qc000023.htm

6 Quantum Cryptography

Quantum systems can be used to achieve cryptographic tasks, such as secret (secure) communication. In cryptography (see, for example, Salomaa _[119]) it is very difficult, if not impossible, to prove by experiment that a cryptographic protocol is secure: who knows whether an eavesdropper (spy, competitor) managed to beat the system? For example, the bit-commitment method, thought for a while to be secure through quantum methods, was proven to be insecure, cf. Mayers _[95 ] and Lo, Chau _[90].

The only confidence one can hope to achieve relies on mathematical arguments, the so-called proofs of security.

As usual in quantum mechanics scenarios, Alice and Bob are widely separated and wish to communicate. They are connected by an ordinary bi-directional open channel and a uni-directional quantum channel, directed from Alice to Bob. The quantum channel allows Alice to send single qubits (e.g. photons) to Bob who can measure their quantum state. An eavesdropper, Eve, is able to intercept and measure the qubits, then pass them on to Bob.

Given two orthonormal bases

where

and

, Alice and Bob can agree to associate

and

with

and

with

. For each bit, Alice pseudo-randomly uses one of these bases and Bob also pseudo-randomly selects a basis for measuring the received qubit. After the bits have been transmitted, Alice and Bob inform each other (using the open channel) of the basis they used to prepare and measure each qubit. In this way, they find out when they used the same basis, which happens on average half of the time, and retain only those results.

If Eve measures the qubits transmitted by Alice, then she uses the correct basis on average half of the time. Therefore, assuming that

qubits are sent by Alice,

bits are received by Bob without any disturbance. In

cases, Bob will also use the correct basis.

What about the other

qubits sent by Alice? Since

and

the probability to find a qubit represented by the state

in the state

Taking into account that

the same result will be obtained by interchanging the two bases. Consequently, if

qubits are disturbed by Eve, then half of them are measured by Bob with the correct basis and

qubits will be projected by Bob’s measurement back onto original state. Eve’s tentative interception disturbs only a quarter of the message retained by Bob. Alice and Bob can now detect Eve’s presence by pseudo-randomly choosing

bits of the string and announcing over the open channel the values they have. If they agree on all these bits, then the probability that no eavesdropper was present is

. The undisclosed bits represent the “secret key".

The above scenario is extremely “theoretical", as it assumes that one possible strategy for Eve – she may deliberately not intercept all qubits (after all, she knows everything about quantum key distribution, doesn’t she?). Noise may influence the trio “communication" as well. There are various subtle methods to address these issues, but we are not going to enter into details (see, for example, Gruska _[67]). It is important to observe that variants of the quantum key distribution are feasible with current technology.

In what follows we will illustrate the use of quantum gates for to obtain the so-called “quantum teleportation", that is transmitting qubits without sending qubits!

What does this mean? Is it a pun? According to Bennett, (a co-author of a 1993 paper that proposed quantum teleportation, [17]) “It’s a means by which you can take apart an unknown quantum state into classical information and purely quantum information, send them through two separate channels, put them back together, and get back the original quantum state".

Teleportation, as it is commonly understood, is a fictional procedure of transferring an object from one location to another location in a three stage process: (a) dissociation, (b) information transmission, (c) reconstitution. The point is that, in contrast with fax transmission – where the original object remains intact at the initial location, only an approximate replica is constructed at destination, in teleportation the original object is distroyed after enough information about it has been extracted, the object is not traversing in any way the space between locations, but it is reconstructed, as an exact replica, at the destination.

Quantum teleportation allows for the transmission of quantum information to a distant location. The objective is to transmit the quantum state of a particle using classical bits and reconstruct the state at the receiver.

Let’s assume that Alice wishes to communicate with Bob a single qubit in an unkown state

; she wants to make the transmission through classical channels. Alice cannot know with certainty the state as any measurement she may perform may change it; she cannot clone it because of the no cloning result! So, it seems that the only way to send Bob the qubit is to send him the physical qubit, or to swap the state into another quantum system and then send Bob that system.