Bridging Discrete and Continuous
Variables in Quantum Information
Jonas S. Neergaard-Nielsen, Makoto Takeuchi, Kentaro Wakui, Hiroki Takahashi,
Kazuhiro Hayasaka, Masahiro Takeoka and Masahide Sasaki
The fundamental unit in quantum information is the qubit—a coherent superposition of two orthogonal
quantum states. In optics, this abstract
object is often realized by discrete degrees
of freedom of a single photon such as
horizontal/vertical polarization or two
separate time bins. A complementary, but
less prominent, paradigm of quantum information is that of continuous variables
(CV) where information is encoded in
continuous degrees of freedom, typically the phase quadratures of an optical
beam. CV experiments often make use
of squeezed light where the quantum
fluctuations of one of the quadratures has
been reduced below the vacuum level. 1
A few years ago, a fascinating new
kind of nonclassical state was demonstrated—the squeezed single photon (also
nicknamed the “Schrödinger kitten”). 2 Its
name already hints at it living in an overlap between the two different paradigms.
This year, we went a step further and developed a method for generating arbitrary
superpositions of such a squeezed single
photon and an ordinary squeezed vacuum
state. The method is based on a special
ambiguous photon subtraction technique.
We showed that by carefully adjusting the
control parameters, any desired superposition of the two states can be prepared. 3
Since the squeezed vacuum and
squeezed photon states are orthogonal,
these superpositions are essentially qubit
states and, in that sense, this experiment
bridges the two regimes of discrete and
continuous variables. When looked at
individually, the generated states are
best understood in terms of continuous
variables; they consist of contributions
from a range of different photon number
states and therefore occupy a larger space
than the two-dimensional qubit space.
However, they can all be reduced to the
same squeezed vacuum/squeezed photon
superposition space, so when considered
The data in this image can be viewed in more detail in the interactive “qubit space flight” applet available online.
A range of squeezed vacuum/squeezed photon superposition states.
together as a class of quantum states,
this qubit description gives a better overview of their relationship to each other.
The qubit state can be conveniently
visualized on a Poincaré (or Bloch) sphere
with the two basis states represented on
the north and south poles. Conversely,
squeezed states, for example, are usually
represented by their continuous distributions in a phase space of the two
conjugate variables. This distribution,
known as the Wigner function, can be
inferred from a homodyne tomographic
measurement. In the figure, we have
plotted the measured Wigner functions
of all the generated states, inserted at
their appropriate locations in the Poincaré
sphere. This mixed picture illustrates the
complementarity of the two different
representations and may become a general
tool for analysis of such complex states.
Apart from the fundamental interest
of this demonstration of a new class of
photonic quantum states in an intermediate regime, it is also an important encouragement for the further development
of a special hybrid scheme of quantum
computation that uses coherent state
superpositions as its basic unit of operation. 4 Furthermore, it may provide a path
toward more efficient quantum communication by taking advantage of the high
capacity obtainable with coherent states. t
to view the video that accompanies
J.S. Neergaard-Nielsen ( firstname.lastname@example.org), Makoto Takeuchi, Kentaro Wakui, Hiroki Takahashi, Kazuhiro Hayasaka, Masahiro Takeoka and Masahide
Sasaki are with the National Institute of Information
and Communications Technology in Tokyo, Japan.
1. J.L. O’Brien et al. Nature Photonics 3, 687-95 (2009).
2. A. Ourjoumtsev et al. Phys. Rev. Lett. 97, 083604 (2006).
3. J.S. Neergaard-Nielsen et al. Phys. Rev. Lett. 105,
4. T.C. Ralph et al. Phys. Rev. A 68, 042319 (2003).
46 | OPN Optics & Photonics News