Security in quantum key distribution protocols.

M8 Mhlambululi Mafu and Francesco Petruccione, “Upper bound on the accessible information for the six-state quantum key distribution protocol,” in Proceedings of the 57th South African Institute of Physics conference (2012). Mhlambululi Mafu and Francesco Petruccione, “Derivation of the quantum bit error rate for the BB84 protocol based on the covariant cloning machine,” 57th Conference of the South African Institute of Physics (2012).

Introduction

In this class of protocols, information is encoded in continuous variables of the electromagnetic field. Despite the challenges that come with developing unconditional safety evidence, much progress has been made in the past two decades.

Setting the scene

As a result of this mismatch, an eavesdropper can learn part of the key shared by Alice and Bob, thus making some schemes insecure over large distances. Symmetric key cryptography is unconditionally secure only if the symmetric distributed key is unconditionally secure, while the security of the asymmetric scheme rests on the assumption that factoring integers turns out to be a difficult task.

Thesis outline

The purpose of this chapter is to discuss the basic concepts of mathematical foundations of quantum mechanics for finite dimensions. The basic principles of linear algebra as used in quantum information theory will also be outlined.

Hilbert Space

In the following chapters we also provide definitions and basic formulations of other tools that are needed. If |xi and |yi are arbitrary vectors in the Hilbert space H, then the inner product is denoted as.

Tensor product

Schmidt decomposition The Schmidt decomposition is a mathematical tool used to analyze a pure state and its partial trace [58]. If the Schmidt number is greater than one, then the bipartite pure state is entangled [59].

Linear Operator

Then there exists an additional system R and a pure state |ψARi on the common system AR such that ρA = trR|ψihψ|AR. The Schmidt number is an important property of a compound quantum system in the sense that it quantifies the amount of entanglement between systems A and system B.

States

Pure States

Mixed States

Entangled States

Quantum operations

Unitary operations

The state |ψi of a quantum system with the density matrix ρ can be transformed by applying a unitary operator U.

Quantum Channel

Measurements

Projective Measurements
Positive Operator-Value Measurements (POVM)
Stinespring’s Dilation
Shannon entropy
Joint entropy
Relative Entropy/Kullback-Leibler distance

In the case of two independent variables XandY, the characteristic additivity of the Shannon entropy, that is, H(X×Y) = H(X) +H(Y) [76, 60]. In terms of the Kullback-Leibler distance in Equation (3.1.6), the relative entropy between two quantum systems with density operators ρ and σ can be written as [61].

Quantum Information Measures

Conditional entropy
Mutual Information
Conditional mutual information
von Neumann entropy
R´ enyi entropies

If we consider the composite system ρAB, the von Neumann entropy is expressed as S(A, B) =−trAB(ρABlog2ρAB). 3.2.8) Suppose we want to consider the correlations between the classical system X and the quantum system B, which is described as the classical-quantum state ρXB =P. 3.2.9) Classical conditional entropy is expressed as. 3.2.10) Therefore, the mutual information between X and B is expressed as.

Distance measures

Trace distance (classical)

Trace distance (quantum)

Fidelity (quantum)

If ρ and σ vary, the quantum fidelity F (ρ, σ) reduces to the classical fidelity F (ri, si) between the eigenvalue distributions ri and si for ρ and σ. This equation shows that the fidelity between two mixed states can be interpreted as the maximum overlap between two purifications of those states.

Introduction

Quantum features

Detection of Measurements
Uncertainty principle
No-Cloning Theorem
Non-orthogonality principle

The uncertainty principle states that a measurement of one observable quantum intrinsically creates an uncertainty in other properties of the system. Then there is no measurement of any kind that can reliably determine which of the two non-orthogonal quantum states has been measured [16].

QKD schemes

Prepare and Measure (P&M) scheme

In a P&M scheme, Alice encodes some classical information into a set of quantum states and sends it via an insecure quantum channel to Bob.

Entanglement-Based (EB) scheme

QKD procedure

Quantum Phase

Afterwards, Alice keeps a record of signal choices, Bob keeps a record of his base choices and the corresponding measurement result.

Classical phase

Security in QKD

Security definition
Security requirements
Infinite-length key security in QKD
Finite-length key security

The term εcor is the maximum probability that the protocol deviates from the behavior of the correct protocol. The second term εPA is directly related to the failure probability of the privacy enhancement procedure.

Tsallis entropy

Similar to the concavity property (0<α <1) of the R´enyi entropy, for two probability measure functions p and r, the concavity property of the Tsallis entropy can be written as.

On the uncertainty relation

This may be due to a large difference that exists between the Shannon and the Tsallis entropy, that is, the Shannon entropy and R'enyi entropy are additive for independent probability distributions while the Tsallis entropy is non-additive or is pseudo-additive [130] . Similarly, if a system consists of two subsystems that are independent of each other, say A and B with the probability distribution pi and pj respectively, then the R´enyi entropy Hα(p) is equal to the sum of Hα(pi ) and Hα( pj) i.e.,. 5.3.7) Therefore, this non-additivity property appears to be a challenge to immediately connect the Tsallis entropy with these applications in quantum information.

Tsallis entropy and Uncertainty Relations

Therefore, we aim to investigate whether the non-extending property of the Tsallis entropy will ever make a difference to the requirements for B instead of using the Shannon entropy. This provides an irreducible lower bound (generalized uncertainty measure) for the uncertainty of the simultaneous measurement of observables when using the Tsallis entropy to express the quantum uncertainty relation.

Conclusion

The secret key rate for the six-state protocol via R´enyi entropies is presented in Ref [84]. Therefore, in this chapter we present achievable key length bounds for the B92 protocol [138], which includes a preprocessing step using uncertainty relations [134] and R´enyi entropies [139].

The B92 QKD Protocol

The first measurement projects onto the basis |ψ+i, which consists of the vectors {|ψ−i,|ψ˜−i}, where |ψ˜−i is orthogonal to |ψ−i. In particular, if the statistic λm is obtained by measuring m samples of ρAB (i.e., the entangled state shared by Alice and Bob) according to a positive operator-valued measure (POVM) with d possible outcomes, and if λ∞(ρAB) denotes the perfect statistic in the limit of infinitely many measurements, so for any state ρAB (see also Sect.

Definitions

R´ enyi entropy

Alice randomly chooses a function F from a bi-universal hash function and sends Bob a description of F. Using the result in [107], it is found that the achievable length of the secret key that can be computed from X out of two universal hash functions, F can be expressed as.

Smooth Min-and Max-entropy

Min- and Max-entropy

Results

Bound on the secure key rate

The maximum probability of failure of the protocol is ε= 10−5 and the probability of failure of the error correction procedure is εEC= 10−10 [M2].

Bound on the achievable key length

By introducing the printout of the classical communication C and without loss of generality, we have. Using the data processing inequality [38] and the uncertainty relation in equation (6.4.2) we get.

Conclusion

In this chapter, we present an experimental demonstration of the B92 QKD protocol [16] using the id300 Clavis2 system from idQuantique. Therefore, in this chapter we show the possibility of implementing the B92 protocol using the id3100 Clavis2 system.

The B92 QKD protocol

An unconditional security proof for the B92 protocol implemented by a strong phase reference pulse instead of the weak pulse assumption was shown by Koashi in 2004 [17]. In particular, an experiment of the B92 protocol reaching a distance of 122 km of standard telecommunication fiber by Gobby et al.

Plug and Play scheme

The great advantage of the Plug and Play system is that it does not require additional optical settings during operation. The implementation of the B92 protocol on a Plug and Play system (which is an interferometric layout) is justified since the original B92 protocol was based on an interferometric layout [16].

Experimental setup

Notably, Plug and Play has also been used as part of the devices in the SECOQC QKD network in Vienna [30]. Finally, the key generation rates for the B92 protocol were compared to the key generation rates for the BB84 protocol to determine the usefulness of the B92 implementation.

Results and Discussion

This probability is evaluated by equation (7.5.4); P refers to the dark number probability and V is the quantum channel visibility in percent. QBER is an important parameter in QKD because it is used for exploration. security in QKD protocols [14].

Table 7.1: Experimentally measured QKD parameters for the set-up shown in Fig 7.2.

Conclusion

In Figure 7.4 we show the variation of Shannon mutual information between Alice and BobI(A:B) and between Alice and Eve I(A :E) against optical loss. As security measures, we determine from experimental data the average error rate, the mutual information shared between the sender and receiver, and the secret key generation rate per photon.

Figure 7.4: The Shannon mutual information between Alice and Bob I(A : B ) and between Alice and Eve I(A : E) against optical loss

Theory of MUB protocols

In this chapter, we present an experimental study of higher-dimensional QKD protocol based on mutually unbiased bases (MUBs), implemented using photons carrying orbital angular momentum (OAM). However, their actual performance in terms of secure key rate depends on whether the amount of noise in higher-dimensional implementations grows faster with increasing dimension than their robustness to noise.

Mutually unbiased bases

The eigenbases belonging to different operators in the set {Z, XZl|l d− 1}} form a complete set of MUBs for every prime number d as the dimension of the Hilbert space. In general, for the main dimensions, the standard basis consists of the dLG modes, while the remaining dbasis concern the overlaps of the LG modes.

Filter based MUB QKD protocol

The error rate refers to the probability that Alice sends the state |φ(β,k)i, while Bob receives an orthogonal state |φ(β,k0)i. Using a full set of MUBs leads to an increase in the acceptable error rate in which we can still extract a reasonable secret key without compromising the security of the protocol.

Experimental Setup

61 parties then compare a small part of their measurements to obtain an estimate of the average error rate. The limit on the tolerable error rate that is safe for secret key generation can be improved by implementing a full set of (d+ 1) MUBs [168, 160].

Results and Discussion

For each permutation of Alice and Bob's projective measurements in the EB scheme, the individual count rates and coincidence count rates were recorded, and the normalized total probabilities calculated for d = 2, 3, 4, and 5 are given in Figure 8.4. The curves indicate the theoretical secret key rate as a function of the average error rate plotted using equation (8.4.3).

Figure 8.3: (a) Cross-sectional intensity profiles of the field recorded on the CCD for permutations of the first basis’s states encoded on SLM A and SLM B

Conclusion

The probability of coincidence can be calculated as an expectation value of the operator Ps⊗Ps0 with respect to the state |ψi (cp. Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to prove entanglement.