Unconditional Continuous Variable Dense Coding

Abstract

We investigate the conditions under which unconditional dense coding can be achieved using continuous variable entanglement. We consider the effect of entanglement impurity and detector efficiency and discuss experimental verification. We conclude that the requirements for a strong demonstration are not as stringent as previously thought and are within the reach of present technology.

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arXiv:quant-ph/0208117v1 17 Aug 2002
Unconditional Continuous Variable Dense Coding.
T.C.Ralph
Centre for Quantum Computer Technology,
Department of Physics, The University of Queensland,
St Lucia 4072 Australia
E-mail: ralph@physics.uq.edu.au
E.H.Huntington
School of Electrical Engineering,
University College,
University of New South Wales,
Australian Defence Force Academy
Canberra ACT 2600 Australia
(February 2002)
Abstract
We investigate the conditions under which unconditional dense coding can
be achieved using continuous variable entanglement. We consider the effect
of entanglement impurity and detector efficiency and discuss experimental
verification. We conclude that the requirements for a strong demonstration
are not as stringent as previously thought and are within the reach of present
technology.
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I. INTRODUCTION
The classical channel capacity of a quantum channel can be enhanced if the sender and
recipient of the information, Alice and Bob repectively, share an entangled state. This effect
is known as quantum dense coding [1] and can be thought of as the converse problem to
quantum teleportation [2] where, effectively, the quantum capacity of a classical channel is
enhanced by the use of entanglement.
Dense coding was originally introduced for discrete variables and an experimental demon-
stration of the effect has been made using photonic polarization entanglement [3]. One
drawback of this demonstration was, due to the low efficiency of entanglement production
and detection, the demonstration was conditional on Bob detecting a pair of photons, a
rare event. In contrast a dense coding scheme based on continuous variables, such as the
quadrature amplitudes of a light field, which has recently been proposed, would in princi-
ple demonstrate an unconditional improvement in classical channel capacity [4]. Ultimately
this scheme can beat, under certain conditions, the maximum channel capacity given by
Fock state encoding. However the conditions for this strong violation found in [4] required
unrealistic levels of squeezing.
A number of groups have taken steps towards the experimental implementation of this
scheme [5,6]. In these experiments increased signal to noise was demonstrated with the
addition of entanglement and conclusions were drawn about the violation of coherent state
classical capacity, based on the results of Ref [4]. However unit entanglement state purity
and detection efficiency were assumed in [4], which is unlikely to have been the case experi-
mentally. Also no attempt was made to experimentally quantify the number of quanta used
in the communication channel. There is thus a need for a more detailed analysis.
In this paper we make such an investigation. We come to the rather surprizing conclusion
that in fact the conditions required for a strong demonstration of the effect, i.e. beating the
ultimate channel capacity given by Fock state encoding, are not as stringent as previously
thought, even taking into account lack of state purity and non-unit detection efficiency.
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II. IDEAL CHANNEL CAPACITIES
We begin by rederiving the channel capacities of Gaussian quantum channels and contin-
uous variable dense coding using quadrature spectral variances. Such variances are directly
measureable in an experiment. The Shannon capacity [7] of a communication channel with
Gaussian noise of power (variance) N and Gaussian distributed signal power S operating at
the bandwidth limit is
C =1
2log2[1 +S
N](1)
Eq.1 can be used to calculate the channel capacities of quantum states with Gaussian prob-
ability distributions such as coherent states and squeezed states [8,9]. Consider first a signal
composed of a Gaussian distribution of coherent state amplitudes all with the same quadra-
ture angle (see Fig.1 (a)). The signal power Vsis given by the variance of the distribution.
The noise is given by the intrinsic quantum noise of the coherent states and is defined to be
Vn= 1. Because the quadrature angle of the signal is known, homodyne detection can in
principle detect the the signal without further penalty thus the measured signal to noise is
S/N = Vs/Vn= Vs.
In general the average photon number per bandwidth per second of a light beam is given
by
¯ n =1
4(V++ V−) −1
2
(2)
where V+(V−) are the variances of the maximum (minimum) quadrature projections of the
noise ellipse of the state. These projections are orthogonal quadratures, such as amplitude
and phase, and obey the uncertainty principle V+V−≥ 1. In the above example one
quadrature is made up of signal plus quantum noise such that V+= Vs+ 1 whilst the
orthogonal quadrature is just quantum noise so V−= 1. Hence ¯ n = 1/4Vsand so the channel
capacity of a coherent state with single quadrature encoding and homodyne detection is
Cc= log2[√1 + 4¯ n](3)
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Establishing in an experiment that a particular optical mode has this capacity would involve:
(i) measuring the quadrature amplitude variances of the beam, V+and V−, (ii) calibrating
Alice’s signal variance and (iii) measuring Bob’s signal to noise. If these measurments agreed
with the theoretical conditions above then Shannons theorem tells us that an encoding
scheme exists which could realize the channel capacity of Eq.3. An example of such an
encoding is given in [10].
For photon numbers ¯ n > 2 improved channel capacity can be obtained by encoding sym-
metrically on both quadratures and detecting both quadratures simultaneously using hetero-
dyne detection or dual homodyne detection (see Fig.1(b)). Because of the non-commutation
of orthogonal quaratures there is a penalty for their simultaneous detection which reduces
the signal to noise of each quadrature to S/N = 1/2Vs. Also because there is signal on both
quadratures the average photon number of the beam is now ¯ n = 1/2Vs. On the other hand
the total channel capacity will now be the sum of the two independent channels carried by
the two quadratures. Thus the channel capacity for a coherent state with dual quadrature
encoding and heterodyne detection is
Cch=1
2log2[1 +S
N
+
] +1
2log2[1 +S
N
−
]
= log2[1 + ¯ n] (4)
which exceeds that of the homodyne technique (Eq.3) for ¯ n > 2.
The above channel capacities are the best achievable if we restrict ourselves to a semi-
classical treatment of light. However the channel capacity of the homodyne technique
(Fig.1(a)) can be improved by the use of non-classical, squeezed light. With squeezed light
the noise variance of the encoded quadrature can be reduced such that Vne< 1, whilst the
noise of the unencoded quadrature is increased such that Vnu≥ 1/Vne. As a result the signal
to noise is improved to S/N = Vs/Vnewhilst the photon number is now given by Eq.2 but
with V+= Vs+ Vneand V−= 1/Vnewhere a pure (i.e. minimum uncertainty) squeezed
state has been assumed. Maximizing the signal to noise for fixed ¯ n leads to S/N = 4(¯ n+¯ n2)
for a squeezed quadrature variance of Vne,opt= 1/(1 + 2¯ n). Hence the channel capacity for
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a squeezed beam with homodyne detection is
Csh= log2[1 + 2¯ n] (5)
which exceeds both coherent homodyne and heterodyne for all values of ¯ n.
A final improvement in channel capacity can be obtained by allowing non-Gaussian
states. The absolute maximum channel capacity for a single mode is given by the Holevo
bound and can be realized by encoding in a maximum entropy ensemble of Fock states and
using photon number detection [8,9,11]. This ultimate channel capacity is
CFock= (1 + ¯ n)log2[(1 + ¯ n)] − ¯ nlog2[¯ n](6)
which is the maximal channel capacity at all values of ¯ n.
We now turn to dense coding. At low average photon numbers the single channel capaci-
ties are always best. However, we will find that for sufficiently high average photon numbers
dense coding can give superior capacities. The set-up is depicted in Fig.1(c). Entanglement
is generated in the standard way by mixing two squeezed states, with their squeezing ellipses
orthogonal,on a 50:50 beamsplitter [12]. One half of the entangled pair is sent to Alice who
encodes on both quadratures in the manner of coherent heterodyne. She sends the beam on
to Bob who has also received the other half of the entangled pair. He uses a dual homodyne
technique to measure both quadratures of the beam from Alice but injects his entangled
beam into the empty port of the dual homodyne beamsplitter. The resulting signal to noise
for the two quadrature channels is S/N = 1/2(Vs/Vne), where now Vneis the variance of the
squeezed quadrature of the beams used to create the entanglement. The photon number is
just that of the beam carrying the signal (the cost of distributing the entanglement is not
taken into account) and so is given by Eq.2 with V+= 1/2Vs+ 1/4Vneand V−= 1/Vne.
Once again pure squeezed states are assumed. Maximizing the signal to noise for fixed ¯ n
gives S/N = ¯ n + ¯ n2for a squeezed quadrature variance of Vne,opt = 1/(1 + 2¯ n). So the
optimum channel capacity for dense coding is
Copt
dc= log2[1 + ¯ n + ¯ n2] (7)
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