2.4. Short Range Transmission via a Single Path Quantum Channel
As previously noted in
Section 1, we will first consider the transmission occurring via a single path quantum channel, where the received radiation field consists of the coherent signal emitted by the aforementioned laser, superimposed with the ever present thermal radiation field [
6,
8,
16]. This situation was recently illustrated in ([
23], Section II), which forms the foundation of the following illustration. Due to the short range nature of the link and the relatively weak thermal radiation field, the interaction between the coherent signal and the surrounding environment remains minimal.
Let
represent the complex amplitudes of the coherent states comprising the thermal radiation field, and let
equal to
denote the average number of thermal noise photons. Then, during the observation time interval
,
, the statistical operator in Glauber’s “
P-representation” is given by ([
8], Equations (4.6) and (4.10), p. 134f.)
Given the Boltzmann constant
, the temperature
T, and the reduced Planck constant
ℏ, the average number
of thermal photons is determined by the Planck distribution law [
8,
16]
Let
denote the associated Laguerre polynomial of degree
n [
29], and let the standard Laguerre polynomial of degree
n, written as
, be defined as
[
29]. Also, let
denote the factorial of
. Assuming
is the average number of signal photons present in the coherent radiation field emitted by the laser when
equals 1, i.e., when the laser is active, and
is the average number of thermal photons present in the thermal radiation field, and computing in the number state basis
, the matrix element
in the
nth row and
mth column of the matrix representation of
as defined in (
3) is given by
in the case where
, that is, within the upper triangular submatrix including the main diagonal of the said matrix representation of
, cf. e.g., [
8,
16] for a similar representation. For the case where
, corresponding to the strict lower triangular submatrix excluding the main diagonal of the matrix representation of
, we obtain
see also, e.g., [
8,
16].
When a non-vanishing coherent signal is present in the received radiation field, that is, when
equals 1, Equations (
5) and (
6) reveal the existence of nonzero interference terms called coherences, indicated by the nonzero off-diagonal elements of
. Clearly, the quantum mechanical density operator
corresponds to a scenario without decoherence and with a known carrier phase of the quantity
appearing in (
5). Since the quantum mechanical density operator
is an observable and, thus, a Hermitian operator, it admits a diagonalization, yielding a complete orthonormal system of eigenkets that constitute a new basis of the Hilbert space, distinct from the number state basis.
If decoherence, an unknown carrier phase, or both dominate the transmission scenario, the phase of the quantity
has to be revisited. The said phase, which is then appropriately modelled as a uniformly distributed random variable, must be averaged out during the data detection process. As a result, all interference terms, that is, all off-diagonal elements in the matrix representation of
with respect to the number state basis, vanish. The resulting statistical operator then becomes diagonal, having the matrix representation [
8]
When the coherent signal is absent, i.e., when
equals 0, Equation (
3) simplifies to [
8,
16]
representing the Bose–Einstein distribution of thermal photons within the thermal radiation field. Clearly,
is diagonal when expressed in the number state basis, which, therefore, forms the set of its eigenkets. Then, no interference terms, i.e., no coherences, are present, indicating that the carrier phase of the thermal radiation is unknown and can be accurately modelled as a uniformly distributed random variable.
According to (
5) as well as (
6) and (
8), the operators
and
possess different eigenstates and, therefore, do not commute. In contrast,
of (
7) and
of (
8) share the same complete orthonormal system of eigenkets, namely, the number states, and are thus compatible. These observations govern the design of the optimum soft output quantum data detector.
Since the considered quantum channel has one single path, no memory effects need to be taken into account when designing the optimum quantum data detector. Therefore, quantum data detection can be performed independently within each observation time interval.
We begin with the
ith observation time interval
, supposing that
equal to 0 and
equal to 1 are equally probable. Using the Kronecker delta
, and defining the cost
equal to
for deciding in favor of hypothesis
when the transmitted codebit
had the value
m, where
, under the maximum a-posteriori probability (MAP) concept ([
16], Equation (3.1499), p. 561), and considering the Hermitian measurement operator
corresponding to hypothesis
, the average cost of the measurement strategy during the
ith observation time interval
is given by ([
8], Equation (1.3), p. 90)
The average cost
determined by (
9) is 1 less the trace of the Hermitian detection operator
measured by applying the measurement operator
, weighted by the a-priori probability
of the codebit
assuming the value 1.
Let
be the eigenkets and
,
, the corresponding real eigenvalues of the said Hermitian detection operator
given by (
10). The optimum measurement operators are then given by [
6,
8]
and
If the measurement at the input of the quantum receiver, represented by the trace
of
, yields the eigenstate
with eigenvalue
, the detected codebit is
equal to 1. Otherwise, the detected codebit is
equal to 0. Therefore, since
is equal to the eigenvalue
with
the optimum soft output in the
ith observation time interval
is hence represented by the LLR
Using the sign function
, the MAP detection rule applied by the optimum soft output quantum data detector is expressed as
Nota bene that in the special case where
is given by (
7) and
by (
8), Equation (
14) simplifies to the well-known expression for
for equiprobable bits, i.e.,
equal to
[
16].
2.5. Multipath Transmission and Reception with Decoherence
In particular, scenarios involving longer transmission links often experience multipath transmission and reception. In such cases, the quantum channel is characterized by multiple paths, through which the transmitted signals arrive at the receiver. This situation has been illustrated very recently for the first time, cf. e.g., [
18,
19,
20].
Consider a quantum channel modelled as a linear multipath channel that causes uncorrelated scattering of the signal photons emitted by the laser. These scatterers are randomly located and statistically independent, forming a continuum of uncorrelated scatterers similar to what occurs in mobile propagation channels [
17,
18]. The channel impulse response (CIR) of this multipath channel, parameterized by the delay
, is denoted as
[
18]. We assume that the CIR
is nonzero only for
within the delay time interval
, where
. Under these conditions, ISI arises at the quantum receiver. Defining the integer number of transmission paths of the multipath quantum channel by
the said multipath quantum channel is referred to as a
-path channel. The signal arriving at the receiver can be expressed by the ket vector [
18]
The previously mentioned uncorrelated scattering induces decoherence [
13,
14,
15,
30], which contributes to the loss of phase coherence between the transmitter and the receiver, see, e.g., [
18,
19,
20]. As a result, we should assume that no phase coherence is preserved between the transmitter and receiver ([
6], p. 1590; [
18]; [
8], p. 171). Consequently, the zero phase angles
appearing in the aforementioned phase factors mentioned in
Section 2.2 are considered unknown and modelled as uniformly distributed random variables with the constant probability density function of
. Moreover, the uncorrelated scattering inherent to the multipath quantum channel further contributes to the degradation of phase coherence, cf. e.g., [
18] and the references therein.
In the absence of phase coherence, the optimum quantum receiver employs photon counting at its front end ([
6], p. 1590). Assume that the transmitter and the receiver are synchronized in time, meaning that the quantum receiver operates over mutually disjoint and consecutive observation time intervals
of duration
, each beginning with the arrival of the corresponding codebit
[
18]. Using the number operator
and denoting mathematical expectation by
, the average number of signal photons
received during the observation time interval
is given by
, after averaging over all uniformly distributed phase differences
, and we obtain
Due to decoherence and averaging over
, the second integral vanishes, while the first integral remains unaffected. Therefore, Equation (
21) simplifies to [
18]
which becomes
supposing that only
channel coefficients
,
, can assume nonzero values.
In the case of a single path channel where
equals 1, only the first term
of Equation (
22) remains. The scenario where
equals 1 has already been discussed, for instance, in ([
6], p. 1589f.; [
8], pp. 171–176). Moreover, if
is a nonnegative random variable with unit expected value, the result is a fading single path turbulent channel, also known as a turbulence channel, which was already investigated from a quantum communications perspective several decades ago ([
31], pp. 1630–1644).
For
, the ISI term
in Equation (
22) does not vanish [
18]. As a result, the average number of received signal photons
of (
22) depends not only on the current codebit
, but also on the
preceding codebits
,
…
. Therefore, Equation (
22) describes a finite state machine (FSM) that operates according to a Markov chain model [
16,
18,
32].
In the context of Mealy modelling, cf. e.g., [
32], the previously described FSM consists of
states
, defined as
at each time index
. Since the term
can take on either of two distinct values, 0 or
, each of the
states
gives rise to two transitions, labelled by the hypothetical codebit value
. This results in a total of
distinct transitions, which lead to the
states
at the next time step
[
18]. The behavior of the said FSM is conveniently represented by a trellis diagram, often abbreviated as trellis [
16,
18,
33]. Accordingly, the
possible transitions correspond to the
different values that the average number of received signal photons
, as defined in (
22), can assume during each observation time interval
, with
. For this reason, we will denote
as
going forward, identifying these quantities as the
distinct hypotheses employed in the quantum data detection scheme [
18]. Note that the said hypotheses
do not necessarily match the actual number of received signal photons
observed in the
ith observation time interval [
18].
Figure 2 shows the detail of a trellis representing a quantum channel with
equal to 3 possibly non-vanishing channel coefficients
,
and
, thus, when applying Mealy modelling, having
equal to 4 states at each time instant. The two states
equal to
and
equal to
at time instant
i are connected to the two states
equal to
and
equal to
at time instant
by a total of
equal to four transitions. The said states and the aforementioned four transitions, all represented in black color, form the first of the two butterflies of the considered trellis. The two states
equal to
and
equal to
at time instant
i are connected to the two states
equal to
and
equal to
at time instant
by a total of
equal to four transitions. These states and the associated four transitions, all represented in green color, form the second of the two butterflies of the considered trellis.
To support trellis initialization in the state equal to and trellis termination in the state equal to , a choice known to improve data detection performance, we introduce initial and terminal tailbits. Specifically, we set equal to 1 for all . All other codebits , for and , are assumed to be 0.
The schematic representation of the transmitted bit vector used when applying the trellis based quantum data detectors, which set out from the Mealy modelling of the FSM representing the multipath quantum channel, is depicted in
Figure 3. As pointed out above, the said transmitted bit vector consists of
N codebits,
initial tailbits and
terminal tailbits, hence, containing
bits.
At the output of the quantum channel, the radiation field resulting from laser emission is linearly superimposed with the omnipresent background thermal noise radiation field, which is in a thermal state characterized by Bose–Einstein distributed thermal noise photons. As already mentioned above,
denotes the average number of thermal noise photons and
represents the complex amplitudes of the associated coherent states present in the thermal field. The said linear superposition is modelled by the map
Given
where
is a zero phase angle, the corresponding statistical operator in Glauber’s “
P-representation” with
defined as ([
8], Equations (4.6) and (4.10), p. 134f.)
is denoted by
during the observation time interval
,
. Comparing (
3) with (
25), respectively (
26), the term
in (
3) has been replaced by
, which explicitly considers the multipath reception by the state
and the codebit
during the observation time interval
, cf. e.g., (
24).
Since the function
of (
25) used in (
26) is always nonnegative, the associated laser light is often referred to as “classical.” In contrast, light, for which
takes on negative values in some regions is sometimes labelled “nonclassical” ([
34], p. 60). However, this distinction may be misleading. For instance, a Fock state is entirely nonclassical by nature ([
34], p. 58), and, fundamentally, all states of light are governed by quantum mechanics ([
34], pp. 60, 150). Thus, the binary classification of light as either “classical” or “nonclassical” may not be entirely appropriate. Considering that Max Planck’s treatment of thermal radiation on 14 December 1900, marked the beginning of quantum mechanics, cf. e.g., ([
35], pp. 217, 277), and that laser light was theoretically introduced by Albert Einstein in 1917 on the basis of quantum principles, it seems inconsistent not to regard both thermal radiation and laser light as “nonclassical”.
With the regular Laguerre polynomial
of degree
, we find
neglecting any phase coherence, see also ([
18], p. 2; [
8], p. 136f.).
Equation (
27) motivates the application of a photon counting detector at the quantum receiver front-end. Nota bene that the received electromagnetic radiation field cannot be in all number states
at the same time, the photon counter can essentially be considered as a parallel bank of projection operators
defined as
, each one adapted to a particular number state
and, hence, firing if that particular number of photons
was measured. In other words, this photon counting detector measures the number
of photons present at the receiver during the observation time interval
[
18]. Using the trace operation, denoted by
, and the projection operator
defined as
,
is the conditional probability of detecting
photons at the photon counting detector, given the transmission of the codeword
. Equation (
28) describes this probability as a Laguerre distribution, see also ([
5], Equation (4.71), p. 174). Notably,
coincides with the transition probability
associated with the FSM introduced earlier [
18]. It follows that Equation (
28) assumes the use of an ideal photon counting detector, such as the one discussed in ([
36], p. 231f.), as the front-end component of the optimum quantum data detection scheme.
If the photon counting detector cannot precisely count the exact number of photons but is only able to differentiate among zero photons, one photon, or more than one photon, we may use
,
and
equal to
instead of
for arbitrary
[
18]. Accounting for real-world effects such as dark counts with the mathematical expectation
and quantum efficiency
, in Equation (
28)
should be replaced by
and
has to be replaced by
[
18].
Based on the above discussion, the actual number of received signal photons
in the
ith observation time interval
results from a linear superposition of
Poisson distributed random variables
, each with mean
. Consequently,
itself follows a Poisson distribution [
18,
37]. It is important to note that Poisson distributed random variables are, by definition, mutually independent [
37]. Moreover, the thermal noise photons, governed by the Bose–Einstein distribution, follow a memoryless geometric distribution [
18,
37]. As a result, measurements taken across any two disjoint observation time intervals are statistically independent. This insight aligns with Forney’s classical considerations [
33].
Of course, the said independence does not exempt from the memory of the multipath quantum channel. However, this is taken into full account by modelling the multipath quantum channel using a Markov chain yielding the above illustrated FSM. The basic concept has been known for many decades in classical communications. Exploiting Bohr’s correspondence principle, the FSM model is transferred to the quantum domain.
The likelihood function
, which denotes the conditional probability of observing the photon count vector
given the transmission of the codeword
including the terminal tailbits, is defined as follows. Let
represent the tensor product of the corresponding
statistical operators
, and
the tensor product of the associated projection measurement operators
. Then, the likelihood function is given by
This expression reflects the joint probability of measuring the specific photon count sequence
conditioned on the transmission of the bit sequence
, via the tensor product structure of the quantum states and measurements across all observation intervals. We obtain
This result is fully analog with classical wireless communications over multipath channels, cf. e.g., ([
16], Section 3.9, pp. 468–666). This statement is also valid for the derivations below.
When the codebits
are treated as outcomes of statistically independent random experiments, the a-priori probability
of a specific codeword
including the fixed-value terminal tailbits set to 1 is given by the regular product of the individual a-priori probabilities
Applying the definition of conditional probability as found in ([
16], Definition 3.25, p. 317), the corresponding a-posteriori probability is obtained by
where
denotes the total probability of observing the photon count vector
, accounting for all possible codewords. Using the definition of the conditional probability ([
16], Definition 3.25, p. 317; [
38], Equation (
5), p. 5), we yield
from (
31) after multiplying by the joint a-priori probability
of (
32).
Equation (
34) is a result of the definition of joint conditional probabilities, sometimes mistakenly called Bayes’s rule. A detailed discussion of this approach can e.g., be found in ([
16], pp. 619–638).
The likelihood function
and its logarithmic counterpart
as well as the a-posteriori probability
, or its logarithmic form
constitute the foundation of the quantum data detection process that directly follows photon counting and serves to mitigate ISI [
18]. The term
is referred to as the metric increment. In summary, optimum quantum data detection leverages either the likelihood function
, as defined in (
31), or the a-posteriori probability
, see (
34). Since classical channel decoding follows the quantum detection step, cf.
Figure 1, producing LLRs at the quantum data detector output, assembled into the LLR vector
, is advantageous.
Referring to optimum quantum data detection as first presented in e.g., [
18,
19,
20], like in
Section 2.4, we set out from the
ith observation time interval
. Again, with the Kronecker symbol
, with the cost
equal to
of the decision in favor of hypothesis
when the codebit
with value
m was transmitted,
, and with the Hermitian measurement operator
reflecting the hypothesis
, in the case of the MAP concept [
16] the average cost of the observational strategy in the said
ith observation time interval
is
which is a modified version of (
9), replacing
used in (
9) by
, hence taking the FSM related to the ISI into account explicitly.
Each
is diagonal in the Fock basis, with the Fock state vectors
acting as eigenkets, and the associated eigenvalues given by
, which are equal to
. As an example, consider the Hermitian detection operator in Equation (
37), corresponding to
and given by
with the eigenvalues
Accordingly, the optimum measurement operators that minimize
of (
37) and are naturally realized by the ideal photon counter discussed in
Section 2 are
given by
and
given by
[
8].
At the conclusion of the measurement within the
ith observation time interval
, the exact Fock state
of the received radiation field, as present in the ideal photon counter, is known. Consequently, only the projector
within
or
contributes a nonzero eigenvalue
. This nonzero eigenvalue
is then evaluated using the conventional classical likelihood ratio test. Nonetheless, due to the presence of ISI, the decision made in the
ith observation time interval
not only depends on earlier transmitted codebits but also influences the decisions in following intervals. As a result, and as reflected in Equation (
34), the classical likelihood ratio test must be modified, accordingly, taking the following form
This result is an extended variant of Helstrom’s theory [
8], as expressed in Equation (
34). Hence, the optimum MAP symbol by symbol quantum data detection strategy is expressed as
being a classical likelihood ratio test because we left the quantum regime at the end of the quantum measurement process. However, due to the occurrence of ISI, modelled by the FSM discussed above and, thus, by a Markov chain, (
41) requires an appropriate detection scheme, e.g., the trellis based BCJR algorithm [
16,
18,
39]. It seems that this approach has been first introduced by [
18].
Correspondingly, also taking ISI and Markov chain modelling into account explicitly, with (
34), the optimum MAP sequence quantum data detection strategy is presented by
which can be implemented by the trellis based Viterbi algorithm [
16,
18].
Maximizing the likelihood function
over all possibly transmitted codewords
will yield the maximum-likelihood (ML) sequence detector, which can be realized by the soft output Viterbi algorithm (SOVA) [
16,
18]. Maximizing the a-posteriori probability
over all possibly transmitted codewords
will yield the MAP sequence detector. Since a-priori information is processed in the latter case, the corresponding variant of the SOVA is termed APRI-SOVA [
40]. Both the SOVA and the APRI-SOVA are considered optimum because they maximize the probability of correct sequence detection, which is equivalent to minimizing the sequence error probability, provided that the metric increments discussed above are used. Details of the mode of operation of the SOVA and the APRI-SOVA have been well known [
16,
18,
33,
40].
ML symbol by symbol detection focuses on maximizing the conditional probabilities of the observed data given the symbol,
, while MAP symbol by symbol detection aims to maximize the posterior probabilities of the symbol given the observed data,
[
16]. To reduce computational complexity, both detection methods are typically implemented in the log domain, similar to optimum sequence detection. Accordingly, ML symbol by symbol detection employs metric increments
, whereas MAP symbol by symbol detection uses metric increments of the form
[
16,
18]. Both approaches can be efficiently realized using the BCJR algorithm [
16,
18,
39], which is considered optimum as it maximizes the probability of correct symbol detection, equivalently, minimizing symbol error probability, when the metric increments
are used. The operational details of the BCJR algorithm are well-established, see e.g., [
16,
39]. It is important to note that optimum symbol by symbol detection schemes implemented in the log domain apply the Jacobian logarithm ([
16], p. 630f.), which leads to the so-called “log-ML” and “log-MAP” algorithms [
16]. Approximations of the Jacobian logarithm yield variants known as “max∗-log-ML” and “max∗-log-MAP”, as well as “max-log-ML” and “max-log-MAP”, respectively [
16].
Optimum sequence quantum data detectors, as well as optimum symbol by symbol quantum data detectors, operate as “forward/backward” algorithms that conceptually consist of four key processing units. These are
The transition metric unit, which computes the metric increments for ML detection, or for MAP detection.
The forward recursion unit, responsible for calculating the forward path metrics.
The backward recursion unit, which either computes the backward path metrics or performs traceback to identify the most likely path through the trellis.
The LLR generation unit, which produces the LLR vector mentioned earlier.
Importantly, only the computations within the transition metric unit differ from those in, e.g., classical mobile radio, such as mobile radio detection. The other three units can be reused without modification.
For the SOVA and the APRI-SOVA algorithms, only approximate LLRs can be generated, using three distinct LLR computation methods known as the “SIMPLE RULE (SR)”, the “HUBER RULE (HR)”, and the “BATTAIL RULE (BR)” ([
16], pp. 619–638). The generation of LLRs according to these three rules has been known for over three decades. A detailed derivation and discussion of SR, HR, and BR as well as of the BCJR algorithm can be, e.g., found in ([
16], pp. 619–638). The reader is referred to these references.
In contrast, all optimum symbol by symbol quantum data detectors produce LLR vectors containing exact LLRs when implemented as log-ML or log-MAP algorithms, and approximate, though very close to exact, LLRs in the cases of max∗-log-ML, the max∗-log-MAP, the max-log-ML and the max-log-MAP implementations.
Trellis-based reduced state quantum data detection sets out from the now well-known trellis-based optimum sequence quantum data detection, respectively the trellis based optimum symbol by symbol quantum data detection discussed above. Like in the cases of these optimum data detection schemes, the reduced state quantum data detection continues until all LLRs to be determined and, consequently, all the corresponding data symbols have been detected. Trellis based reduced state quantum data detection requires small modifications and extensions of the transition metric unit and the forward recursion unit, which we will discuss in what follows. Since the said modifications and extensions only concern the transition metric unit and the forward recursion unit, which are identical in the case of the max-log-MAP symbol by symbol detection and the SOVA, the novelties presented are identically applicable to
the trellis-based reduced state sequence quantum data detection, which is formed by setting out from the SOVA, termed rsSOVA with the “prefix rs” indicating the reduced state version, respectively from the APRI-SOVA, termed rsAPRI-SOVA, and
the trellis-based reduced state symbol by symbol quantum data detection, which is formed by setting out from the max-log-MAP symbol by symbol detection, termed rsmax-log-MAP, the max-log-ML symbol by symbol detection, termed rsmax-log-ML, the max∗-log-MAP symbol by symbol detection, termed rsmax∗-log-MAP, the max∗-log-ML symbol by symbol detection, termed rsmax∗-log-ML, the log-MAP symbol by symbol detection, termed rslog-MAP and the log-ML symbol by symbol detection, termed rslog-ML, the latter two differing from the max∗-log-ML symbol by symbol detection and the max-log-MAP symbol by symbol detection only in the used version of the Jacobian logarithm but not in the processing of the trellis states and the trellis transitions.
Suppose that we require reduced state quantum data detection with only
trellis states,
,
,
. Using
explicitly, (
22) can be written as follows
Reduced state data detection requires that the
binary data symbols
,
⋯
,
contained in the third term on the right side of (
43) are known to the quantum data detector prior to the
trellis transitions
. This requirement calls for preliminary decisions
,
⋯
,
needed for the transition metric computation and hence in the forward recursion. The said preliminary decisions can be accomplished by recursive processing and adds iteratively one single decision per forward recursion step. Therefore, reduced state quantum data detection requires the modification of the transition metrics computation by using
of (
43), which relies on the preliminary decided binary data symbols
,
…
,
and the extension of the forward recursion by adding preliminary decision taking.
In the reduced state trellis, there are
trellis transitions
, which originate at those
states
leading to the
possible forward recursion path metrics
denoting the path metric of the state
at the time instant
i. Also, there are
trellis transitions
, which originate at those
states
leading to the
possible forward recursion path metrics
denoting the path metric of the state
at the time instant
i.
Figure 4 depicts a butterfly of the trellis with
states per time instant, considered in trellis based reduced state quantum data detection, which comprises the two states
of (
44) and
of (
45) at the time instant
i and the two states
equal to
and
equal to
at the time instant
. The butterfly consists of two transitions originating at the state
at time instant
i and two transitions originating at the state
at time instant
i. Two of these four transitions terminate at the state
equal to
at time instant
and the remaining two of these four transitions terminate at the state
equal to
at time instant
.
When summing up all the above-mentioned possible forward recursion path metrics by iteratively applying the Jacobian logarithm, we yield the forward recursion estimate of the logarithmic probability . Next, when summing up all the possible forward recursion path metrics by iteratively applying the Jacobian logarithm, we yield the forward recursion estimate of the logarithmic probability . Subtracting from yields the forward recursion LLR of , which is to be used for the preliminary decision . Since we now have the viable decision , we may continue with the next iteration step in the forward recursion.
The implementation complexity of reduced state algorithms is highly dependent on the chosen architecture, especially the chosen hardware/software co-design strategy. Such aspects are beyond the scope of this manuscript and will be left to a future study.
While state reduction is an advantageous strategy for managing detection complexity, Kohonen maps can be employed for resource-efficient implementations, and near optimum performance in classical mobile communication systems [
21]. The Kohonen maps-based sequence detector, also referred to as “Kohonen maps based detector” or “Kohonen detector”, is also applicable to quantum channels. It consists of four processing units, namely the metric calculation unit, the metric aggregation unit, the trace back unit and the LLR calculation unit, cf.
Figure 5 [
21]. The combination of these four units follows the signal processing principle of an APRI-SOVA by applying a neural network with a one-dimensional linear structure in both the input plane and the output plane using distinct features of the Kohonen map by appropriately adjusting the weights and the distance function [
21]. More specifically, the neural network adopted by the Kohonen maps-based detector is a hybrid of learning vector quantization (LVQ) and Kohonen maps, both belonging to the family of soft competitive learning clustering algorithms [
41]. This approach enables leveraging neural networks to perform near optimum sequence detection by efficiently utilizing existing hardware resources [
21]. Training the Kohonen map by incorporating LLR feedback has been shown to improve the overall performance of the Kohonen maps-based detector [
21].
In order for the Kohonen maps-based detector to be applied in quantum communication channels, the aforementioned FSM has to be modelled by means of a Moore machine. In the case of Moore modelling, the FSM contains unique states equal to at each time instant . For this reason, we will use initial and terminal tailbits, setting equal to 1 for all .
The schematic representation of the transmitted bit vector used when applying the Kohonen map based quantum data detector, which sets out from the Moore modelling of the FSM representing the multipath quantum channel, is shown in
Figure 6. As pointed out above, the said transmitted bit vector consists of
N codebits,
initial tailbits and
terminal tailbits, hence, containing
bits.
Each possible state at a given time instant
i is associated with a state number, i.e., a state index
. Interpreting the FSM as a Moore machine involves a doubling of the number of states and a reinterpretation of the transition metrics, as they no longer depend on transitions, i.e., both the initial states and the target states, but solely on the target states. We further denote the set of possible states in the Mealy case by
and the corresponding set in the Moore case by
. The remodelling as a Moore machine necessitates the introduction of a mapping function in order to assign the previously determined transition metrics
, for the Mealy case to transition metrics for the Moore case
, respectively,
,
. We define a surjective mapping
to assign the transition of a Mealy state pair to a Moore state. For each
the mapping is defined as
where
denotes the floor function (in German “Gaußklammer”). Furthermore,
denotes the state index of the initial state and
the state index of the target state in the case of Mealy modelling.
Once the transition metrics according to Moore modelling are obtained and summarized in a
matrix
, the Kohonen detector operates similarly to the case of classical wireless communication systems. At this stage, the Kohonen map serves as a state mapper allowing for an adequate add-compare-select (ACS) operation for the permissible state transitions [
21]. In what follows,
denotes a possible predecessor state with its state index
at the time instant
i of the successor state
with state index
at the time instant
. The output neuron for the state
with state number
at the time instant
is configured with the weight vector
using
where
denotes the path metric of the state with state number
at the time instant
i. Any state
can be reached by two distinct predecessors,
and
say. In order to determine the path metrics at the time instant
for each possible successor state
, solely the weights of the Kohonen map must be examined. We obtain
which can be arranged in the
path metrics row vectors
We will illustrate the above discussions of the soft output Kohonen maps-based quantum data detection in what follows. Let the states with the state indices
and
refer to the predecessor states at time instant
of the state
with the state index
at time instant
i. Furthermore, let the state indices
and
refer to the predecessor states at time instant
of the state
with the state index
at time instant
i. As shown in
Figure 7, the state
with the state index
,
,
…
and
denoting the particular bit values associated with this state
, and the state
with the state index
,
,
…
and
denoting the particular bit values associated with the state
, are the predecessor states at time instant
i of the state
with the state index
at time instant
. The aforementioned states
of (
52),
of (
53) and
of (
54) are part of the trellis representing the multipath quantum channel with
possibly non-vanishing channel coefficients
,
, setting out from the Moore modelling applied in the soft output Kohonen maps based quantum data detector. In
Figure 7, the average numbers
and
of signal photons at the photon counter are associated with the appropriate states. Also, the recursive computation of the path metrics
by the soft output Kohonen map based quantum data detector is also illustrated.
The calculated path metrics are arranged in the
matrix
for further processing. The optimum path is found in the trace back unit by again utilizing the Kohonen map as a state mapper, yet in a reverse-oriented manner. The weights of the neurons representing the possible predecessor states
are configured with
Denoting the state index of the survivor state at time instant
by
, its predecessor state is found according to
The remaining predecessor states can be determined recursively. All predecessors along the survivor path can be arranged in the vector
of length
. Similarly to the case of SOVA, respectively the APRI-SOVA, approximate LLRs can be generated using the Kohonen detector [
21].
Owing to its simplicity and promising performance, the authors choose to begin with the aforementioned “SIMPLE RULE” (SR). According to the SR, approximate LLRs are built based on the path metric differences of the survivor and concurrent predecessor states [
16]. In the case of Moore modelling of the FSM, the path metric difference at time instant
is given by
where
denotes the path metric of the concurrent predecessor at time instant
i. The structure of the Kohonen map employed in the generation of LLRs mirrors that of the trace back unit, with the sole modification being the inversion of the sign of the concurrent predecessor state’s path metric and the insertion of zeros instead of
for impermissible state transitions, i.e.,
The path metric difference
represents the distinct metric deviation that allows to distinguish between whether the FSM was in the predecessor state
or in the concurrent predecessor state
. Due to the structure of FSM modelling and the influence of ISI, these two states differ by only a single symbol, specifically, the one at time instant
. Therefore, the path metric difference is interpreted as a reliability measure for that particular symbol at time instant
. The LLRs are found recursively using
The updates according to the “HUBER RULE” (HR) and the “BATTAIL RULE” (BR) can be performed in accordance with [
16].
In order to improve the adaptability of the Kohonen map-based detector, a novel updating procedure was also introduced, where, instead of investigating spatial distribution of FSM states, a probability based neighborhood function is calculated [
21]. These probabilites are calculated based on the LLRs, which can be filtered and fed back in order to update the Kohonen map. Since Kohonen maps follow a “winner takes most” approach, not only the weights of the winning output neuron are updated, but also those within a defined “winning neighborhood”
. The probability of each state
at each time instant
is governed by the a-priori probabilities of the
most recently received codebits, which are supposed to be mutually independent. It is thus obtained by
Given the state probability (
67) and the probability threshold
, which must be exceeded if the weights of a neuron shall be updated, the neighborhood of the winning output neuron is given by
This purely local perspective can be improved by using a “forward/backward” approach to obtain the optimum state probabilities [
21].
By combining two megatrends, namely artificial intelligence and quantum communications, soft output quantum data detectors using Kohonen maps may provide low-complexity, high-performance detection methods that enhance the scalability and efficiency of next-generation communication networks.