Effect on a QKD Chain Going Through a Generic Pauli Channel Together with a Correction Based on the 3-qubit Code

Quantum cryptography is an emergent area in quantum technologies to set secure communication. BB84 protocol is a procedure to exchange a secure key to encode communication in the form of a chain of quantum bits. In this work, we model the non-ideal transmission of such a key through a parametric family of noisy channels: Pauli channels (including Bit-flipping and Dephasing noises). Assuming that Quantum Key Distribution is performed under such protocol, we analyse the fidelity of the transmission and the probability distribution for the imperfect outcomes. The analysis also includes the application of the quantum error correction algorithm: the 3-qubit code. Finally, we add the presence of an eavesdropper getting his conditional success rate to become unnoticed. The agreement in the basis selection between the transmitter and receiver is assumed as under a tentative reconciliation procedure. Outcomes show just positive effects on the transmission between sender and receiver, but a mild improvement for the success probability for the eavesdropper.


INTRODUCTION
In recent years, a new communication era came up during applied quantum physics development.It is mainly because quantum communication guarantees the confidentiality of a message, and this kind of security is sustained by quantum mechanics laws, rather than classical communication and cryptography.The advantage of quantum cryptography on classical one is because a quantum state cannot be measured without fluctuations on the state of photons being transmitted.It means, if an eavesdropper wants to interfere with the message, it will be irremediably modified, representing an amazing way to protect the information inside a channel.
Quantum communication systems exploit the advantages of quantum mechanics providing absolute randomness and security.It improves the quality and capacity of information transmission in contrast to classical binary systems.The general model of communication through a quantum channel consists of a quantum system with an encoder codifying a message from its original form.It is reached by using a previous shared key, together with a receiver which decodes the message into its original form.In the quantum version, the settlement of such key is referred as Quantum Key Distribution (QKD) [1].Thus, information sent over quantum channels is carried by encoded quantum states.This model represents the allowed physical transformation of a quantum system.
In this paper, the most typical QKD protocol, BB84 [1], will be selected to analyse the effect of different types of quantum channels, and its possible control with Quantum Error Correction (QEC) [2] to realize propositive and effective ways to improve the communication system during the key sharing.Also, error correction protocols should be analysed in the context to quantify their effectiveness in the secret quantum sharing.To secure errorless information being transmitted from a sender to a receiver, QEC provides, together with quantum processing, the tools for the success.Those errors can be generated due decoherence, noisy channels, or other factors.As comparison, classic correction errors employ redundancy, which consists in repeating the same pattern of information for bits comparison at the decoding.When the repeated information is stored and those copies are found being different because of an error, the majority vote is taken.In the quantum version, errors in two qubits are less probable than just one on the qubits.Therefore, the most common QEC algorithm, the 3-qubit code [2], assumes just one error on the block of information being sent through the channel.Analogous to the classical code, information is repeated to proceed with the comparison of all states.This comparison is only viable due to the no-cloning theorem [3], which forces the states to be composed and entangled.
Concretely, the topics considered in this work are the effects of parameterized imperfect quantum channels during the QKD, together with the impact of the error correction protocols.For error correction under the 3-qubit code, we will make a more realistic physical outlook generating non-ideal errors as coming from general quantum channels in the form of Pauli channels [4].In addition, basic concepts are reviewed to understand the idea behind QEC, its logic, and application form.The structure of this work is as follows.Second section depicts the action of Pauli channels on a cryptographic quantum key in the form of a string with well-characterized states in the bases stated by the BB84 protocol.Such effect is quantitatively analysed in terms of the fidelity of the process and the probability distribution for the deformed states arisen from the channel.Third section deals with the comparative analysis for the corrective effect of the 3-qubit code in QEC for the previous process.Fourth section finally comparatively analyses the impact on the eavesdropper success without and with the use of 3-qubit code as common strategy used by all parties receiving information.Last section states the conclusions.

EFFECT OF A GENERIC PAULI CHANNEL ON A STRING OF QUBITS
Pauli's quantum channels [5] whose noise can be easily modelled.This model maps out the effect of general quantum noise in a real channel.For a single qubit system, Pauli's channels can be expressed in terms of the Kraus operators K i = √ α i σ i as: Furthermore, to analyse the channel's effects on a multiple qubit string message or key, a generalization of the previous expression is needed.For an n-qubit channel, taking into account that σ i is a Hermitian operator (σ † i = σ i ), the following nesting expression is used (there, σ k s are the Pauli operators and the identity, whose type is identified by the subscript s; the superscript k indicates the party on which it is being applied): Considering the input state, ρ in , for QKD purposes is a pure state, it takes the form of a chain of qubits: Fidelity [2] is the probability to get the same state at the output as at the input.It is get using the density matrix as F = ψ in | ρ out |ψ in .Such expression, from (2), has two identical sides in each term of the sum, so the equation could be compressed using the square of the module of the entries of Pauli matrices.But the square of any non-zero diagonal entry is always equal to 1, so the fidelity expression can be simplified.Working on the binomial development, we get a condensed equation involving just the channel's coefficients α i , for i = 1, 2. It means: Because the effect of a general channel as (1) is to generate mixed states because the noise added (thus involving other arbitrary modified strings |ψ l = |l 1 l 2 ...l n ), it is also necessary to know the related probability of getting them: there, we assume l is the base-10 version of the string: l = 2 0 l 1 + 2 1 l 2 + ... + 2 n−1 l n .In the last expression, there are only two outcomes different from zero for the squared absolute value: l k = j k for σ 0 , σ 3 , and l k = j k for σ 1 , σ 2 .In both cases such value is one.Thus, if i = j then α 1 , α 2 will be cancelled; otherwise, if i = j, then α 0 , α 3 will be cancelled.Then, regarding the Kronecker delta: δ i, j = 1 if i = j, and otherwise 0, it is easy to show that: being D the number of qubits becoming different after transiting the channel.Again, the relevant channel parameter dependence reduces to Because the BB84 protocol alternatively employs the basis {|+ , |− }, qubit strings sent over the channel can alternate between the {|0 , |1 } and {|+ , |− } basis.Then, we should consider alternatively ρin using only strings of qubits using jk = +, −.After, we will analyse strings combining both basis.Thus, in order to understand the general behaviour of the channel, we analyse the cases with | jk ∈ {|+ , |− }.Just as (3), we consider an input state in the form of an n-qubit string in such basis: The σy Then, for this case, it is easy to realize from (4) that the channel's Fidelity in this case is expressed as: Similarly, the same treatment for the corresponding probability Pl shows that: Again, the relevant channel dependence is comprised in Σ = α 2 + α 3 .It is due the application of Hadamard gates to the Pauli operators, giving the non-zero diagonal matrix entries only for σ 0 and σ 1 .Therefore, α 0 and α 1 could be the only coefficients remaining, which by substituting the initial condition ∑ 3 i=o α i = 1 gives the last expression.Now, we will define the global fidelity and probability for a mixed base qubit chain modelling the BB84 protocol: Which will become simply the product of both fidelities and probabilities respectively: where n i is the total number of qubits of each basis and D i is the number of qubits becoming different after transiting the channel in the different bases; 1 for {|0 , |1 } and 2 for the {|+ , |− } basis.
To graphically realize the last outcomes, Fig. 1a plots the fidelity F 1/2 in ( 4) and ( 8) as function of Σ = α 1 + α 2 or Σ = α 2 + α 3 .Both sigma's are the sum of a couple of Pauli's channels parameters as it has been proved.Their domain is [0, 1] and each curve shows for certain number of qubits n being transmitted via the channel.The plot shows how the fidelity drops while Σ, Σ give a noisier channel, and even drops faster when the number of qubits transmitted increases.While, Fig. 1b, shows the probability to find at the output a certain modified chain of qubits as function of the different qubits ratio with respect of those n in the input: x = D n , D n .When Σ, Σ is near its minimum value (just as the transparent channel) the probability of measuring different qubits at the end of the channel is practically zero; hence, the probability of measuring the same input state at the output is almost 1.However, when Σ, Σ is nearly to 1 (the noisiest channels), the probability of measuring different output states becomes maximum.
Because we are interested on the effect of the Pauli's channel family in a QKD process, a parametric space representation first used in [5] will be considered.Thus, as a reference, Fig. 2a and c) α 1 = α 2 = α 3 = 1/3, the Central ICO channel, an important channel appearing in Teleportation [6] combined with Indefinite Causal Order [7].
Then, in such space, a unitary fidelity could be represented by combining each one of the basis (the most common scenario will have an even number of qubits generated on each basis): 2b coloured in agreement with the colour bar below, where red/blue means the lower/higher fidelity.Due to the symmetry exhibited by the last expression by changing α 1 ↔ α 3 , we only show the upper half of the plot to show the distribution of F inside.Note that the regions with the lowest fidelities correspond to the channels with the largest values of α 2 .It is easily explained because the channel generating transformations with the form σ 2 ρσ 2 in one of their mixtures, has an effect on both bases.

CHANNEL ENHANCEMENT THROUGH QUANTUM ERROR CORRECTION
In the current section, we inquiry how the 3-qubit code [2] in QEC can improve the fidelity, the efficiency of the algorithm, and how it guarantees the correct transmission of information through Pauli channels.This code adds two extra qubits for each qubit in the chain to generate redundancy.It lets to realize if there is an error and exactly where Using this procedure, it is possible to correct the error in the key through two methodologies: the first consists on adding other two ancilla qubits to the circuit, which will check the parity between the three initial qubits throughout C a NOT b gates between pairs of the main group of qubits and the ancilla ones [8].The other method [2] is based upon the use of a projection operator, which does not change the state, but it only leaves the information of the qubit having the error.The procedure properly works for bit-flip and dephasing errors (modified by applying Hadamard gates at the beginning and at the end of the correction procedure).Note that while Bit-flipping noise affects more notably to the states |0 , |1 , Dephasing noise does on |+ , |− .It was already noticed in the analysis of Pauli channels in the previous section.The fidelity of both procedures is well-known, which can be written in terms of p (the probability that one qubit becomes changed) as: there, p corresponds with Σ, Σ parameters before analysed.In Fig. 3a, we see the effect of QEC on the fidelity of a string of qubits transiting across the channel as compared with Figure 1.This figure represents the fidelity for a string of qubits with length n.By comparing these results with the system without QEC, we infer the improvement on the fidelity which does not drop to zero immediately.It is well-known that unitary fidelity (for just one qubit) improves for p = Σ, Σ ≤ 1 2 .Instead, for p = Σ, Σ > 1 2 , fidelity speedup its dropping to zero as compared with Figure 1a.
In an alternative presentation, Fig. 3b shows the effect of using the 3-qubit code on the entire unitary fidelity F for the current process for all possible Pauli's channels.It becomes: In this case, the highest fidelity area near the origin (transparent channel) increases, together with the lowest fidelity region near from the syndromes α 1 = 1 (Bit-flipping noise), α 3 = 1 (Dephasing noise), and α 2 = 1 (Dephasing and Bit-flipping noise).Such a behaviour is expected in agreement with those aspects around p = 1 2 commented in the previous paragraph for the improvements in the fidelity using the 3-qubits code.In the next section, we will be interested on an individual attack of Eve, an eavesdropper through the channel.

SUCCESS EAVESDROPPING PROBABILITY IN A PAULI NOISING CHANNEL
In the current section, we will analyse the effect on the fidelity due to the presence of an eavesdropper [9], Eve, in an intermediate point of the channel.Eve will be following a steal and replace strategy.Because the channel effect on the state being transmitted will be interrupted, we need to model a partial action of the channel on the states.The most reasonable modelling for the noise being introduced by a Pauli channel characterized by the parameters α i , i = 0,...,3 is a linear transition from the transparent channel.Thus, by introducing a unitary time (or distance) parameterizing the process, t ∈ [0, 1], the α i (t) parameters could be modelled as: thus, channel effect transitioning from the transparency (t = 0) until its final form in t = 1 as function of t.Note that ∑ 3 i=0 α i (t) = 1 still.Be aware that we are using the same symbols for the final values α i , i = 0, ..., 3; Σ, and Σ and their corresponding time-dependent functions α i , Σ, Σ , just differing in the time dependence remarking, thus α i ≡ α i (1).
Under this approach, the unitary probability for the eavesdropper success, as instance for one qubit | j k in the {|0 , |1 } basis, relative to the event when Alice and Bob select the same basis in the QKD process (it means, the conditional probability to the event of Alice and Bob agreeing their basis) can be obtained as follows.First, qubit | j k enters in the beginning of the channel during a time τ ∈ [0, 1] when Eve intercepts the qubit, and measures the same qubit sent by Alice: , by following the calculations in the previous section.After, Eve replaces the qubit with another identical state ρ m = | j k j k |.It travels through the channel to Bob during the remaining 1 − τ time.Similarly, the probability of such an event is Thus, the unitary success probability should be averaged on the interval τ ∈ [0, 1] (an additional factor 1  2 should be introduced by the probability election of the same measurement basis than Alice): A similar calculation to get P 2 fulfills for the basis {|+ , |− } just substituting Σ(t) and Σ with Σ (t) = α 2 (τ) + α 3 (τ) and Σ respectively.Then, finally, the mean unitary success probability (considering both bases) could be expressed as: In Fig. 4a we can see the behaviour of the conditional success probability P for an eavesdropper between the communication of Alice and Bob, when it is assumed they agree in the basis election.Note that it does not surpass P = 1 2 , the corresponding value in the ideal BB84 protocol occurring as it is expected near from the transparent channel.Also, note the differences in the range as compared with the Fig. 2b for F .P drops to one half of it.Alice and Bob would detect the eavesdropper, otherwise he would fail in more than one half of the times.Then, it is noticed that if channel is characterized with larger values for α 0 , the success of the eavesdropper will drop dramatically (but together with the fidelity F ). Finally, we will be interested on the last analysis, the success of a eavesdropper but employing the 3-qubit code for both, Bob and the eavesdropper.In this case, we will consider each qubit is repeated three times and the eavesdropper knows such a strategy between Alice and Bob.The eavesdropper will try also to apply the same 3-qubit code in τ to get an improved rate of discrimination, upon the correct election of the measurement basis with that of Alice.At the end, Bob also will apply the same code to improve his own discrimination outcome.
The analysis should be performed just changing the success performances for one qubit in (16) with those corresponding with the application of the 3-qubit code in (13).It means for the basis: The same should be performed for the basis {|+ , |− } to get P 3−QC 2 in terms of Σ .Then, the average probability of success for the eavesdropper regarding the agreement in the Alice and Bob bases becomes: Fig. 4b shows such a probability as function of the channel parameters as before.By comparison, we observe that the effect of the 3-qubit code on the success probability for the eavesdropper slightly improves when the channel's is near from the origin of tetrahedron (more transparent channels).For channels near to α 2 ≈ 1, the eavesdropper's probability of success still drops to its minimal value notably almost without change and in fact worse than the case with QEC.It is advisable from Fig. 5, comparing P i and P 3−QC i , observing a little gap near from Σ, Σ ≈ 1.It shows that 3-qubit code does not help to the eavesdropper for channels deeply characterized by α 2 → 1.Thus, for critical values of α i ≈ 1, i = 1, 3, the channels have a similar behaviour than those without the 3-qubit code; however, they increase the success for the eavesdropper when the noise of the channel is mild or at least far from α i = 1, i = 1, 2, 3, but still below from the value for the ideal BB84 protocol.

CONCLUSIONS
We analysed the effect of imperfect Pauli's Channel on a string of qubits along the QKD process.The distortion in the qubits depends on two main factors, the number of qubits and the channel α-parameters.Clearly, by increasing the number of qubits n, the global fidelity becomes reduced, meaning we are obtaining a distorted string of qubits at the channel output, but still proportional to each single qubit.Despite, we are constructed an averaged (geometrical) fidelity to represent the two possible bases used in the BB84 protocol.By tracking that fidelity on the parametric space of Pauli's Channels (including the most emblematic channels) to advise the effect of those parameters on the string, we first noticed that regions with the largest values of α 2 exhibited the lowest fidelities due to their impact on both bases used in the underlying protocol.Moreover, the last errors introduced by the channels noise become reduced by using the 3-qubit correction code when the relevant derived parameters for the process Σ, Σ becomes lower than 1  2 , enhancing the communication capacity for the corresponding channels.Otherwise, fidelity become reduced worsen the noise independently of the basis being used.In the event of the presence of an eavesdropper, noise could shadow his presence.Despite, we are obtained the mean success probability of a potential eavesdropper through any point of the channel.Again, such probability become reduced as function of a higher α 2 characterizing to the channel.The 3-qubit code slightly improves the success probability for the eavesdropper, but obviously not superseding the ideal BB84 corresponding values.As for the fidelity of the process, the effect of the 3-qubit code becomes counterproductive if Σ, Σ 0.77, as Fig. 5 shows.In general, we are observed that for the nearest transparency (more common channels), the 3-qubit code improves the fidelity of transmission, but when an eavesdropper also used it, it helps him to become unnoticed.
possible values | jk of each qubit of the chain can be either | 0 = |+ or | 1 = |− .Then, because they fulfill the property | jk = H | j k (being H the Hadamard operator) and the properties:

FIGURE 1 .
FIGURE 1.(a) Fidelity for an increasing number of qubits chain size as function of Σ, Σ ; and (b) corresponding probability (represented in logarithmic scale) distribution of the relative fraction of qubits changed on the chain as function of Σ, Σ .

FIGURE 2 .
FIGURE 2. (a) Pauli's Channels parametric space with some emblematic channels, and (b) Unitary fidelity for all possible Pauli's channels represented in colour on that space.

FIGURE 3 .
FIGURE 3. Improved fidelities by using proper 3-qubit codes for both basis.(a) Fidelity for an increasing number of qubits in a fix basis, and (b) Unitary fidelity for all possible Pauli's channels represented in colour on the parametric space.

FIGURE 4 .
FIGURE 4. Pauli channel behaviour with the presence of an eavesdropper (Eve) between Alice and Bob.(a) Eavesdropper success probability P in colour under the noisy Pauli channels effect.(b) Eavesdropper success probability P 3−QC applying the 3-qubit code for both, Bob and the eavesdropper.

FIGURE
FIGURE Pauli channel behaviour with the presence of an eavesdropper (Eve) between Alice and Bob.(a) Eavesdropper success probability P in colour under the noisy Pauli channels effect.(b) Eavesdropper success probability P 3−QC applying the 3-qubit code for both, Bob and the eavesdropper.