TY - JOUR

T1 - Cartesian and polar Schmidt bases for down-converted photons

T2 - how high dimensional entanglement protects the shared information from non-ideal measurements

AU - Miatto, Filippo

AU - Brougham, Thomas

AU - Yao, Alison

PY - 2012/7/12

Y1 - 2012/7/12

N2 - We derive an analytical form of the Schmidt modes of spontaneous parametric down-conversion (SPDC) biphotons in both Cartesian and polar coordinates. We show that these correspond to Hermite- Gauss (HG) or Laguerre-Gauss (LG) modes only for a specific value of their width, and we show how such value depends on the experimental parameters. The Schmidt modes that we explicitly derive allow one to set up an optimised projection basis that maximises the mutual information gained from a joint measurement. The possibility of doing so with LG modes makes it possible to take advantage of the properties of orbital angular momentum eigenmodes. We derive a general entropic entanglement measure using the R ́enyi entropy as a function of the Schmidt number, K, and then retrieve the von Neumann entropy, S. Using the relation between S and K we show that, for highly entangled states, a non-ideal measurement basis does not degrade the number of shared bits by a large extent. More specifically, given a non-ideal measurement which corresponds to the loss of a fraction of the total number of modes, we can quantify the experimental parameters needed to generate an entangled SPDC state with a sufficiently high dimensionality to retain any given fraction of shared bits.

AB - We derive an analytical form of the Schmidt modes of spontaneous parametric down-conversion (SPDC) biphotons in both Cartesian and polar coordinates. We show that these correspond to Hermite- Gauss (HG) or Laguerre-Gauss (LG) modes only for a specific value of their width, and we show how such value depends on the experimental parameters. The Schmidt modes that we explicitly derive allow one to set up an optimised projection basis that maximises the mutual information gained from a joint measurement. The possibility of doing so with LG modes makes it possible to take advantage of the properties of orbital angular momentum eigenmodes. We derive a general entropic entanglement measure using the R ́enyi entropy as a function of the Schmidt number, K, and then retrieve the von Neumann entropy, S. Using the relation between S and K we show that, for highly entangled states, a non-ideal measurement basis does not degrade the number of shared bits by a large extent. More specifically, given a non-ideal measurement which corresponds to the loss of a fraction of the total number of modes, we can quantify the experimental parameters needed to generate an entangled SPDC state with a sufficiently high dimensionality to retain any given fraction of shared bits.

KW - high dimensional quantum entanglement

KW - photons

KW - polar coordinates

UR - http://www.scopus.com/inward/record.url?scp=84865604827&partnerID=8YFLogxK

U2 - 10.1140/epjd/e2012-30063-y

DO - 10.1140/epjd/e2012-30063-y

M3 - Article

VL - 66

SP - 183

EP - 190

JO - European Physical Journal D: Atomic, Molecular, Optical and Plasma Physics

JF - European Physical Journal D: Atomic, Molecular, Optical and Plasma Physics

SN - 1434-6060

IS - 7

ER -