Quantum key distribution without a shared reference frame
C. E. R. Souza, C. V. S. Borges, and A. Z. Khoury
Instituto de Física, Universidade Federal Fluminense, Niteroi, RJ 24210-346, Brazil
J. A. O. Huguenin
Departamento de Ciências Exatas, Polo Universitário de Volta Redonda-UFF, Avenida dos Trabalhadores 420, Vila Santa Cecília,
Volta Redonda, RJ 27250-125, Brazil
L. Aolita and S. P. Walborn
*
Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, Brazil
Received 12 September 2007; published 27 March 2008
We report a simple quantum-key-distribution experiment in which Alice and Bob do not need to share a
common polarization direction in order to send information. Logical qubits are encoded into nonseparable
states of polarization and first-order transverse spatial modes of the same photon.
DOI: 10.1103/PhysRevA.77.032345 PACS numbers: 03.67.Dd, 03.67.Hk
I. INTRODUCTION
It has been shown that the transmission of quantum infor-
mation generally requires a shared reference frame SRF
and that this presents a considerable amount of overhead; an
infinite amount of information must be exchanged in order to
establish a perfect SRF 1,2. The type of reference frame
required depends not only on the physical system used, but
also on how information is encoded.
There exist protocols for quantum communication without
SRFs 3–8. Generally, they follow the same idea as
decoherence-free subspaces 9: encode rotation-invariant
“logical” qubit states into composite states of two or more
physical qubits. For example, the states
|0
L
=
1
2
|0
1
|1
2
- |1
1
|0
2
1a
and
|1
L
=
1
2
|0
1
|0
2
+ |1
1
|1
2
1b
are angular momentum eigenstates with zero eigenvalue, and
are thus invariant under bilateral R
y
R
y
rotations ro-
tations around the y axis in the Bloch sphere, where
R
y
|0 → cos
2
|0 + sin
2
|1 , 2a
R
y
|1 → cos
2
|1 - sin
2
|0 . 2b
For any rotation there exist states that are invariant, provided
that the rotation acts collectively on the qubits. This is only
approximately true for closely spaced ions in the same trap
or closely spaced photons traveling in the same optical fiber,
for example.
In Ref. 10, it was shown that two-qubit states defined in
the polarization and transverse spatial degrees of freedom of
the same photon satisfy the “collective condition” perfectly.
For quantum communication using polarization of photons,
the required reference frame is a well-established axis in the
plane transverse to the propagation direction. The lack of this
reference frame can be expressed as a random R
y
rotation
of the polarization degree of freedom. The same is true for
the first-order Hermite-Gaussian HG transverse spatial
modes 10,11. Thus, qubits 1 and 2 in Eq. 1a and 1b can
be represented by the polarization and HG mode of a single
photon by making the identification |0
1
|H, |1
1
|V,
|0
2
|h, |1
2
|v, where H and V stand for horizontal and
vertical polarization, and h and v stand for horizontal HG
01
and vertical HG
10
HG modes 11,12. For the sake of sim-
plicity, we consider different Hilbert spaces for the different
degrees of freedom of the same photon, which, although a
slight abuse of notation, allows us to make a simple analogy
with multiqubit entangled states.
Here we report an experimental investigation of a SRF-
free Bennett-Brassard 1984 BB84 key distribution protocol
using these two degrees of freedom of the same photon.
First, let us briefly summarize the BB84 quantum-key-
distribution protocol in the context of photon polarization
13,14. Traditionally, Alice sends photons to Bob, each po-
larized in one of four directions |H, |V, | + |H
+ |V /
2, or |-|H - |V /
2, and records which polar-
ization state she sent. Bob then measures randomly in either
the H / V or + / - basis, recording each basis chosen and the
corresponding result. Using classical communication, Alice
and Bob sift through their results, keeping only those cases
in which Bob measured in the “correct basis.” Bob’s sifted
results should coincide with each polarization that Alice sent
and will serve as a key in a classical cryptography protocol.
They can check the error rate in their key strings to deter-
mine the security achieved and can apply classical error cor-
rection and privacy amplification techniques 14. The lack
of a SRF in the BB84 protocol results in a larger quantum bit
error rate, which may compromise the success of the key
distribution. We note that a procedure for a rotation-invariant *
swalborn@if.ufrj.br
PHYSICAL REVIEW A 77, 032345 2008
1050-2947/2008/773/0323454 ©2008 The American Physical Society 032345-1