Non-Gaussianity as a power-up for quantum communication and estimation Gerardo Adesso

G. Adesso  page 2 Continuous Variable Quantum Information  Prof. Fabrizio Illuminati  Prof. Silvio De Siena -Postdocs: Dr. G. Adesso Dr. F. Dell’Anno -Students: L. Albano Farias (former PhD) L.A.M. Souza (visiting from Minas Quantum Theory group, University of Salerno

G. Adesso  page 3 In order to implement quantum interfaces one needs to be able to: Entangle multiple nodes Entangle multiple nodes Engineer high fidelity channels for the transmission of information Engineer high fidelity channels for the transmission of information Teleport, store and retrieve states between light and matter Teleport, store and retrieve states between light and matter Light Atomic ensembles Continuous variables

G. Adesso  page 4 Continuous variable systems We need a common language: canonically conjugate observables e.g. position q and momentum p of massive particles… … or phase/amplitude quadratures of light (q=“electric field”, p=“magnetic field”) … or collective spin components of atomic ensembles (q=Sx/, p=Sy/ )

G. Adesso  page 5 Gaussian states  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 entanglement, thanks to the symplectic formalism  Valid resources for experimental implementations of continuous variable protocols  Crucial role and remarkable control in quantum optics -coherent states -squeezed states -thermal states A.Porzio

G. Adesso  page 6 „85% of our continuous variable papers in five years dealt with Gaussian states and their entanglement properties“ …probably it‘s enough… Bipartite entanglement versus global and local degrees of information PRL 2004a Localization and scaling of multimode entanglement under symmetry PRL 2004b Equivalence between entanglement and optimal coherent-state teleportation fidelity PRL 2005 Linear optical schemes to engineer multimode entangled Gaussian states PRL 2006 Monogamy constraints and genuine multipartite entanglement measure PRL 2007a, PRL 2007b

G. Adesso  page 7 Limitations of Gaussian states and operations Things you cannot do without some non-Gaussian aid  Distill Gaussian entanglement  Perform universal quantum computation with Gaussian cluster states  Violate (loophole-free) Bell tests with homodyne detection …… Things that work better with non-Gaussian states …it happens in the best families…  Entanglement (Gaussian states have the minimum one for given 2 nd moments)  Optimal cloning of coherent states  Information/disturbance trade-off  …

G. Adesso  page 8 This talk Things you cannot do without some non-Gaussian aid  Distill Gaussian entanglement  Perform universal quantum computation with Gaussian cluster states  Violate (loophole-free) Bell tests with homodyne detection …… Things that work better with non-Gaussian states  Entanglement (Gaussian states have the minimum one for given 2 nd moments)  Optimal cloning of coherent states  Information/disturbance trade-off  …  Quantum teleportation of classical and nonclassical states  Quantum estimation of loss in bosonic channels

G. Adesso  page 9 Non-Gaussian states are entering into reality  Single-mode Photon-Added Coherent State A. Zavatta, S. Viciani, and M. Bellini, Science 306, 660 (2004)  Single-mode Photon-Subtracted Squeezed State J. Wenger, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett. 92, (2004)  Two-mode Photon-Subtracted Squeezed State A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett. 98, (2007)  Single-mode Photon-Added/Subtracted Thermal State V. Parigi, A. Zavatta, M. Kim, and M. Bellini, Science 317, 1890 (2007)  Single-mode Squeezed Cat-like State A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, Nature 448, 784 (2007)  Fock states various groups: Lvovsky PRL 2001, Yamamoto NJP 2006, Grangier PRL 2006 & Nature 2007, […] For a review on multiphoton quantum optics and quantum state engineering, see: F. Dell’Anno, S. De Siena, and F. Illuminati, Phys. Rep. 428, 53 (2006) M. Bellini

G. Adesso  page 10  It was known that photon-subtracted deGaussified twin-beam states are better resources (at fixed squeezing) for teleporting coherent states Opatrny et al., PRA 61, (2000); Cochrane et al., PRA 65, (2002); Olivares et al., PRA 67, (2003); Kitagawa et al., PRA 73, (2006). Continuous variable teleportation with non-Gaussian resources  We performed a systematic study to identify the optimal non-Gaussian resources giving rise to the highest fidelity in various conditions F. Dell’Anno et al., Phys. Rev. A 76, (2007) F. Dell’Anno et al., Eur.Phys.J.–ST 160, 115 (2008)

G. Adesso  page 11 Selecting performant non-Gaussian resources Strategy Strategy – Exploiting the free parameters ( ,  |  |) for optimization

G. Adesso  page 12 Optimized fidelity: results squeezed Bell-like states  The optimal resources are squeezed Bell-like states with  Sharp improvement over Gaussian twin beams ( I ) and over photon-subtracted states ( II ) for various inputs  Robust against thermal noise PURE MIXED SBs SCs TBs

G. Adesso  page 13 Understanding optimization  We demonstrated that the best non-Gaussian resources for teleportation have to be endowed with three crucial features: -More entanglement -More non-Gaussianity (distance from the reference Gaussian state with equal 2 nd moments) Genoni, Paris, Banaszek, PRA More “squeezed-vacuum affinity” (overlap with Gaussian twin beams) squeezed Bell-like The squeezed Bell-like resource states realize the best interplay ! SBs SCs TBs

G. Adesso  page 14 How to engineer squeezed Bell-like states F. Dell’Anno et al., Phys. Rev. A 76, (2007)

G. Adesso  page 15 Quantum estimation of loss in bosonic channels  The dissipative channel is described by a master equation: d  /d  = tan  L a [ , with: L a [  ] = 2 a  a † – a † a  –  a † a.  To gain a control over the channels one needs to estimate  (0,  /2)  This consists in two steps: 1.Devising the optimal input state 2.Determining the optimal measurement on the output  After repeating N times, one constructs an estimator for   Optimal estimation  minimum variance of the estimator inout  Monras & Paris, PRL 2007

G. Adesso  page 16 Quantum Estimation Theory (basics) Problem Problem – Optimal input probe states, at a fixed energy, which maximize H There is an ultimate bound on the precision of estimation (Heisenberg limit): H ≤ 4 n In our case, for any probe with energy (mean photon number) n, H ≤ 4 n …can it be saturated? M. Paris M. Paris A. Smerzi

G. Adesso  page 17 Heisenberg-limited estimation of loss Apart from the conceptual solution, there is also a technological simplification as the optimal measurement for Fock probes only involves photon counting (no adaptive estimation required) Fock states Fock states for any n and any loss achieve H=4n: optimal estimation  The best possible (coherent and/or squeezed) Gaussian states never saturate the limit for finite loss  Optimal measurement on the best Gaussian probes involves additional squeezing, displacement and photon counting Monras & Paris, PRL 2007 G. Adesso et al. arXiv: FOCK Gaussian

G. Adesso  page 18 Loss estimation with low-energy probes  Fock states have a discrete energy  Ideal probes should have low energy (0<n≤1), in order not to alter the channel  Can non-Gaussian states be better resources also in this regime? Optimized superpositions of the vacuum and the first low-lying Fock states (n≤K) outperform the best Gaussian probes already for K=2 0 p 8 p 4 p 2 3 p 8 p f H n = p 8 p 4 p 2 3 p 8 p f H n = p 8 p 4 p 2 3 p 8 p f H n = 0.75 G. Adesso et al. arXiv: Gaussian NG: K=1 NG: K=2 NG: K=3

G. Adesso  page 19 Loss estimation with low-energy probes (continued)  Fock states have a discrete energy  Ideal probes should have low energy (0<n≤1), in order not to alter the channel  Can non-Gaussian states be better resources also in this case? Optimized superpositions of the vacuum and the first low-lying Fock states (n≤K) outperform the best Gaussian probes already for K=2 G. Adesso et al. arXiv: Gaussian NG: K=1 NG: K=2 NG: K=3  Adding terms to the superposition and optimizing the weights makes the process rapidly converging towards the Heisenberg limit  K=2 remains an optimal workpoint  Physical understanding: optimal photonic qutrit-like probe states with K=2 can be recast quite generally as truncations of photon-subtracted deGaussified twin beams

G. Adesso  page 20 „We identified optimized non-Gaussian states yielding significant improvements to primitive informational tasks“ TELEPORTATION ESTIMATION

G. Adesso  page 21  On the experimental side: Instigating to refine techniques with the precise aim of engineering and control selected classes of non-Gaussian states  On the theoretical side: Analysis of non-Gaussian entanglement, possibly deriving quantitative bounds for some special cases which rely on experimentally accessible functionals of the state  On a more general, fundamental ground: Conceiving truly novel protocols from scratch, possibly with no qubit analogue, able to exploit the “infinite” possibilities of continuous variable systems and non-Gaussian states for the manipulation and transmission of quantum information Some possible future directions

G. Adesso  page 22 Thank  On the experimental side: Instigating to refine techniques with the precise aim of engineering and control selected classes of non-Gaussian states  On the theoretical side: Analysis of non-Gaussian entanglement, possibly deriving quantitative bounds for some special cases which rely on experimentally accessible functionals of the state  On a more general, fundamental ground: Conceiving truly novel protocols from scratch, possibly with no qubit analogue, able to exploit the “infinite” possibilities of continuous variable systems and non-Gaussian states for the manipulation and transmission of quantum information Some possible future directions G. Adesso  page 21

G. Adesso – Multipartite entanglement for quantum communication Nottingham, July 1 st 2008