Gerardo Adesso University of Nottingham Davide Girolami University of Nottingham Alessio Serafini University College London arxiv:1203.5116; Phys. Rev. Lett. (in press)
A family of useful additive entropies They converge to von Neumann entropy for α 1 The case α = 2 is simply related to the state purity
Very natural: ground and thermal states of all physical systems in the harmonic approximation regime Relevant theoretical testbeds for the study of structural properties of correlations, thanks to the symplectic formalism Preferred resources for experimental unconditional implementations of most continuous variable protocols Crucial role and remarkable control in quantum optics & metrology - coherent states - squeezed states - continuous variable cluster states -...
see e.g. GA & Illuminati, J Phys A 2007; Weedbrook et al. RMP 2012 Information (or lack thereof) Purity/Linear entropy Von Neumann entropy Rényi, Tsallis entropies Bipartite entanglement (Logarithmic) negativity Gaussian entanglement of formation Genuine multipartite entanglement Gaussian tangle/contangle Bipartite general quantum correlations Gaussian quantum discord Multipartite general quantum correlations N/A
Suppose we have a three-mode pure Gaussian state We want to study entanglement in the various partitions We only want to use measures derived from von Neumann entropy
Bipartite entanglement: one vs two modes use marginal von Neumann entropy
Reduced bipartite entanglement: one vs one modes use Gaussian entanglement of formation
Genuine tripartite entanglement: one vs one vs one modes Cannot use anything based on that entropy: monogamy is violated by Eof!
No unified treatment of correlations is available!
Why is von Neumann entropy S so special anyway? It satisfies a very important inequality for general tripartite systems: a "potent hammer" in quantum information theory (M. Nielsen) Strong subadditivity inequality see e.g. Lieb & Ruskai, J Math Phys 1973; Wehrl RMP 1978
Can we find another entropy that Satisfies the strong subadditivity? Has a clear physical meaning? Allows us to define valid measures of all types of correlations?
Can we find another entropy that Satisfies the strong subadditivity? Has a clear physical meaning? Allows us to define valid measures of all types of correlations? YES for Gaussian states
Wigner distribution where is a phase-space vector, and is the covariance matrix of elements which (up to local displacements) completely characterises the n-mode Gaussian state ρ
see e.g. Buzek, Keitel & Knight PRA 1995 Let s quantify information in the Gaussian state ρ in terms of the continuous Shannon entropy of its Wigner distribution ρ W ρ S? (ρ) H(W ρ )
see e.g. Buzek, Keitel & Knight PRA 1995 Let s quantify information in the Gaussian state ρ in terms of the continuous Shannon entropy of its Wigner distribution The Rényi-2 entropy emerges naturally!
Let s measure the distance between two Gaussian states in terms of the distinguishability between their Wigner functions 1 2
Suppose now the second state is the product of the marginals of the first one. 1 2
Suppose now the second state is the product of the marginals of the first one. 1 2 this is 0 only for Gaussian states!
Suppose now the second state is the product of the marginals of the first one. 1 2 this is 0 for all and only Gaussian states!
In fact the stronger property holds as well in the Gaussian world... Strong subadditivity inequality Proof: straightforward using the Fisher-Hadamard matrix inequalities. see e.g. Horn & Johnson, Matrix Analysis
We have found another entropy for Gaussian states that Satisfies the strong subadditivity? Has a clear physical meaning? Allows us to define valid measures of all types of correlations? ρ W ρ S 2 (ρ) H(W ρ )
Information (or lack thereof) Rényi-2 entropy: S 2 ρ = 1 log det γ 2 Total correlations Rényi-2 mutual information: I 2 ρ A:B = S 2 ρ A + S 2 ρ B S 2 ρ AB Classical correlations One-way GR2 classical correlations: J 2 ρ A B = sup S Π B η 2 ρ A dη p B η S 2 ρ A η Quantum correlations GR2 quantum discord: D 2 ρ A B = I 2 ρ A:B J 2 ρ A B Bipartite entanglement GR2 entanglement: E 2 ψ AB = S 2 ρ A ; then extend via Gaussian convex roof
see e.g. Coffman, Kundu & Wootters PRA 2000 Let ρ AB Z be a general n-mode Gaussian state. Then Z E 2 ρ A: B Z E 2 ρ A:k is satisfied k=b GR2 entanglement is monogamous! + + +
see e.g. Coffman, Kundu & Wootters PRA 2000 Let ρ ABC be a fully inseparable 3-mode pure Gaussian state. Then E 2 ρ A:B:C = E 2 ρ A: BC E 2 ρ A:B E 2 ρ A:C defines a measure of genuine tripartite entanglement (residual GR2 entanglement) E 2 ρ A:B:C is invariant under mode permutations! (This is the measure )
see e.g. Coffman, Kundu & Wootters PRA 2000 Let ρ ABC be a fully inseparable 3-mode pure Gaussian state. Then E 2 ρ A:B:C = E 2 ρ A: BC E 2 ρ A:B E 2 ρ A:C defines a measure of genuine tripartite entanglement (residual GR2 entanglement) E 2 ρ A:B:C is invariant under mode permutations! (This is the measure ) The GR2 discord is also monogamous for pure 3-mode Gaussian states and D 2 ρ A BC D 2 ρ A B D 2 ρ A C = E 2 ρ A: BC E 2 ρ A:B E 2 ρ A:C
We have found another entropy for Gaussian states that Satisfies the strong subadditivity? Has a clear physical meaning? Allows us to define valid measures of all types of correlations? ρ W ρ S 2 (ρ) H(W ρ )
Information (or lack thereof) Rényi-2 entropy: S 2 ρ = 1 log det γ 2 Total correlations Rényi-2 mutual information: I 2 ρ A:B = S 2 ρ A + S 2 ρ B S 2 ρ AB Classical correlations One-way GR2 classical correlations: J 2 ρ A B = sup S Π B η 2 ρ A dη p B η S 2 ρ A η Quantum correlations GR2 quantum discord: D 2 ρ A B = I 2 ρ A:B J 2 ρ A B Bipartite entanglement GR2 entanglement: E 2 ψ AB = S 2 ρ A ; then extend via Gaussian convex roof Genuine tripartite entanglement Residual GR2 entanglement = residual GR2 discord (for ρ ABC pure)
Information (or lack thereof) Rényi-2 entropy: S 2 ρ = 1 log det γ 2 Total correlations Rényi-2 mutual information: I 2 ρ A:B = S 2 ρ A + S 2 ρ B S 2 ρ AB Classical correlations One-way GR2 classical correlations: J 2 ρ A B = sup S Π B η 2 ρ A dη p B η S 2 ρ A η Quantum correlations GR2 quantum discord: D 2 ρ A B = I 2 ρ A:B J 2 ρ A B Bipartite entanglement GR2 entanglement: E 2 ψ AB = S 2 ρ A ; then extend via Gaussian convex roof Genuine tripartite entanglement Residual GR2 entanglement = residual GR2 discord (for ρ ABC pure)
Rényi-2 entropy provides a natural measure of information for arbitrary Gaussian states It is operationally linked to the phase space sampling entropy of the Wigner distribution It satisfies the strong subadditivity inequality It leads us to define bona fide quantifiers of all types of correlations, which are shown to satisfy relevant properties such as monogamy These should be adopted as the measures of choice for studies of Gaussian quantum information theory arxiv:1203.5116; Phys. Rev. Lett. (in press)
Are quantum tasks involving Gaussian states always optimally achieved by Gaussian operations if the figure of merit is based on Rényi-2 entropy? E.g. we know that the bosonic additivity conjecture is verified with such an entropy (Giovannetti et al) What about cloning of coherent states? Information/disturbance tradeoff? The approach we followed is not necessarily Gauss-centered. Just use for any state the Wehrl entropy (Shannon continuous entropy of the Husimi Q function) and we might extend our results to arbitrary non-gaussian states [work in progress]
Thank you!!!