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
ρ=∑ipiρi⊗σi\rho = \sum\limits_i p_i \rho_i \otimes \sigma_i
\rho\neq \sum\limits_i p_i \rho_i \otimes \sigma_i
ρ≠∑ipiρi⊗σi\rho\neq \sum\limits_i p_i \rho_i \otimes \sigma_i

LOCC

Non-Local 

\Lambda
Λ\Lambda

Entanglement :

Purity :

\rho = \frac{1 }{d}\mathbb{I}_d
ρ=1dId\rho = \frac{1 }{d}\mathbb{I}_d
\rho \neq \frac{1 }{d}\mathbb{I}_d
ρ≠1dId\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 |
ρ=∑ipi∣i⟩⟨i∣\rho = \sum\limits_i p_i | i \rangle \langle i |
\rho\neq \sum\limits_i p_i | i \rangle \langle i |
ρ≠∑ipi∣i⟩⟨i∣\rho\neq \sum\limits_i p_i | i \rangle \langle i |

Incoherent operations

Coherence creating operations

Fixed basis

\{ | i \rangle \}
{∣i⟩}\{ | i \rangle \}
| i \rangle \langle i |
∣i⟩⟨i∣| i \rangle \langle i |
| i \rangle \langle i |
∣i⟩⟨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}
I\mathcal{I}
\delta = \sum\limits_i p_i | i \rangle\langle i |
δ=∑ipi∣i⟩⟨i∣ \delta = \sum\limits_i p_i | i \rangle\langle i |
  • Set of Incoherent operations
\Lambda_{\mathcal{I}} : \mathcal{I} \rightarrow \mathcal{I}
ΛI:I→I\Lambda_{\mathcal{I}} : \mathcal{I} \rightarrow \mathcal{I}
\mathcal{O}
O\mathcal{O}

Measure of coherence

  • Relative entropy of coherence 
C_r(\rho)
Cr(ρ)C_r(\rho)
C_r(\rho) = \min\limits_{\delta\in \mathcal{I}} S( \rho \| \delta)
Cr(ρ)=minδ∈IS(ρ∥δ)C_r(\rho) = \min\limits_{\delta\in \mathcal{I}} S( \rho \| \delta)
= S(\Delta(\rho)) - S(\rho),^1
=S(Δ(ρ))−S(ρ),1= S(\Delta(\rho)) - S(\rho),^1
\text{where } \Delta \text{ dephases } \rho \text{ (deletes off-diagonal terms)}
where Δ dephases ρ (deletes off-diagonal terms)\text{where } \Delta \text{ dephases } \rho \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}
∙ There exists a state ∣ΦM⟩,for which Cr(ΦM) is maximized\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}{\sqrt{M}} | i \rangle
∣ΦM⟩=∑i=1M1M∣i⟩| \Phi_M \rangle = \sum\limits_{i =1 }^M \frac{1}{\sqrt{M}} | i \rangle

Important operational Tasks

  • Coherence distillation
\rho^{\otimes n} \rightarrow \Phi_2 ^{\otimes m}
ρ⊗n→Φ2⊗m \rho^{\otimes n} \rightarrow \Phi_2 ^{\otimes m}
  • Coherence dilution
\Phi_2^{\otimes n} \rightarrow \rho ^{\otimes m}
Φ2⊗n→ρ⊗m\Phi_2^{\otimes n} \rightarrow \rho ^{\otimes m}

Optimal Asymptotic Rate

R := \frac{m}{n} = C_r(\rho)
R:=mn=Cr(ρ)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)
Cf(ρ):=min{pi,ψi}∑ipiCr(ψi)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
where, ρ=∑ipiψi\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}
(∣ψ⟩AB)⊗n\left(| \psi \rangle^{AB} \right)^{\otimes n}
\Lambda \text{ (unrestricted)}
Λ (unrestricted)\Lambda \text{ (unrestricted)}
\Lambda_{\mathcal{I}}
ΛI\Lambda_{\mathcal{I}}

One-way classical communication

\Phi_2^{\otimes m}
Φ2⊗m\Phi_2^{\otimes m}

Why assisted?

  • The distributed scenario models how to localize coherence on an inaccessible remote target without a quantum channel. 
  • One-way communication is easier to model because there is no feedback to consider.
  • When Bob's state is a qubit and the shared state is pure, then 2-way communication offers no advantage over 1-way communication.
  • This can be relavent to some biological systems where coherence is important, open-destination quantum metrology where coherence improves precision. Or any protocol that requires a highly coherent initial state.

Assisted distillation of coherence

Optimal rate
S(\Delta(\rho^B))^1
S(Δ(ρB))1S(\Delta(\rho^B))^1
n \rightarrow \infty
n→∞n \rightarrow \infty
n = 1
n=1n = 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}
ψAB\psi^{AB}
\{ P^A_i\}
{PiA}\{ P^A_i\}

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

\Lambda_i
Λi\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⁡{PiA}imax⁡M∈N{log⁡2M:max⁡{ΛiB}iF2(∑ipiΛiB(ρiB),ΦMB′)≥1−ϵ} \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
max⁡{pi,ψiB}imax⁡M∈N{log⁡2M:max⁡{ΛiB}iF2(∑ipiΛiB(ψiB),ΦMB′)≥1−ϵ}\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})
where, ∑ipiψi=ρB=TrA(ψAB)\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})
piρiB=TrA((PiA⊗IB)ψAB)p_i \rho_i^B = \text{Tr}_A((P_i^A \otimes \mathbb{I}^B ) \psi^{AB})
\rho_i^B
ρiB\rho_i^B
\rho_i^B
ρiB\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 \}
{pi,ψi}\{ 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
Cc(ψ,ϵ):=maxM∈N{log2M:maxΛ∈OF2(Λ(ψ),ΦM)≥1−ϵ}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}))
maxψ‾∈b∗(ψ,ϵ)Smin(Δ(ψ‾))−δ≤Cc(ψ,ϵ)≤maxψ‾∈b∗(ψ,2ϵ)Smin(Δ(ψ‾))\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))
Smin(ρ):=−log2(λmax(ρ))S_{min}(\rho) := - \log_2(\lambda_{max}(\rho))
b_*(\rho, \epsilon) = \{ \overline{\psi} \text{ s.t. } F(\rho, \overline{\psi}) \geq 1 - \epsilon \}
b∗(ρ,ϵ)={ψ‾ s.t. F(ρ,ψ‾)≥1−ϵ}b_*(\rho, \epsilon) = \{ \overline{\psi} \text{ s.t. } F(\rho, \overline{\psi}) \geq 1 - \epsilon \}
\text{ If } \Delta(\Phi_M) \succ \Delta(\psi) :
 If Δ(ΦM)≻Δ(ψ):\text{ If } \Delta(\Phi_M) \succ \Delta(\psi) :

Proof Sketch direct

\psi
ψ\psi
\Phi_M
ΦM\Phi_M
\Lambda_{\mathcal{I}}
ΛI\Lambda_{\mathcal{I}}
=S_{min}(\Delta(\psi))
=Smin(Δ(ψ))=S_{min}(\Delta(\psi))
\log_2M
log2M\log_2M
M
MM
Winter, A., & Yang, D. : PRL 116.12, 120404, (2016)
\cdot \psi
⋅ψ\cdot \psi
\cdot \Lambda(
⋅Λ(\cdot \Lambda(
\cdot \Phi_{\overline{M}}
⋅ΦM‾\cdot \Phi_{\overline{M}}
\cdot\overline{\psi}
⋅ψ‾\cdot\overline{\psi}
\epsilon
ϵ\epsilon
\epsilon^{\prime}
ϵ′\epsilon^{\prime}
\text{with } \epsilon^{\prime} \text{ error, optimal}
with ϵ′ error, optimal\text{with } \epsilon^{\prime} \text{ error, optimal}
\text{rate } M^*, \text{ will be max }
rate M∗, will be max \text{rate } M^*, \text{ will be max }
\text{over}
over\text{over}
\text{ball}
ball\text{ball}
\text{around } \psi
around ψ\text{around } \psi
\log_2M^* \geq \max\limits_{\overline{\psi} \in b_*(\psi, \epsilon)}S_{min}(\Delta(\overline{\psi}))
log2M∗≥maxψ‾∈b∗(ψ,ϵ)Smin(Δ(ψ‾)) \log_2M^* \geq \max\limits_{\overline{\psi} \in b_*(\psi, \epsilon)}S_{min}(\Delta(\overline{\psi}))
\psi
ψ\psi
)
))
\Lambda
Λ\Lambda
\Lambda
Λ\Lambda

Proof Sketch: Converse

\text{Starting with an operator identity :}
Starting with an operator identity :\text{Starting with an operator identity :}
\Phi_M \delta \Phi_M = \frac{1}{M}\Phi_M
ΦMδΦM=1MΦM\Phi_M \delta \Phi_M = \frac{1}{M}\Phi_M
\delta \in \mathcal{I}
δ∈I\delta \in \mathcal{I}
\delta
δ\delta
\bullet \text{ Assume the optimal rate with error } \epsilon \text{ is } M, \text{ acheived by } \Lambda
∙ Assume the optimal rate with error ϵ is M, acheived by Λ\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\}
log⁡2M≤min⁡γ∈I{−log⁡2Tr(ψ‾γ)}\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
ψ‾ depends on Λ,ΦM,ψ\overline{\psi} \text{ depends on } \Lambda, \Phi_M, \psi
\overline{\psi}
ψ‾\overline{\psi}
\cdot \psi
⋅ψ\cdot \psi
\cdot \overline{\psi}
⋅ψ‾\cdot \overline{\psi}
= \max\limits_i \langle i | \overline{\psi} | i \rangle
=max⁡i⟨i∣ψ‾∣i⟩= \max\limits_i \langle i | \overline{\psi} | i \rangle
= S_{min}(\Delta(\overline{\psi}))
=Smin(Δ(ψ‾))= S_{min}(\Delta(\overline{\psi}))
2\epsilon
2ϵ2\epsilon
\log_2 M \leq \max\limits_{\overline{\psi} \in b_*(\psi, 2\epsilon)} S_{min}(\Delta(\overline{\psi}))
log⁡2M≤max⁡ψ‾∈b∗(ψ,2ϵ)Smin(Δ(ψ‾))\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
Cc(E,ϵ):=max⁡M∈N{log⁡2M:max⁡{Λi}iF2(∑ipiΛi(ψi),ΦM)≥1−ϵ}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}
∙ Let E={pi,ψi} be a pure state ensemble\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}
∙ We define the optimal concentration rate for E as\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}})
max⁡E‾∈b(E,ϵ)FminΔ(E‾)−δ≤Cc(E,ϵ)≤max⁡E‾∈b(E,2ϵ)FminΔ(E‾)\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))
where, FminΔ(E)=min⁡ψi∈ESmin(Δ(ψi))\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,}
∙ The coherence of assistance is defined as,\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
Da(ρB)=max⁡Eρ={pi,ψi}i∑ipiS(Δ(ψi))=DcA∣B(ψAB)=S(Δ(ρB))1D_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,}
∙ We define the one-shot coherence of assistance as,\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)
Ca(ρ,ϵ)=max⁡EρCc(Eρ,ϵ)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})
lim⁡ϵ→0lim⁡n→∞1nCa(ρ⊗n,ϵ)=lim⁡n→∞1nDa(ρ⊗n)\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.
  • Our converse proof works for entanglement also and we are trying to generalize to arbitrary resource. 
  • Similar questions are open for when 2-way communication is allowed

Thank you!

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