Operational quantification of continuous variable correlations

Transcription

1 Operational quantification of continuous variable correlations Carles Rodó, Gerardo Adesso and Anna Sanpera Grup de Física Teòrica Universitat Autònoma de Barcelona, E Bellaterra, Spain. YEP-07 (Young European Physicists meeting 2007) Frauenchiemsee, Munich, Germany; 17-21th September 2007.

2 Outline Continuous Variable (CV) states. Gaussians and non-gaussians Bit-correlations and Cryptography Operational quantification of continuous variable correlations

3 Distribution functions Continuous Variable (CV) states. Gaussians and non-gaussians CV systems: a system is a continuous variable system if it posses two canonical conjugated degrees of freedom i.e. there exists 2 observables that fulfill the Canonical Comutation Relations (CCR) so they posses a continuous spectra and live in an infinite dimensional Hilbert space. Examples: the position-momentum of a massive particle, the quadratures of an electromagnetic field or the collective spin of a polarized ensamble of atoms we call them modes. CCR imposes the symplectic scalar product that defines the geometry of the Phase-Spase. For N modes they read: Any CV state represented by a density operator (infinite dimensional) can be mapped to a quasi-distribution function in phase-space. We can have several choices: (1932) Wigner function (symmetrical ordered), (1963) P-function (normal orderd), (1965) Q-function (antinormal ordered), others We use here the Wigner function.

4 Transforms Continuous Variable (CV) states. Gaussians and non-gaussians Density Wigner Characteristic Weyl-Fourier transform (WF-T) Symplectic-Fourier transform (SF-T) We can build our Wigner quasi-distribution and recover the state

5 Gaussian states Continuous Variable (CV) states. Gaussians and non-gaussians Gaussian states are thoses states whose Wigner function is gaussian!!! Density normalization characterizes the state Wigner Characteristic Displacement Vector (DV) first moments & Covariance Matrix (CM) second moments CM DV

6 Examples of Gaussian states Continuous Variable (CV) states. Gaussians and non-gaussians Coherent state Thermal state Vacumm, squeezed, multimodes,

7 Schematic pictures for Gaussian states dimension Hilbert space Phase space stucture description positivity (hermiticity) operations spectra pure states purity, fidelity, PPT separability 1x1 and 1xN modes

8 Schematic pictures for NON-Gaussian states dimension Hilbert space Phase space stucture description more moments!! infinite positivity (hermiticity) operations spectra pure states purity, fidelity, PPT separability 1x1 and 1xN modes

9 Examples of Non-Gaussian states Continuous Variable (CV) states. Gaussians and non-gaussians Photonic qubit states (superpositions of Gaussians): Ideal 2 photon substraction from a Two Mode Squeezed state (de-gaussified states): Photon added states (de-gaussified states): vacuum it has negative values -> it s not Gaussian

10 Entanglement in Non-Gaussian states Continuous Variable (CV) states. Gaussians and non-gaussians We can use Logarithmic Negativity which is monotone (good map, monotonic, faithfulness, normalized) and additive. Photonic qubit states: Ideal 2 photon substraction from a Two Mode Squeezed state: M. M. Wolf et al., Phys. Rev. Lett. 96, (2006) non-gaussian states are more entangled quant-ph/ in general it s not easy to calculate the negativity for arbitrary non-gaussian states!!

11 Cryptography with entangled Gaussian states Bit-correlations and Cryptography A ABE AB E pure mixed B we use the standard form of a symmetric bipartite 1x1 mode gaussian state M. Navascues et al., Phys. Rev. Lett. 94, (2005) quant-ph/ C. Rodó et al., Open Sys. Inf. Dyn 14, 69 (2007) quant-ph/ DV vs CV in cryptography extracting bit correlations from quantum states

12 Extracting bit-correlations Bit-correlations and Cryptography Alice & Bob will perform homodyne measures (gaussian operations) to they own x-quadrature for each copy. They will asociate the bit 0/1 (+/-) to a positive/negative outcome result an so they will end up with a list of classically correlated bits. Clearly the higher the entanglement of Alice & Bob s state is the higher correlations will be. Once they have the list of SECURE classically correlated bits they can perform Maurer s advantage distillation protocol to distill a RANDOM secret key. Bit strength correlation bit strength correlation correlations in the state (entanglement if the state is pure) the ideal case would be an EPR state (TMS with infinite squeezing but energetically impossible) because then we have perfect correlations and so a maximally entangles state

13 Definition Operational quantification of continuous variable correlations Bit quadrature correlations, Q: (operationally quantifies correlations, clas+quan of any CV state, and so entanglement for pure states) Properties: expected value (average over the state)

14 Examples Operational quantification of continuous variable correlations Gaussian states Origin: product states Red line: pure states Blue line: separable states Green line: perfectly correlated states normalized negativity randomly generated symmetric Gaussian states Pure states: Q is an entanglement monotone Mixed states: Q majorizes entanglement

15 Non-Gaussian states Operational quantification of continuous variable correlations Photonic qubit states: Mixtures of Gaussian states: Photon substracted states: monotonic A. Kitagawa et al., Phys. Rev. A. 73, (2006) quant-ph/ Experimental de-gaussified states: A. Ourjoumtsev et al., Phys. Rev. Lett. 98, (2005) quant-ph/

16 Final remarks Operational quantification of continuous variable correlations Everything is ok? NO. This quantity is taking into account only up to second order correlations. Photonic qutrit state: For this state Q=0 nevertheless it s an entangled state, so our operational quantification of correlations is underestimating entanglement. Is this a problem? +or-. For all relevant non-gaussian states studied and produced nowadays our results are satisfactory. We also proved that if Q=0 (either the state is product or) then the corresponding CM of the state is separable. Summarizing For Gaussian states all is fine (for pure state Q is a monotone, for mixed states it majorizes entanglement). For non-gaussian states provided entanglement is mostly in the smalls moments Q quantifies clas+quant correlations presents in the state.

17 Thanks for your attention! C. Rodó et al quant-ph/

18 Experimental setups