Quantifying Resources How to measure resourcefulness
Abstract¶ How to quantify how much resource a given object has.
Keywords: Resource Theories monotones. ¶ Resource Measures ¶ Let R = ( F , O ) \mathcal{R} = (\mathfrak{F}, \mathfrak{O}) R = ( F , O ) F \mathfrak{F} F O \mathfrak{O} O
A function f f f ρ \rho ρ f : ρ → R f: \rho \rightarrow \mathbb{R} f : ρ → R
f ( ρ ) ≥ f ( E ( ρ ) ) f(\rho) \geq f(\mathcal{E}(\rho)) f ( ρ ) ≥ f ( E ( ρ )) where E ∈ F \mathcal{E} \in \mathfrak{F} E ∈ F monotonic under allowed operations. Physically, this means that the quantifier for ρ \rho ρ ρ \rho ρ E ( ρ ) \mathcal{E}(\rho) E ( ρ )
Resource Measures Features ¶
A resource measure f f f
f ( ρ ) = 0 ⟺ ρ ∈ O . f(\rho) = 0 \Longleftrightarrow \rho \in \mathfrak{O}. f ( ρ ) = 0 ⟺ ρ ∈ O . The resource measure is only zero if and only if ρ \rho ρ ρ \rho ρ
A resource measure f f f
f ( ρ ) ≤ p f ( ρ 1 ) + ( 1 − p ) f ( ρ 2 ) f(\rho) \leq pf(\rho_1) + (1-p)f(\rho_2) f ( ρ ) ≤ p f ( ρ 1 ) + ( 1 − p ) f ( ρ 2 ) where ρ = p ρ 1 + ( 1 − p ) ρ 2 \rho = p\rho_1 + (1-p)\rho_2 ρ = p ρ 1 + ( 1 − p ) ρ 2 0 ≤ ρ ≤ 1 0\leq \rho \leq 1 0 ≤ ρ ≤ 1
This means that taking a mixture of two states does not make it more resourceful.
A resource measure f f f
f ( ρ ⊗ σ ) = f ( ρ ) ⊗ f ( σ ) , f(\rho \otimes \sigma) = f(\rho) \otimes f(\sigma), f ( ρ ⊗ σ ) = f ( ρ ) ⊗ f ( σ ) , Complete Sets of Monotones ¶ Resource measures do not always capture the preorder on the set of states imposed by the allowed operations. It is possible that
f ( ρ ) ≥ f ( σ ) f(\rho) \geq f(\sigma) f ( ρ ) ≥ f ( σ ) even if ∄ E ∈ F \nexists~~\mathcal{E}\in\mathfrak{F} ∄ E ∈ F E ( ρ ) = σ \mathcal{E}(\rho) = \sigma E ( ρ ) = σ
A set of resource measures { f i } \{f_{i}\} { f i }
f i ( ρ ) ≥ f i ( σ ) ∀ i ⟺ ρ ≺ σ . f_{i}(\rho) \geq f_{i}(\sigma)~\forall~i~~\Longleftrightarrow~ \rho \prec \sigma. f i ( ρ ) ≥ f i ( σ ) ∀ i ⟺ ρ ≺ σ . Complete set of monotones therefore fully capture the preorder on the set of states.
Distance Based Resource Measures ¶ Distance based resource measures are frequently used in Convex resource theories. The aim is to quantify the amount of resource in an object based on its distance from the set of free states.
A distance measure is a measure that is contractive under quantum channels. A function d d d R ≥ 0 \mathbb{R}_{\geq 0} R ≥ 0 d : H A ⊗ H B → R ≥ 0 d: \mathcal{H}_{A} \otimes \mathcal{H}_{B} \rightarrow \mathbb{R}_{\geq 0} d : H A ⊗ H B → R ≥ 0 E \mathcal{E} E
d ( ρ , σ ) ≥ d ( E ( ρ ) , E ( σ ) ) . d(\rho, \sigma) \geq d(\mathcal{E}(\rho), \mathcal{E}(\sigma)). d ( ρ , σ ) ≥ d ( E ( ρ ) , E ( σ )) . Resource quantifiers using distances measures usually take the form
f ( ρ ) = min σ ∈ O d ( ρ , σ ) f(\rho) = \min_{\sigma \in \mathfrak{O}} d(\rho, \sigma) f ( ρ ) = σ ∈ O min d ( ρ , σ ) Resource Measures Examples ¶ Robustness Measures ¶ Robustness measures quantify the resource of an object by measuring how much of another object needs to be mixed into it to make to not resourceful.
Let F \mathfrak{F} F T \mathfrak{T} T
The robustness of a state ρ \rho ρ T \mathfrak{T} T
R ( ρ ) = min σ ∈ T { s ≥ 0 : ρ + s σ 1 + s ∈ F } \mathcal{R}(\rho) = \min_{\sigma \in \mathfrak{T}} \bigg\{ s \geq 0 ~ : \frac{\rho + s \sigma}{1 + s} \in \mathfrak{F} \bigg\} R ( ρ ) = σ ∈ T min { s ≥ 0 : 1 + s ρ + s σ ∈ F } Different robustness measures are named for different set of states T \mathfrak{T} T
Let F \mathfrak{F} F
The absolute robustness of a state ρ \rho ρ
R a ( ρ ) = min σ ∈ F { s ≥ 0 : ρ + s σ 1 + s ∈ F } \mathcal{R}_{a}(\rho) = \min_{\sigma \in \mathfrak{F}} \bigg\{ s \geq 0 ~ : \frac{\rho + s \sigma}{1 + s} \in \mathfrak{F} \bigg\} R a ( ρ ) = σ ∈ F min { s ≥ 0 : 1 + s ρ + s σ ∈ F } This is a measure of the minimum amount of free state that must be mixed with ρ \rho ρ
Properties
R a ( ρ ) ≥ 0 \mathcal{R}_{a}(\rho) \geq 0 R a ( ρ ) ≥ 0 R a ( ρ ) = 0 \mathcal{R}_{a}(\rho) = 0 R a ( ρ ) = 0 ρ ∈ F \rho \in \mathfrak{F} ρ ∈ F Let F \mathfrak{F} F D \mathfrak{D} D
The generalised robustness of state ρ \rho ρ
R g ( ρ ) = min σ ∈ D { s ≥ 0 : ρ + s σ 1 + s ∈ F } \mathcal{R}_{g}(\rho) = \min_{\sigma \in \mathfrak{D}} \bigg\{ s \geq 0 ~ : \frac{\rho + s \sigma}{1 + s} \in \mathfrak{F} \bigg\} R g ( ρ ) = σ ∈ D min { s ≥ 0 : 1 + s ρ + s σ ∈ F } This is a measure of the minimum amount of any state that must be mixed with ρ \rho ρ
Properties
R g ( ρ ) ≥ 0 \mathcal{R}_{g}(\rho) \geq 0 R g ( ρ ) ≥ 0 R g ( ρ ) = 0 \mathcal{R}_{g}(\rho) = 0 R g ( ρ ) = 0 ρ ∈ F \rho \in \mathfrak{F} ρ ∈ F min σ ∈ F D max ( ρ ∣ ∣ σ ) = log 2 ( 1 + R g ( ρ ) ) \min_{\sigma \in \mathfrak{F}} D_{\textrm{max}}(\rho \vert \vert \sigma) = \log_2(1+R_g(\rho)) min σ ∈ F D max ( ρ ∣∣ σ ) = log 2 ( 1 + R g ( ρ )) Weight Measures ¶ Weight measures quantify the resource of an object by decomposing a state into a convex mixture of a resourceful state and free state, the resourcefulness is then quantified by “how much” of a general state is present within the state as compared to a resourceless state.
Let F \mathfrak{F} F D \mathfrak{D} D
The weight of a state ρ \rho ρ
W ( ρ ) = { w ≥ 0 : ρ = w ρ G + ( 1 − w ) σ , ρ G ∈ D , σ ∈ F } . \mathcal{W}(\rho) = \big\{ w \geq 0: \rho = w \rho_{G} + (1-w) \sigma, \rho_{G}\in \mathcal{D}, \sigma \in \mathfrak{F} \big\}. W ( ρ ) = { w ≥ 0 : ρ = w ρ G + ( 1 − w ) σ , ρ G ∈ D , σ ∈ F } . Entropic Measures ¶ Entropy functions are frequently used in resource theories to quantify how far a given resourceful state is from the set of free states. Often, this includes a minimisation over the set of free states.
For example, let F \mathfrak{F} F quantum relative entropy S ( ρ ∣ ∣ σ ) S(\rho \vert \vert \sigma) S ( ρ ∣∣ σ )
R ( ρ ) = min σ ∈ F S ( ρ ∣ ∣ σ ) . \mathcal{R}(\rho) = \min_{ \sigma \in \mathcal{F} } S(\rho \vert \vert \sigma). R ( ρ ) = σ ∈ F min S ( ρ ∣∣ σ ) .