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=i∑piρi⊗σi
\rho\neq \sum\limits_i p_i \rho_i \otimes \sigma_i
ρ=i∑piρi⊗σi
LOCC
Non-Local
\Lambda
Λ
Entanglement :
Purity :
\rho = \frac{1 }{d}\mathbb{I}_d
ρ=d1Id
\rho \neq \frac{1 }{d}\mathbb{I}_d
ρ=d1Id
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=i∑pi∣i⟩⟨i∣
\rho\neq \sum\limits_i p_i | i \rangle \langle i |
ρ=i∑pi∣i⟩⟨i∣
Incoherent operations
Coherence creating operations
Fixed basis
\{ | i \rangle \}
{∣i⟩}
| i \rangle \langle i |
∣i⟩⟨i∣
| i \rangle \langle i |
∣i⟩⟨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
\delta = \sum\limits_i p_i | i \rangle\langle i |
δ=i∑pi∣i⟩⟨i∣
- Set of Incoherent operations
\Lambda_{\mathcal{I}} : \mathcal{I} \rightarrow \mathcal{I}
ΛI:I→I
\mathcal{O}
O
Measure of coherence
- Relative entropy of coherence
C_r(\rho)
Cr(ρ)
C_r(\rho) = \min\limits_{\delta\in \mathcal{I}} S( \rho \| \delta)
Cr(ρ)=δ∈IminS(ρ∥δ)
= S(\Delta(\rho)) - S(\rho),^1
=S(Δ(ρ))−S(ρ),1
\text{where } \Delta \text{ dephases } \rho \text{ (deletes off-diagonal terms)}
where Δ dephases ρ (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
| \Phi_M \rangle = \sum\limits_{i =1 }^M \frac{1}{\sqrt{M}} | i \rangle
∣ΦM⟩=i=1∑MM1∣i⟩
Important operational Tasks
- Coherence distillation
\rho^{\otimes n} \rightarrow \Phi_2 ^{\otimes m}
ρ⊗n→Φ2⊗m
- Coherence dilution
\Phi_2^{\otimes n} \rightarrow \rho ^{\otimes m}
Φ2⊗n→ρ⊗m
Optimal Asymptotic Rate
R := \frac{m}{n} = C_r(\rho)
R:=nm=Cr(ρ)
Optimal Asymptotic Rate
C_f(\rho) := \min\limits_{ \{ p_i, \psi_i \}}\sum_i p_i C_r(\psi_i)
Cf(ρ):={pi,ψi}min∑ipiCr(ψi)
\text{where, }\rho = \sum\limits_i p_i \psi_i
where, ρ=i∑piψi
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
\Lambda \text{ (unrestricted)}
Λ (unrestricted)
\Lambda_{\mathcal{I}}
ΛI
One-way classical communication
\Phi_2^{\otimes m}
Φ2⊗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))1
n \rightarrow \infty
n→∞
n = 1
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
- Alice performs POVM on her part of the shared state and sends the result to Bob.
\psi^{AB}
ψAB
\{ P^A_i\}
{PiA}
2. Bob performs an Incoherent operation on his part of the state based on Alice's outcome.
\Lambda_i
Λi
\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
{PiA}imaxM∈Nmax{log2M:{ΛiB}imaxF2(i∑piΛiB(ρiB),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
{pi,ψiB}imaxM∈Nmax{log2M:{ΛiB}imaxF2(i∑piΛiB(ψiB),ΦMB′)≥1−ϵ}
\text{where, }\sum\limits_i p_i \psi_i = \rho^B = \text{Tr}_A(\psi^{AB})
where, i∑piψi=ρB=TrA(ψAB)
p_i \rho_i^B = \text{Tr}_A((P_i^A \otimes \mathbb{I}^B ) \psi^{AB})
piρiB=TrA((PiA⊗IB)ψAB)
\rho_i^B
ρiB
\rho_i^B
ρiB
- 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}
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):=M∈Nmax{log2M:Λ∈OmaxF2(Λ(ψ),ΦM)≥1−ϵ}
\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}))
ψ∈b∗(ψ,ϵ)maxSmin(Δ(ψ))−δ≤Cc(ψ,ϵ)≤ψ∈b∗(ψ,2ϵ)maxSmin(Δ(ψ))
S_{min}(\rho) := - \log_2(\lambda_{max}(\rho))
Smin(ρ):=−log2(λmax(ρ))
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}
\text{ If } \Delta(\Phi_M) \succ \Delta(\psi) :
If Δ(ΦM)≻Δ(ψ):
Proof Sketch direct
\psi
ψ
\Phi_M
ΦM
\Lambda_{\mathcal{I}}
ΛI
=S_{min}(\Delta(\psi))
=Smin(Δ(ψ))
\log_2M
log2M
M
M
Winter, A., & Yang, D. : PRL 116.12, 120404, (2016)
\cdot \psi
⋅ψ
\cdot \Lambda(
⋅Λ(
\cdot \Phi_{\overline{M}}
⋅ΦM
\cdot\overline{\psi}
⋅ψ
\epsilon
ϵ
\epsilon^{\prime}
ϵ′
\text{with } \epsilon^{\prime} \text{ error, optimal}
with ϵ′ error, optimal
\text{rate } M^*, \text{ will be max }
rate M∗, will be max
\text{over}
over
\text{ball}
ball
\text{around } \psi
around ψ
\log_2M^* \geq \max\limits_{\overline{\psi} \in b_*(\psi, \epsilon)}S_{min}(\Delta(\overline{\psi}))
log2M∗≥ψ∈b∗(ψ,ϵ)maxSmin(Δ(ψ))
\psi
ψ
)
)
\Lambda
Λ
\Lambda
Λ
Proof Sketch: Converse
\text{Starting with an operator identity :}
Starting with an operator identity :
\Phi_M \delta \Phi_M = \frac{1}{M}\Phi_M
ΦMδΦM=M1ΦM
\delta \in \mathcal{I}
δ∈I
\delta
δ
\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
\log_2M \leq \min\limits_{\gamma \in \mathcal{I}} \left\{ -\log_2\text{Tr}(\overline{\psi}\gamma) \right\}
log2M≤γ∈Imin{−log2Tr(ψγ)}
\overline{\psi} \text{ depends on } \Lambda, \Phi_M, \psi
ψ depends on Λ,ΦM,ψ
\overline{\psi}
ψ
\cdot \psi
⋅ψ
\cdot \overline{\psi}
⋅ψ
= \max\limits_i \langle i | \overline{\psi} | i \rangle
=imax⟨i∣ψ∣i⟩
= S_{min}(\Delta(\overline{\psi}))
=Smin(Δ(ψ))
2\epsilon
2ϵ
\log_2 M \leq \max\limits_{\overline{\psi} \in b_*(\psi, 2\epsilon)} S_{min}(\Delta(\overline{\psi}))
log2M≤ψ∈b∗(ψ,2ϵ)maxSmin(Δ(ψ))
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):=M∈Nmax{log2M:{Λi}imaxF2(i∑piΛi(ψi),ΦM)≥1−ϵ}
\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{ We define the optimal concentration rate for } \mathfrak{E} \text{ as}
∙ We define the optimal concentration rate for E 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}})
E∈b(E,ϵ)maxFminΔ(E)−δ≤Cc(E,ϵ)≤E∈b(E,2ϵ)maxFminΔ(E)
\text{where, } F^{\Delta}_{min}(\mathfrak{E}) = \min\limits_{\psi_i \in \mathfrak{E}} S_{min}(\Delta(\psi_i))
where, FminΔ(E)=ψi∈EminSmin(Δ(ψi))
Asymptotic limit
\bullet \text{ The coherence of assistance is defined as,}
∙ 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)=Eρ={pi,ψi}imaxi∑piS(Δ(ψi))=DcA∣B(ψAB)=S(Δ(ρB))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,
C_a(\rho, \epsilon) = \max\limits_{\mathfrak{E}_{\rho}} C_{c}(\mathfrak{E}_{\rho}, \epsilon)
Ca(ρ,ϵ)=EρmaxCc(Eρ,ϵ)
\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})
ϵ→0limn→∞limn1Ca(ρ⊗n,ϵ)=n→∞limn1Da(ρ⊗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!
One-shot assisted concentration of coherence Madhav Krishnan V arxiv:1804.06554 with Eric Chitambar & Min-Hsiu Hsieh
One Shot Assisted Concentration of Coherence
By madhav_krishnan
One Shot Assisted Concentration of Coherence
Seminar USyd
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