One-shot assisted concentration of coherence

Madhav Krishnan V

arxiv:1804.06554 with Eric Chitambar & Min-Hsiu Hsieh

What is a Resource Theory?

Full Theory

Free Entities

Expensive Entities

\rho = \sum\limits_i p_i \rho_i \otimes \sigma_i
\rho\neq \sum\limits_i p_i \rho_i \otimes \sigma_i

LOCC

Non-Local 

\Lambda

Entanglement :

Purity :

\rho = \frac{1 }{d}\mathbb{I}_d
\rho \neq \frac{1 }{d}\mathbb{I}_d

Noisy Ops

Purity increasing ops

Generalized Resource theory

Anshu, A., Hsieh, M. H., & Jain, R :  arXiv preprint arXiv:1708.00381. (2017)
  • Using some broad constraints on convexity and preservation of tensor product structures, a large number of resource theories can be cast in a unified framework
  • Entanglement
  • Purity
  • Coherence
  • Thermodynamics
  • Non-uniformity
  • Randomness extractors
  • Contextuality
  • Assymetry

Resource theory of coherence

Free Entities

Expensive Entities

\rho = \sum\limits_i p_i | i \rangle \langle i |
\rho\neq \sum\limits_i p_i | i \rangle \langle i |

Incoherent operations

Coherence creating operations

Fixed basis

\{ | i \rangle \}
| i \rangle \langle i |
| i \rangle \langle i |

Connection to Entanglement

  • Trade off between entanglement and coherence has been studied in Physical review letters 117.2 (2016): 020402.
  • A connection was made between coherence and entanglement in the resource distribution scenario in Physical review letters 116.7 (2016): 070402.

Definitions

  • Set of Incoherent states
\mathcal{I}
\delta = \sum\limits_i p_i | i \rangle\langle i |
  • Set of Incoherent operations
\Lambda : \mathcal{I} \rightarrow \mathcal{I}
\mathcal{O}

Measure of coherence

  • Relative entropy of coherence 
C_r(\rho)
C_r(\rho) = \min\limits_{\delta\in \mathcal{I}} S( \rho \| \delta)
= S(\Delta(\rho)) - S(\rho),^1
\text{where } \Delta \text{ deletes off-diagonal terms}
1.Winter, A., & Yang, D. : PRL 116.12, 120404, (2016)

Maximally coherent state

\bullet \text{ There exists a state } | \Phi_M \rangle, \text{for which }C_r(\Phi_M) \text{ is maximized}
| \Phi_M \rangle = \sum\limits_{i =1 }^M \frac{1}{M} | i \rangle

Important operational Tasks

  • Coherence distillation
\rho^{\otimes n} \rightarrow \Phi_2 ^{\otimes m}
  • Coherence dilution
\Phi_2^{\otimes n} \rightarrow \rho ^{\otimes m}

Optimal Asymptotic Rate

R := \frac{m}{n} = C_r(\rho)

Optimal Asymptotic Rate

C_f(\rho) := \min\limits_{ \{ p_i, \psi_i \}}\sum_i p_i C_r(\psi_i)
\text{where, }\rho = \sum\limits_i p_i \psi_i
Winter, A., & Yang, D. : PRL 116.12, 120404, (2016)

Assisted distillation of coherence

Alice

Bob

\left(| \psi \rangle^{AB} \right)^{\otimes n}
\Lambda \text{ (unrestricted)}
\Lambda \in \mathcal{O}

One-way classical communication

\Phi_2^{\otimes m}

Assisted distillation of coherence

Optimal rate
S(\Delta(\rho^B))^1
n \rightarrow \infty
n = 1
1. Chitambar, E., et al. Physical review letters 116.7 (2016): 070402.
?

Method of the proof

  • The assisted distillation task can be broken down into two parts
  1. Alice performs POVM            on her part of the shared state          and sends the result to Bob.
\psi^{AB}
\{ P^A_i\}

2. Bob  performs an Incoherent operation           on his part of the state based on              Alice's outcome. 

\Lambda_i
\max\limits_{\lbrace P^A_i \rbrace_i}\max\limits_{M \in \mathbb{N}} \left\lbrace \log_2 M : \max\limits_{\lbrace \Lambda^{B}_i \rbrace_i} F^{2}\left( \sum\limits_{i}p_i \Lambda_i^{B}(\rho_{i}^{B}), \Phi_{M}^{B^{\prime}} \right) \geq 1 - \epsilon \right\rbrace
\max\limits_{\lbrace p_i, \psi^{B}_i \rbrace_i}\max\limits_{M \in \mathbb{N}} \left\lbrace \log_2 M : \max\limits_{\lbrace \Lambda^{B}_i \rbrace_i} F^{2}\left( \sum\limits_{i}p_i \Lambda_i^{B}(\psi_{i}^{B}), \Phi_{M}^{B^{\prime}} \right) \geq 1 - \epsilon \right\rbrace
\text{where, }\sum\limits_i p_i \psi_i = \rho^B = \text{Tr}_A(\psi^{AB})
p_i \rho_i^B = \text{Tr}_A((P_i^A \otimes \mathbb{I}^B ) \psi^{AB})
\rho_i^B
\rho_i^B
  • So the best strategy is for Alice to create an optimal ensemble on Bob's side and for Bob to optimally distill this ensemble
  • So, we need to calculate the optimal concentration rate for an ensemble of pure states
\{ p_i, \psi_i \}

Pure state concentration

C_c(\psi, \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
\max\limits_{\overline{\psi} \in b_*(\psi, \epsilon)} S_{min}(\Delta(\overline{\psi})) - \delta \leq C_{c}(\psi, \epsilon) \leq \max\limits_{\overline{\psi} \in b_*(\psi, 2\epsilon)} S_{min}(\Delta(\overline{\psi}))
S_{min}(\rho) := - \log_2(\lambda_{max}(\rho))
b_*(\rho, \epsilon) = \{ \overline{\psi} \text{ s.t. } F^2(\rho, \overline{\psi}) \geq 1 - \epsilon \}

Proof of pure state concentration

\bullet \text{ If } \Delta(\Phi_M) \succ \Delta(\psi), \text{ then there exists a } \Lambda \text{ s.t,}^1
1. Physical review letters 116.12 (2016): 120404
\Lambda(\psi) = \Phi_M
\because \text{ spec}(\Phi_M) = ( \frac{1}{M},...,\frac{1}{M}), \text{ this is equivalent to,}
\frac{1}{M} \geq \max\limits_i \langle i | \psi | i \rangle = \lambda_{max}(\Delta(\psi))
\Rightarrow -\log_2(\lambda_{max}(\Delta(\psi))) = S_{min}(\Delta(\psi)) \geq \log_2 M
\bullet \text{ For any } \overline{\psi}, \text{ such that } F^2(\overline{\psi}, \psi ) \geq 1 - \epsilon, \text{there exists a } \Lambda , \text{ s.t.,}
\Lambda(\overline{\psi}) = \Phi_{\overline{M}}, \text{ where } \overline{M} = S_{min}(\Delta(\overline{\psi})) - \delta
\bullet \text{ If we apply the same } \Lambda \text{ on } \psi, \text{ we will get,}
1 - \epsilon \leq F^2(\overline{\psi}, \psi ) \leq F^2(\Lambda(\overline{\psi} ), \Lambda(\psi ))
= F^2( \Phi_{\overline{M}}, \Lambda(\psi))
\bullet \text{ So, we have},
\max\limits_{\overline{\psi} \in b_*(\psi, \epsilon)} S_{min}(\Delta(\overline{\psi})) - \delta \leq C_{c}(\psi, \epsilon)

For the converse,

\bullet \text{ Let } M \text{ be the maximum of all } \epsilon \text{ acheivable rates for concentration}
\text{then there exists a } \Lambda \text{ s.t., } F^2(\Lambda(\psi), \Phi_M) \geq 1 - \epsilon
\bullet \text{ Note that for any incoherent state } \gamma,
\Phi_M \Lambda(\gamma) \Phi_M = \frac{1}{M} \Phi_M
\Rightarrow \log_2M \leq -\log_2 \text{Tr}(\Phi_M \Lambda(\psi) \Phi_M \Lambda(\gamma) )
= -\log_2\text{Tr}(\Lambda^*(\Phi_M \Lambda(\psi) \Phi_M ) \gamma )
= -\log_2\text{Tr}(Q \gamma )
\log_2M \leq -\log_2\text{Tr}(Q\gamma)
\leq -\log_2\text{Tr}(\sqrt{Q} \psi \sqrt{Q} \gamma)
\leq -\log_2\text{Tr}(\overline{\psi}\gamma)
| \overline{\psi} \rangle \equiv \frac{\sqrt{Q} | \psi \rangle} {\sqrt{\langle \psi | Q | \psi \rangle }}
\bullet \text{ As } \gamma \text{ is an arbitrary incoherent state, we can write,}
\log_2M \leq \min\limits_{\gamma \in \mathcal{I}} \left\lbrace -\log_2 \text{Tr}( \overline{\psi}\gamma ) \right\rbrace
= -\log_2(\lambda_{max}(\Delta(\overline{\psi}))
= S_{min}(\Delta(\overline{\psi}))
\bullet \text{ Now we just have to show that } \overline{\psi} \in b_*(\psi, 2\epsilon)
\sqrt{\text{Tr}(Q\psi)} = \sqrt{\text{Tr}(\Phi_M \Lambda(\psi) \Phi_M \Lambda(\psi))} = \langle \Phi_M | \Lambda(\psi) | \Phi_M \rangle
= F^2(\Lambda(\psi) , \Phi_M ) \geq 1 - \epsilon
F(\psi, \overline{\psi}) = \frac{\langle \psi| \sqrt{Q}|\psi\rangle }{\sqrt{\langle \psi | Q | \psi \rangle }} \geq \frac{\langle \psi |Q | \psi \rangle}{\sqrt{\langle \psi | Q | \psi \rangle}} = \sqrt{\text{Tr}(Q\psi)} > 1 - \epsilon
\bullet \text{ So, } F^2(\overline{\psi}, \psi) > 1 - 2 \epsilon, \text{ hence proving the converse.}

Pure state concentration

C_c(\psi, \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
\max\limits_{\overline{\psi} \in b_*(\psi, \epsilon)} S_{min}(\Delta(\overline{\psi})) - \delta \leq C_{c}(\psi, \epsilon) \leq \max\limits_{\overline{\psi} \in b_*(\psi, 2\epsilon)} S_{min}(\Delta(\overline{\psi}))
S_{min}(\rho) := - \log_2(\lambda_{max}(\rho))
b_*(\rho, \epsilon) = \{ \overline{\psi} \text{ s.t. } F^2(\rho, \overline{\psi}) \geq 1 - \epsilon \}
\text{ If } \Delta(\Phi_M) \succ \Delta(\psi) :

Proof Sketch direct

\psi
\Phi_M
\Lambda_{\mathcal{I}}
\geq S_{min}(\Delta(\psi))
\log_2M
M
Winter, A., & Yang, D. : PRL 116.12, 120404, (2016)
\cdot \psi
\Phi_M
\Phi_{\overline{M}}
\cdot\overline{\psi}
\epsilon
\epsilon^{\prime}
\text{with } \epsilon^{\prime} \text{error, optimal}
\text{rate } M^*, \text{ will be max }
\text{over}
\text{ball}
\text{around } \Phi_M
\log_2M^* \geq \max\limits_{\overline{\psi} \in b_*(\psi, \epsilon)}S_{min}(\Delta(\overline{\psi}))

Proof Sketch: Converse

\text{Starting with an operator identity :}
\Phi_M \delta \Phi_M = \frac{1}{M}\Phi_M
\delta \in \mathcal{I}
\delta
\bullet \text{ Assume the optimal rate with error } \epsilon \text{ is } M, \text{ acheived by } \Lambda
\log_2M \leq \min\limits_{\gamma \in \mathcal{I}} \left\{ -\log_2\text{Tr}(\overline{\psi}\gamma) \right\}
\overline{\psi} \text{ depends on } \Lambda, \Phi_M, \psi
\overline{\psi}
\psi
\overline{\psi}
= \max\limits_i \langle i | \overline{\psi} | i \rangle
= S_{min}(\Delta(\overline{\psi}))
2\epsilon
\log_2 M \leq \max\limits_{\overline{\psi} \in b_*(\psi, 2\epsilon)} S_{min}(\Delta(\overline{\psi}))

Ensemble concentration

C_c(\mathfrak{E}, \epsilon) := \max\limits_{M \in \mathbb{N}} \left\lbrace \log_2M : \max\limits_{ \lbrace\Lambda_{i}\rbrace_i} F^{2}\left(\sum\limits_{i}p_i\Lambda_i(\psi_i), \Phi_{M} \right) \geq 1 - \epsilon \right\rbrace
\bullet \text{ Let }\mathfrak{E} = \{ p_i, \psi_i \} \text{ be a pure state ensemble}
\bullet \text{ We define the optimal concentration rate for } \mathfrak{E} \text{ as}
\max\limits_{\overline{\mathfrak{E}} \in b(\mathfrak{E}, \epsilon)}F_{min}^{\Delta}(\overline{\mathfrak{E}}) - \delta \leq C_{c}(\mathfrak{E}, \epsilon) \leq \max\limits_{\overline{\mathfrak{E}} \in b(\mathfrak{E}, 2\epsilon)} F_{min}^{\Delta}(\overline{\mathfrak{E}})
\text{where, } F^{\Delta}_{min}(\mathfrak{E}) = \min\limits_{\psi_i \in \mathfrak{E}} S_{min}(\Delta(\psi_i))

Asymptotic limit

\bullet \text{ The coherence of assistance is defined as,}
D_a(\rho^B) = \max\limits_{\mathfrak{E}_{\rho}=\lbrace p_i, \psi_i \rbrace_i} \sum\limits_i p_i S(\Delta(\psi_i)) = D_c^{A | B}(\psi^{AB}) = S(\Delta(\rho^B))^1
Chitambar, E., et al. Physical review letters 116.7 (2016): 070402.
\bullet \text{ We define the one-shot coherence of assistance as,}
C_a(\rho, \epsilon) = \max\limits_{\mathfrak{E}_{\rho}} C_{c}(\mathfrak{E}_{\rho}, \epsilon)
\lim\limits_{\epsilon \rightarrow 0}\lim\limits_{n \rightarrow \infty}\frac{1}{n} C_a(\rho^{\otimes n}, \epsilon) = \lim\limits_{n \rightarrow \infty}\frac{1}{n}D_a(\rho^{\otimes n})

Conclusions

  • We derived bounds on the pure state concentration of coherence 
  • We generalize this bound to an ensemble of pure states and use this to find the one-shot coherence concentration.
  • We recover the correct asymptotic behaviour.
  • We think our converse proof can be generalized to any resource theory with a maximal resource state. 
  • Similar questions are open for when 2-way communication is allowed

Thank you!

Made with Slides.com