsilon) := \max\limits_{M \in \mathbb{N}} \left\lbrace \log_2 M : \max\limits_{\Lambda \in \mathcal{O}} F^2(\Lambda(\psi), \Phi_{M}) \geq 1- \epsilon \right\rbrace
F^2(\Lambda(\psi), \Phi_{M}) \geq 1- \epsilon
\max\limits_{\Lambda \in \mathcal{O}} F^2(\Lambda(\psi), \Phi_{M}) \geq 1- \epsilon
G_{min}(\rho) := \min\limits_{\gamma \in \mathcal F} \{ -\log \text{Tr}(\rho\gamma) \}

Converse

The one-shot distillable resource can be bounded as

R(\rho, \epsilon) \leq \max\limits_{\overline\rho \in b(\rho, 2\sqrt\epsilon)}G_{min}(\overline\rho) \equiv G^{2\sqrt{\epsilon}}_{min}(\rho)

if

\Phi_M \gamma \Phi_M \leq \frac{1}{M} \Phi_M \hspace{0.5cm} \forall \gamma \in \mathcal F
  • Entanglement
  • Coherence 
  • Purity
\checkmark
\checkmark
\checkmark
{\scriptstyle b(\rho, 2\sqrt\epsilon) }
{\scriptstyle b(\rho, \epsilon) = \{ \overline\rho \hspace{0.1cm}: \hspace{0.1cm} F(\rho, \overline\rho ) \geq 1 - \epsilon \} }
{\scriptstyle b(\rho, \epsilon) = \{ \overline\rho \hspace{0.1cm}: \hspace{0.1cm} F(\rho, \overline\rho ) \geq 1 - \epsilon \} }

Proof Sketch

\log M \leq \min\limits_{\gamma \in \mathcal{I}} \left\{ -\log_2\text{Tr}(\overline{\rho}\gamma) \right\}
\overline{\rho} \text{ depends on } \Lambda, \Phi_M, \rho
\overline{\rho}
\cdot \rho
\cdot \overline{\rho}
= G_{min}(\overline \rho)
2\sqrt{\epsilon}
\log_2 M \leq \max\limits_{\overline{\rho} \in b(\rho, 2\sqrt\epsilon)} G_{min}(\overline{\rho})

Assume log M is the optimal rate

\exists \Lambda \in \mathcal O \text{ } s.t. \text{ } F^2(\Lambda(\rho), \Phi_M ) \geq 1 - \epsilon
\Phi_M \Lambda(\gamma) \Phi_M \leq \frac{1}{M} \Phi_M \hspace{0.5cm} \forall \gamma \in \mathcal F
G_{min}(\psi) = S_{min}(\Delta(\psi)) := -\log(\lambda_{max}(\Delta(\psi)))

Coherence

Entanglement

G_{min}(\psi^{AB}) = S_{min}(\rho^B_{\psi})

Pure state improvement

\log_2 M \leq \max\limits_{\overline{\psi} \in b_*(\psi, 2 \epsilon)} G_{min}(\overline{\psi})

Robustness

\mathcal R_f(\rho) := \min\limits_{\pi \in \mathcal F} \{ s \geq 0 : \frac{\rho + s \pi}{1 + s} \in \mathcal F \}
\mathcal R_g(\rho) := \min\limits_{\pi \in \mathcal D (\mathcal{H})} \{ s \geq 0 : \frac{\rho + s \pi}{1 + s} \in \mathcal F \}

Pure state transformation

\psi \rightarrow \phi
F^2(\Lambda(\psi), \phi) \geq 1-\epsilon
\Lambda(\omega) = \text{Tr}(\overline\psi\omega)\phi + (1 - \text{Tr}(\overline\psi\omega))\pi_{\phi}
\overline \psi \text{ optimizes } G_{min}^{\epsilon}(\psi)
\Lambda(\omega) = \text{Tr}(\overline\psi\omega)\phi + (1 - \text{Tr}(\overline\psi\omega))\pi_{\phi}
\text{If } \mathcal F \text{ is convex, } \Lambda \in \mathcal O, \text{ iff}
\pi_{\phi} \text{ optimizes } \mathcal R_f(\phi)
(1 - \text{Tr}(\overline\psi\gamma)) \geq \frac{ \mathcal R_f(\phi)}{1 + \mathcal R_f(\phi) } \hspace{0.1cm}\forall \gamma \in \mathcal F
\iff G_{min}^{\epsilon}(\psi) \geq \log(1 + \mathcal R_f(\phi))
G_{min}(\psi^{AB}) \geq \log(1 + R^E_f(\Psi^{AB}_M) \\ = \log M

One-shot Entanglement distillation

\mathcal R_f(\rho) := \min\limits_{\pi \in \mathcal F} \{ s \geq 0 : \frac{\rho + s \pi}{1 + s} \in \mathcal F \}

Not finite for all resource theories

Smoothing

\mathcal R^{\tilde\epsilon}_f(\rho) := \min\limits_{\pi \in \mathcal F} \{ s \geq 0 : \frac{\rho + s \pi}{1 + s} \in \mathcal F^{\tilde\epsilon} \}

Resource

\tilde\epsilon-
\mathcal F^{\tilde\epsilon} = \{ \rho : C(\rho) \leq \tilde\epsilon \}
C(\rho)

is some resource montone

\mathcal{O}^{\tilde\epsilon} = \{ \Lambda : \Lambda(\gamma) \in \mathcal F^{\tilde\epsilon} \hspace{0.1cm} \forall \gamma \in \mathcal F^{\tilde\epsilon} \}

Summary

For any convex resource theory such that

\Phi_M \gamma \Phi_M \leq \frac{1}{M} \Phi_M \hspace{0.5cm} \forall \gamma \in \mathcal F
R(\rho, \epsilon ) \leq G_{min}^{2\sqrt \epsilon}(\rho)
\rho \hspace{0.1cm}- \hspace{-0.2cm} \rightarrow
\Lambda \in \mathcal{O}
\psi \hspace{0.1cm}- \hspace{-0.2cm} \rightarrow
\Lambda \in \mathcal{O}^{\tilde\epsilon}
\Phi_M
\epsilon
\Phi_M
\epsilon
G_{min}^{\epsilon}(\psi) \geq \log(1 + \mathcal R_f^{\tilde\epsilon}(\Phi_M))

Future Work

  • Find analytical expression for rate from achievablity condition

  • Trade off-between    and 

  • Recover asymptotic limit

\tilde\epsilon
\epsilon
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