Entanglement transformations are irreversible
“analogies between Entanglement and Thermodynamics.
“entropy as the unique function S that encodes all transformations between comparable equilibrium states: X can be transformed into Y adiabatically iff S(X) ≤ S(Y)”
“if S(X) = S(Y) then X and Y must be connected by reversible transformations, e.g. Carnot cycles.”
A second law of entanglement. A unique function that governs all transformations between entangled systems. Like entropy in thermodynamics.
Entanglement counterpart to the second law of thermodynamics. Impossible.
Entanglement theory is not reversible. Under all non-entangling transformations.
“fundamentally different from thermodynamics. And all other known quantum resources. that are reversible under resource non-generating transformations. e.g purity and coherence”
“under Gibbs-preserving maps, thermodynamic transformations of quantum systems are reversible.”
“Reversible entanglement transformations require generation of large amounts of entanglement. as measured by either the negativity or the standard robustness.”
"Alice and Bob. Share a large number of identical copies of a bipartite quantum state. And wish to transform them into as many copies as possible of some target state. All while achieving a vanishingly small error in the asymptotic limit.
/--
Transformation rate:
max ratio m/n that can be achieved in the limit n → ∞
under the condition that n copies of ρ are transformed into m copies of ω
with asymptotically vanishing error.
-/
def R(ρ → ω) := sorry
Separable states on a bipartite system AB: states σAB that admit a decomposition of the form σAB = ∫|ψ⟩⟨ψ|A ⊗ |φ⟩⟨φ|B dμ(ψ,φ), where μ is an appropriate probability measure on the set of pairs of local pure states.
def non_entangling [quantum_channel Λ] {quantum_state σAB} :=
separable σAB → separable Λ(σAB)
R(ρ → ω) depends on the allowed operations. All transformation rates here are wrt NE channels.
def reversibly_interconvertible [quantum_state ρ] [quantum_state ω] :=
R(ρ → ω) * R(ω → ρ) = 1
Distillable entanglement of quantum state
R(ρ → Φ₂) = distillable_entanglement(ρ)
Entanglement cost of quantum state ρ
R(Φ₂ → ρ)⁻¹ = entanglement_cost(ρ)
def entanglement_theory_is_reversible :
∀ ρ, distillable_entanglement ρ = entanglement_cost ρ
/-- Thm. 1 -/
theorem entanglement_theory_irreversible :
∃ ρ, distillable_entanglement ρ ≠ entanglement_cost ρ :=
begin
-- let ω3 := 1/6 * ∑i,j=1..3 (|ii⟩ ⟨ii| − |ii⟩ ⟨jj|)
-- use ω3,
-- distillable_entanglement ω3 = log(3/2)
-- entanglement_cost ω3 = 1
-- entanglement_cost ω3 > distillable_entanglement ω3
sorry
end
reversibility under asymptotically NE channels?
“The theory of entanglement proposed in Plenio2007 may still be reversible.
This theory features asymptotically NE operations: may generate entanglement vanishingly small in the asymptotic limit.
According to what measure should one enforce the generated entanglement to be small?
“There are many asymptotically inequivalent ways to quantify entanglement.”
The choice is crucial to decide between reversibility and irreversibility.
“For some entanglement measures reversibility is possible only with exponential entanglement generation.
In fact, even a minor change of the quantifier from the generalised robustness to the closely related standard robustness of entanglement makes reversibility impossible.
So Plenio2007’s operations, despite generating vanishingly little entanglement with respect to the generalised robustness, create macroscopically large amounts of it as measured by either the negativity or the standard robustness.
Reversibility necessarily leads to macroscopic entanglement generation.”
“Consider a modified version of asymptotic entanglement manipulation. Take n copies of an initial state ρAB. We want to transform them into m copies of a target state ωA’B’ with asymptotically vanishing error.
To define the set of allowed transformations, we fix an entanglement measure M and consider all those transformations Λₙ on n copies of the system AB that are (M,δₙ)-approximately NE, in the sense that M(Λₙ(σAⁿBⁿ)) ≤ δₙ for all separable states σAⁿBⁿ, where the numbers δₙ quantify the magnitude of the entanglement fluctuations at each step of the process.
The maximum ratio m/n that can be achieved in the limit n → ∞ determines the transformation rate under these operations. The modified notion of distillable entanglement is then defined by choosing the maximally entangled bit Φ₂ as the target state, and analogously the modified entanglement cost corresponds to the transformation from Φ₂ to a given state ρ. By choosing a suitable measure of entanglement and setting δₙ = 0 for all n, we recover our original definition of NE transformations.”
“Plenio2007: if M is the generalised robustness of entanglement, entanglement can be manipulated reversibly even if we take δₙ → 0 as n → ∞.
But a completely opposite conclusion is reached when M is taken to be either the standard robustness or the Logarithmic negativity.”
/-- Theorem 2: entanglement theory is irreversible
under_ops_that_generate sub-exponential amounts of entanglement
according to the negativity or the standard robustness Rs𝒮𝒮 -/
theorem entanglement_theory_irreversible
under_ops_that_generate sub-exponential amounts of entanglement
according to the negativity or the standard robustness Rₛ
“Separability is independent of the choice of measure. So our axiomatic assumption of strict no entanglement generation bypasses the above problems completely.
Theorem 2 shows that irreversibility of entanglement is robust to fluctuations in the generated resources.
What makes entanglement theory so special? Where does its irreversibility come from?
Is there a closed expression for the associated entanglement cost?
Does Entanglement catalysis play a role?
Convert quantum state with catalyst
Bug report on Generalized quantum Stein’s lemma does not affect our results.”
ω3 ω3 ω3 ω3
ω3 ω3 ω3 ω3
ω3 ω3 ω3 ω3
distillable_entanglement ω3 = log(3/2)
Φ₂Φ₂Φ₂Φ₂
Φ₂Φ₂Φ₂
entanglement_cost ω3 = 1
ω3ω3ω3ω3
ω3ω3ω3