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We propose and demonstrate an optical implementation of a quantum key distribution protocol, which uses three-non-orthogonal states and six states in total. The proposed scheme improves the protocol that is proposed by Phoenix, Barnett and Chefles [J. Mod. Opt. 47, 507 (2000)]. An additional feature, which we introduce in our scheme, is that we add another detection set; where each detection set has three non-orthogonal states. The inclusion of an additional detection set leads to improved symmetry, increased eavesdropper detection and higher security margin for our protocol.

Quantum key distribution (QKD) is a cryptographic protocol which allows two communicating parties, conventionally called Alice and Bob, to distribute a secret key in such a way that the presence of an eavesdropper, Eve could be revealed [

In our scheme, Alice uses one of the three mutually non-orthogonal states in one set to encode her bit of infor- mation, and on the receiving side, Bob uses one of the three mutually non-orthogonal states of the other set to make measurements. The protocol is realized through the following steps, as shown in

1) Alice randomly and with equal probability prepares and sends quantum signals using any of the two detec- tion sets. Detection set0 is made up of the following mutually non-orthogonal states:

_{1} is made up of these states:

2) Bob performs a positive operator valued measurement (POVM) on each received signal, using any of the two detector sets. He then announces to Alice which detection set he used. They discard signals corresponding to time slots where they used similar detection sets for both preparation and measurement, and keep the rest. This is shown as the first result in the table.

3) On the remaining measured signals, Bob announces the time slots where he received no detections at all. Again, they discard the signals corresponding to these time slots and keep the rest.

4) Alice then announces one of the states she did not send. After this, Bob would then know what state Alice sent, and the signals corresponding to such time slots would be used to construct the secret key.

Time slot | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Alice prepares | ||||||||||

Bob detection set choice | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 |

Result | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |

Bob measures | ||||||||||

Result | 1 | 1 | 0 | 1 | 1 | 1 | 1 | |||

Alice says not in | ||||||||||

Bob says | ||||||||||

Sequence | ||||||||||

Inferred bit | 1 | 1 | 0 | 1 | 1 |

Consider the states of detection set_{0} to be

detection set_{1}, which has a set of states orthogonal to those in detection set_{0}, has the states:

In order to assign bit values shown in _{0} is assigned the basis “0” and detection set_{1} is assigned basis state “1”. The major additional feature, which happens in our protocol, occurs when Alice sends signals in a particular detection set and Bob uses the same detection set to make measurements. During this time slot, they discard their results. This happens in time slots 3, 6 and 8. In time slot 1, Alice prepares the state _{1}, Bob uses detection set_{0} to make the measurements, since they use different detection sets, they use signals from this time slot and they record “1” (yes). Bob then performs some measurements on the signals, which he received. Since Alice has announced the result as not a

In this section we begin by describing some of the most important aspects of information, i.e., measurement of information. Information can be measured according to its degree of uncertainty, i.e., if an event is likely then one gains little information in learning that it finally occurred and the converse is true. This brings us to the concept of entropy [

where

where

where

By introducing an extra detection set, we obtain a discrimination probability which is higher than that of the original scheme (where

Since these elements are positive semi-definite and form an identity, they are positive operator-valued mea- sures (POVMs). The maximum value of the mutual information can be achieved by a POVM with six elements described by

which implies that the system is not in one of the signal states. The signal states are the ones that carry the necessary information from which the secret key would be extracted.

Alice and Bob can detect the presence of an eavesdropper during the classical communication. This is achieved when they realize that the bits sent by Alice and measured by Bob are different. Let’s consider the case where Alice sends the state

If Alice sends the signal state

We have simplified the above equation by applying the fact that

By performing optimal mutual information measurement, it follows that Eve will obtain the results of

that describes the average state received by Bob is written as

that is the state with a similar form as that obtained for the maximum state-discrimination measurement.

For detection set_{0}, the protocol will give a correct key bit if Bob measures_{1}, the protocol will give a correct key bit if Bob measures

where 1/6 is the probability for Bob to have chosen the given measurement and 1/5 is the probability that arises from Alice’s random choice of announcing which state she did not send. This probability is less than the one proposed in [_{0}, if Alice prepares state_{1}.

As compared to the previous scheme, we realize that in our protocol, there is a higher chance that Eve will be detected. If Eve performs an intercept and resend attack, she will introduce errors in Alice’s and Bob’s shared key with the probability

Ultimately the average state received by Bob will be

For Bob, the probabilities of correct detection

and

bability that Bob will generate an incorrect key bit is lower than in the original scheme. This means that most of the time Bob will be able to correctly generate the correct bit.

We provide an optical implementation of this scheme in

In a like manner, on the left arm, between positions 3 and 4, we have a half-wave plate, which is given by the following unitary operator:

In both arms, these unitary operators are responsible for producing the states at position 4. On the right arm, the vertically polarised light at position 7, will be detected by D4, while on the left arm at the same position, the horizontally polarised light will be detected by D1. Between positions 8 and 9 of the right arm, the following unitary operator describes the half-wave plate:

1 | 2 | 3 | 4 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|

while on the left arm the unitary operator is given as

On the right arm, the vertically polarised light at position 10, will be detected by D5, while on the left arm at the same position, the horizontally polarised light will be detected by D2. The horizontally polarised light at position 11 for the right arm will be detected by D6 while the vertically polarised light on the left arm at position 11 will be detected by D3.

It is worth noting that in order to switch from set_{0} to set_{1}, the following unitary is used:

Conversely, the unitary used for switching from set_{1} to set_{0} is given as:

Apart from our proposed protocol being more secure, the additional symmetry that exists in our protocol also allows the various security proofs methods to be applied for instance the post-selection technique which was proposed by Renner and Christandl [

In this work, we have demonstrated an optical implementation of a six-state symmetric QKD protocol. Since our protocol uses additional detection set, this improves its symmetry, increases the discrimination probability and thereby improves the detection of Eve. In our proposed scheme, we have also demonstrated that the Bob’s pro- bability of generating error-free bitstring is lower than that of the original scheme. This shows that our proposed six-state symmetric QKD protocol is more secure.

This work is based on research supported by the South African Research Chair Initiative of the Department of Science and Technology and National Research Foundation.