Bidirectional Secret Communication by Quantum
Collisions
Fabio Antonio Bovino
Elsag Datamat, via Puccini 2, Genova. 16154, Italy
[email protected]
Abstract. A novel secret communication protocol based on quantum entanglement is introduced. We demonstrate that Alice and Bob can perform a bidirectional secret communication
exploiting the “collisions” on linear optical devices between partially shared entangled states.
The protocol is based on the phenomenon of coalescence and anti-coalescence experimented by
photons when they are incident on a 50:50 beam splitter.
Keywords: secret communications, quantum entanglement.
1 Introduction
Interference between different alternatives is in the nature of quantum mechanics [1].
For two photons, the best known example is the superposition on a 50:50 beamsplitter (Hong Ou Mandel –HOM– interferometer): two photons with the same polarization are subjected to a coalescence effect when they are superimposed in time [2].
HOM interferometer is used for Bell States measurements too, and it is the crucial
element in the experiment of teleportation or entanglement swapping.
Multi-particle entanglement has attracted much attention in these years. GHZ
(Greemberger, Horne, Zeilinger) states showed stronger violation of locality. The
generation of multi-particle entangled states is based on interference between independent fields generated by Spontaneous Parametric Down Conversion (SPDC) from
non-linear crystals. As an example four photons GHZ states are created by two pairs
emitted from two different sources.
For most applications high visibility in interference is necessary to increase the fidelity of the produced states. Usually, high visibility can be reached in experiments
involving only a pair of down-converted photons emitted by one source and quantum
correlation between two particles is generally ascribed to the fact that particles involved are either generated by the same source or have interacted at some earlier time.
In the case of two independent sources of down converted pairs stationary fields
cannot be used, unless the bandwidth of the fields is much smaller than that of the
detectors. In other words the coherent length of the down conversion fields must be so
long that, within the detection time period, the phase of the fields is constant. This
limitation can be overcome by using pulsed pump laser with sufficiently narrow temporal width.
E. Corchado et al. (Eds.): CISIS 2008, ASC 53, pp. 280–285, 2009.
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© Springer-Verlag Berlin Heidelberg 2009
Bidirectional Secret Communication by Quantum Collisions
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2 Four-Photons Interference by SPDC
We want to consider now the interference of two down-converted pairs generated by
distinct sources that emit the same state. We take, as source, non-linear crystals, cut
for a Type II emission, so that the photons are emitted in pairs with orthogonal polarizations, and they satisfy the well known phase-matching conditions, i.e. energy and
momentum conservation. In the analysis we select two k-modes (1,3) for the first
source and two k-modes for the second source (2,4), along which the emission can be
considered degenerate, or in other words, the central frequency of the two photons is
half of the central frequency of the pump pulse.
We are interested to the case in which only four photons are detected in the experimental set-up, then we can reduce the total state to
2 ⊗2
Ψ1234
=
η2
2
2
Ψ13
+
η2
2
2
1
1
Ψ24
+ η 2 Ψ13
Ψ24
(1)
where |η|² is the probability of photon-conversion in a single pump pulse: η is proportional to the interaction time, to the χ⁽²⁾ non-linear susceptibility of the non-linearcrystal and to the intensity of the pump field, here assumed classical and un-depleted
during the parametric interaction.
Thus we have a coherent superposition of a double pair emission from the first
source (eq. 2), or a double pair emission from the second source (eq. 3), or the emission of a pair in the first crystal and a pair in the second one (eq. 4):
2
Ψ13
=
η2
2
∫ ∫ ∫ ∫ dω1dω2dω3dω4
× Φ(ω1 + ω2 )Φ(ω3 + ω4 )e−i (ω1 +ω2 )ϕ e−i (ω3 +ω4 )ϕ
[
× [aˆ
]
(ω )] vac
× aˆ1+e (ω1 )aˆ3+o (ω2 ) − aˆ3+e (ω1 )aˆ1+o (ω2 )
+
1e
(ω3 )aˆ3+o (ω4 ) − aˆ3+e (ω3 )aˆ1+o
2
Ψ24
=
η2
2
4
∫ ∫ ∫ ∫ dω1dω2 dω3dω4
× Φ(ω1 + ω2 )Φ(ω3 + ω4 )e −i (ω1 +ω2 )ϕ
[
× [aˆ
]
(ω )] vac
× aˆ 2+e (ω1 )aˆ 4+o (ω2 ) − aˆ2+e (ω1 )aˆ 4+o (ω2 )
+
2e
(ω3 )aˆ4+o (ω4 ) − aˆ2+e (ω3 )aˆ4+o
1
1
Ψ13
Ψ24
=
η2
2
4
]
× aˆ1+e (ω1 )aˆ3+o (ω2 ) − aˆ1+e (ω1 )aˆ3+o (ω2 )
+
2e
(3)
∫ ∫ ∫ ∫ dω1dω2 dω3dω4
× Φ(ω1 + ω2 )Φ(ω3 + ω4 )
[
× [aˆ
(2)
(4)
(ω3 )aˆ4+o (ω4 ) − aˆ2+e (ω3 )aˆ4+o (ω4 )] vac
The function Φ (ω1 + ω2 ) contains the information about the pump field and the parametric interaction, and can be expanded in the form
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Fabio Antonio Bovino
Φ (ω1 + ω 2 ) = Ε p (ω1 + ω 2 )φ (ω1 + ω 2 , ω1 − ω 2 )
(5)
where Ε p (ω1 + ω 2 ) describes the pump field spectrum and φ (ω1 + ω 2 , ω1 − ω 2 ) is the
two photon amplitude for single frequency pumped parametric down-conversion.
Without loss of generality, we impose a normalization condition on Φ (ω1 + ω 2 ) so that
2
∫∫ dω1dω 2 Φ (ω1 + ω 2 ) = 1
(6)
We want to calculate the probability to obtain Anti-Coalescence or Coalescence on
the second Beam-splitter (BS) conditioned to Anti-coalescence at the first one.
For Anti-Coalescence-Anti-Coalescence probability (AA) we obtain:
AA =
5 + 3Cos(4Ω 0ϕ )
20
(7)
For Anti-Coalescence-Coalescence Probability (AC) we obtain:
3Sin 2 (2Ω 0ϕ )
5
AC =
(8)
For Coalescence-Coalescence probability (CC) we obtain:
CC = 5
3 + Cos(4Ω 0ϕ )
20
(9)
If the two sources emit different states, the result is different. In fact for AA probability we obtain:
AA =
3 + Cos(4Ω 0ϕ )
20
(10)
5 − Cos (4Ω 0ϕ )
10
(11)
7 + Cos (4Ω 0ϕ )
20
(12)
For AC probability we obtain:
AC =
For CC probability we obtain:
CC =
3 Four-Photons Interference: Ideal Case
Now, let us consider the interference of two pairs generated by distinct ideal polarization entangled sources that emit the same state, for example two singlet states:
1
1
Ψ13
Ψ24
=
(
)(
)
1 + +
aˆ1e aˆ3o − aˆ1+e aˆ3+o aˆ 2+e aˆ 4+o − aˆ 2+e aˆ 4+o vac
2
For AA, AC, CC probabilities we obtain:
(13)
Bidirectional Secret Communication by Quantum Collisions
1
,
4
AC = 0,
283
AA =
(14)
3
CC =
4
If the two distinct ideal polarization entangled sources emit different states, for example a singlet state the first one and a triplet state the second one, we have:
1
1
Ψ13
Ψ24
=
(
)(
)
1 + +
aˆ1e aˆ3o − aˆ1+e aˆ3+o aˆ 2+e aˆ 4+o + aˆ 2+e aˆ 4+o vac
2
(15)
For AA, AC, CC Probabilities we obtain:
AA = 0,
1
AC = ,
2
1
CC =
2
(16)
4 Bidirectional Secret Communication by Quantum Collision
The last result could be used for a quantum communication protocol to exchange
secret messages between Alice and Bob.
Alice has a Bell’s states synthesizer (i.e. an entangled states source, a phase shifter
and a polarization rotator), a quantum memory and a 50:50 beam-splitter.
Alice codifies binary classical information by two different Bell states, instead she
codifies 1 with singlet state and 0 with triplet state. Then she sends one of the photons
of the state to Bob, and maintains the second one in the quantum memory.
Fig. 1. Set-up used by Alice and Bob to perform the bidirectional secret communication
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Fabio Antonio Bovino
Bob has the same set-up of Alice, so that he is able to codify a binary sequence by
the same two Bell’s states used in the protocol. Bob sends one of the photons of the
state to Alice, and maintains the second one in the quantum memory. Alice and Bob
perform a Bell’s measurement on the two 50:50 beam-splitters.
If Alice and Bob have exchanged the same bit, after the Bell’s measurements the
probability of coalescence-coalescence will be ¾, and the probability of AntiCoalescence-Anti-Coalescence will be ¼.
If Alice and Bob have exchanged a different bit, the probability to obtain Coalescence-Coalescence will be ½ and the probability to obtain Anti-CoalescenceCoalescence will be ½.
After the measurements, Alice and Bob communicate on an authenticated classical
channel the results of the measurements: if the result is Anti-Coalescence-AntiCoalescence, Alice and Bob understand that the same state has been used; if the result
is Anti-Coalescence-Coalescence or Coalescence-Anti-Coalescence, they understand
that they used a different state.
In the case of Coalescence-Coalescence, they have to repeat the procedure until a
useful result is obtained. If we assume that the probability to use the singlet and triplet
states is equal to ½, Alice (Bob) can reconstruct the 37.5% of the message sent by
Bob (Alice).
Fig. 2. Possible outputs of a Bell’s measurement
5 Conclusion
Sources emitting entangled states “on demand” do not exist and the protocol, for now,
can not be exploited. The novelty is the use of quantum properties to encoding and
decoding a message without exchange of a key and the protocol performs a really
quantum cryptographic process. It seems that the fundamental condition is the authentication of the users: if the classical channel is authenticated it is not possible to extract information from the quantum communication channel. The analysis of security
is not complete and comments are welcome.
Bidirectional Secret Communication by Quantum Collisions
References
1. Schrödinger, E.: Die Naturewissenschaften 48, 807 (1935)
2. Hong, C.K., Ou, Z.Y., Mandel, L.: Phys. Rev. Lett. 59, 2044 (1987)
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