Quantum Communications Systems with Thermal Noise

Abstract

The chapter deals with the analysis of Quantum Communications systems in the presence of thermal noise, sometimes called background noise, neglected in the previous chapter. This is achieved by passing from pure states to mixed states, described by density operators. Consequently, the analysis and especially the optimization become very difficult. For binary Quantum Communications systems the optimization is based again on Helstrom’s theory, whereas for multilevel systems the numerical optimization is used, and especially the square root measurement (SRM), which is suboptimal but gives a good approximation of the performance. The SRM technique is applied to the most popular Quantum Communications systems (QAM, PSK, and PPM) seen in the previous chapter in the absence of thermal noise. We will find that, also in the presence of thermal noise, Quantum Communications systems perform better than the classical counterparts.

Appendix

1.1 Alternative Discretization

An alternative discretization of a Glauber density operator is based on the subdivision of the integration region in (8.8) into a finite number $K$ of regions of the complex plane, namely,

$$\begin{aligned} A_{k} \quad \text{ with } \quad \bigcup _{k=1}^KA_{k}=\mathbb {C}\;. \end{aligned}$$

(8.75)

In such a way, one gets the discrete approximation of the density operator

$$\begin{aligned} \rho (\gamma )\simeq \sum _{k=1}^K h_{k}\; |\alpha _{k}\rangle \langle \alpha _{k}| \end{aligned}$$

(8.76)

where

$$ h_{k}= \frac{1}{\pi \mathcal {N}} \int _{ A_{k}}\exp \left( -\frac{|\alpha -\gamma |^2}{\mathcal {N}}\right) \, \mathrm{d}\alpha $$

and $|\alpha _{k}\rangle$ are the Glauber coherent states with complex parameter $\alpha _{k}\in A_{k}$. This integral gives the probability of the state $|\alpha _k\rangle$

$$\begin{aligned} \mathrm {P}[s=|\alpha _k\rangle ]=h_k\;. \end{aligned}$$

(8.77)

The normalization of the probability is verified because the volume determined by the Gaussian profile is unitary

$$\begin{aligned} \frac{1}{\pi \mathcal {N}} \int _{\mathbb {C}} \exp \Bigl (-\frac{|\alpha -\gamma |^2}{\mathcal {N}} \Bigr )\; \, \mathrm{d}\alpha =1\,. \end{aligned}$$

(8.78)

To arrive at a finite form, the Glauber states in (8.76) are approximated by a finite number $N$ of terms, that is,

$$ |\alpha _k\rangle = \mathrm{e}^{-\frac{1}{2}|\alpha _k|^2} \sum _{n=0}^{N-1}{\frac{\alpha _k^n}{\sqrt{n!}}} \;|n\rangle \;. $$

Partition strategy. The partition of the complex plane $\mathbb {C}$ into the regions $A_k$ can be done in several ways. The strategy that will be considered here is:

• (1) partition $\mathbb {C}$ into a finite number $L$ circular rings centered on the nominal state $|\gamma \rangle$

  $$ D_i=\left\{ r_{i-1}^2\le |\alpha -\gamma |^2<r_i^2\right\} , \qquad i=1,2,\ldots ,L $$

  with $r_0 = 0$ and $r_{L}=\infty$.

• (2) subdivide the $i$th ring into $k_i$ equal parts

  $$ A_{ip}=\left\{ \frac{2\pi }{k_i}\,(p-1)\le \arg (\alpha -\gamma )<\frac{2\pi }{k_i}\,p \right\} , \qquad p=1,2,\ldots ,k_i $$

  where for convenience we use the double subscript $ip$ instead of $k$,

• (3) choose the state $|\alpha _{ip}\rangle$, with $\alpha _{ip}$ given by the barycenter of the region $A_{ip}$,

• (4) assume the condition that the states are equally likely.

With such a procedure the number of states is $K=k_1+\cdots k_L$ and the volume determined by the ring $D_i$ results in

$$\begin{aligned} v_i&= \frac{1}{\pi \mathcal {N}} \int _{D_i} \exp \Bigl (-\frac{|\alpha - \gamma |^2}{\mathcal {N}}\Bigr )\; \, \mathrm{d}\alpha \nonumber \\&=\frac{1}{\pi \mathcal {N}} \int _0^{2\pi }\mathrm{d}\phi \int _{r_{i-1}}^{r_i}\mathrm{d}r\,r\, \mathrm{e}^{-r^2/\mathcal {N}} \\&=\frac{1}{2}\left( \mathrm{e}^{-r_{i-1}^2/\mathcal {N}}-\mathrm{e}^{-r_{i}^2/\mathcal {N}}\right) \nonumber \end{aligned}$$

(8.79)

which is subdivided into $k_i$ equal parts, so that the equal probability condition is

$$ \frac{v_i}{k_i}=\frac{1}{K} \quad \text{ with }\quad K=k_1+k_2+\cdots +k_L\,. $$

By imposing such a condition, the radii of the circular rings can be evaluated by the recurrence

$$ r_i=\sqrt{-\mathcal {N}\,\log \left( \mathrm{e}^{-r_{i-1}^2/\mathcal {N}}-k_i/\mathcal {N}\right) } , \qquad i=1,\ldots ,L $$

letting $r_0 = 0$ and $r_{L}=\infty$.

Finally, the barycentric radius of the ring $D_i$ results in

$$ r^b_i=\frac{\sqrt{\mathcal {N}} \left( \mathrm{e}^{-r_{i-1}^2/\mathcal {N}}-\mathrm{e}^{-r_i^2\mathcal {N}}\right) }{\sqrt{\pi } \left( \text{ erf }\left( r_i/\sqrt{\mathcal {N}}\right) -\text{ erf } \left( r_{i-1}/\sqrt{\mathcal {N}}\right) \right) } $$

where erf is the error function. In such a way, the coherent state $|\alpha _{ip}\rangle$ is identified by the complex number

$$ \alpha _{ip} =\gamma + r^b_i\, \mathrm{e}^{\phi _{ip}} , \qquad \phi _{ip}=\frac{2\pi }{k_i}p\;. $$

The number of states in this approximation of the density operator is given by the total number of subdivisions $K=k_1+\cdots +k_L$.

Fig. 8.26

Examples of subdivision in circular rings for the discretization of a Glauber density operator

Example 8.5

We consider two examples of subdivision shown in Fig. 8.26. In both cases the subdivision is into $L=4$ rings and the inner ring is not subdivided and is associated to the nominal state $|\gamma \rangle$. In the first example, the 3 outer rings are subdivided into 5 parts, and we get $K=16$ global subdivisions, while in the second example they are respectively subdivided into 5 , 7 and 9 parts with $K=22$ global subdivisions.

$$\begin{aligned}&\mathrm{number\;of\; rings}{:}\; { L} = 4 \qquad \quad \mathrm {number\; of\; rings}{:}\; { L} = 4 \\&\mathrm{subdivisions\; of\; rings}{:}\; 1 , 5 , 5 , 5 \qquad \quad \mathrm{subdivisions\; of\; rings}{:}\; 1 , 5 , 7 , 9\\&\mathrm{global\; subdivisions}{:}\; { K} = 16 \qquad \quad \mathrm{global\; subdivisions}{:}\; { K} = 22. \end{aligned}$$

We complete the evaluation in the first subdivision, assuming

$$ \mathcal {N}=0.2\quad N_s=2.0\;. $$

With $N=11$ we find the matrix

$$ {\tilde{R}=\left[ \begin{array}{lllllllllll} 0.15696 &{} 0.18636 &{} 0.15597 &{} 0.10627 &{} 0.06254 &{} 0.03284 &{} 0.01572 &{} 0.00696 &{} 0.00288 &{} 0.00113 &{} 0.00042 \\ 0.18636 &{} 0.24576 &{} 0.22704 &{} 0.16979 &{} 0.10909 &{} 0.06222 &{} 0.03217 &{} 0.01530 &{} 0.00676 &{} 0.00280 &{} 0.00109 \\ 0.15597 &{} 0.22704 &{} 0.22990 &{} 0.18741 &{} 0.13065 &{} 0.08054 &{} 0.04485 &{} 0.02290 &{} 0.01084 &{} 0.00479 &{} 0.00199 \\ 0.10627 &{} 0.16979 &{} 0.18741 &{} 0.16562 &{} 0.12463 &{} 0.08265 &{} 0.04937 &{} 0.02697 &{} 0.01363 &{} 0.00643 &{} 0.00285 \\ 0.06254 &{} 0.10909 &{} 0.13065 &{} 0.12463 &{} 0.10082 &{} 0.07162 &{} 0.04570 &{} 0.02661 &{} 0.01431 &{} 0.00716 &{} 0.00337 \\ 0.03284 &{} 0.06222 &{} 0.08054 &{} 0.08265 &{} 0.07162 &{} 0.05431 &{} 0.03688 &{} 0.02280 &{} 0.01298 &{} 0.00688 &{} 0.00341 \\ 0.01572 &{} 0.03217 &{} 0.04485 &{} 0.04937 &{} 0.04570 &{} 0.03688 &{} 0.02657 &{} 0.01737 &{} 0.01044 &{} 0.00583 &{} 0.00304 \\ 0.00696 &{} 0.01530 &{} 0.02290 &{} 0.02697 &{} 0.02661 &{} 0.02280 &{} 0.01737 &{} 0.01198 &{} 0.00758 &{} 0.00444 &{} 0.00243 \\ 0.00288 &{} 0.00676 &{} 0.01084 &{} 0.01363 &{} 0.01431 &{} 0.01298 &{} 0.01044 &{} 0.00758 &{} 0.00502 &{} 0.00308 &{} 0.00176 \\ 0.00113 &{} 0.00280 &{} 0.00479 &{} 0.00643 &{} 0.00716 &{} 0.00688 &{} 0.00583 &{} 0.00444 &{} 0.00308 &{} 0.00197 &{} 0.00117 \\ 0.00042 &{} 0.00109 &{} 0.00199 &{} 0.00285 &{} 0.00337 &{} 0.00341 &{} 0.00304 &{} 0.00243 &{} 0.00176 &{} 0.00117 &{} 0.00072 \end{array}\right] } $$

As done in Example 8.19, we perform the factorization choosing the accuracy 0.001 to neglect the very small eigenvalues. In such a way we get the $11\times 4$ factor

$$ \gamma =Z_h\;\textit{D}_h= \begin{bmatrix} 0.33675&0.19320&0.07563&0.02214 \\ 0.47623&0.13659&-0.00015&-0.01598 \\ 0.47623&-0.00004&-0.05355&-0.01550 \\ 0.38884&-0.11157&-0.04355&0.00685 \\ 0.27495&-0.15774&0.00022&0.01825 \\ 0.17389&-0.14962&0.03924&0.01109 \\ 0.10039&-0.11515&0.05644&-0.00395 \\ 0.05366&-0.07692&0.05419&-0.01563 \\ 0.02683&-0.04614&0.04207&-0.01983 \\ 0.01265&-0.02537&0.02828&-0.01807 \\ 0.00565&-0.01296&0.01704AP20&-0.01361 \end{bmatrix} $$

The comparison with the results of Example 8.19 shows that the numerical values are very close (the difference is of the same order of magnitude as the chosen accuracy) .

1.2 Proof of Proposition 8.2 on Factorization

We exploit the rule of mixed products (ordinary products and Kronecker’s products) seen in Sect. 2.13, Eq. (2.104). For example, for $K=4$ and $i=2$ this rule yields

$$\begin{aligned} \rho _2&=\rho (0)\otimes \rho (\alpha )\otimes \rho (0)\otimes \rho (0)\\&=({\gamma ^0} \,{\gamma ^0}^*)\otimes ({\gamma ^1} \,{\gamma ^1}^*) \otimes ({\gamma ^0} \,{\gamma ^0}^*)\otimes ({\gamma ^0} \,{\gamma ^0}^*)\\&=({\gamma ^0}\otimes {\gamma ^1} \otimes {\gamma ^0} \otimes {\gamma ^0})({\gamma ^0}\otimes {\gamma ^1} \otimes {\gamma ^0} \otimes {\gamma ^0})^* \end{aligned}$$

from which

$$ \rho _2 =\gamma _2\,\gamma _2^*\quad \text{ with }\quad \gamma _2={\gamma ^0}\otimes {\gamma ^1} \otimes {\gamma ^0}\otimes {\gamma ^0} $$

and it is evident how the general result (8.65) can be obtained.

1.3 Simultaneous Diagonalization of $S$ and $T$

We outline a procedure for finding the simultaneous diagonalization (8.72) starting from an arbitrary EID $S=U_S\,\varLambda \,U_S^*$ of the symmetry and using the commutativity $\textit{TS}=\textit{ST}$.

(1):

We assume that, possibly after a reordering, the multiple eigenvalues in $\varLambda$ occur contiguously on the main diagonal. Then $\varLambda$ has the block diagonal form

$$\begin{aligned} \varLambda = \mathrm {diag}[\lambda _1I_1,\ldots ,\lambda _kI_k] \end{aligned}$$

(8.80)

where $\lambda _i$ are the distinct eigenvalues of $S$ and $I_i$ are identity matrices of size given by the multiplicity of $\lambda _i$.

(2):

Since $S$ commutes with $T$, it follows $U_S\varLambda U_S^*T= TU_S\varLambda U_S^*$ and $\varLambda U_S^*TU_S= U_S^*TU_S\varLambda$, so that $\varLambda$ commutes with $V=U_S^*TU_S$.

(3):

Partition the matrix $V=[\,V_{ij}\,]$ according to the partition (8.80) of $\varLambda$. Then, from $\varLambda V=V\varLambda$ we get $\lambda _iV_{ij}=\lambda _jV_{ij}$ and $V_{ij}$ does not vanish only if $i=j$. One concludes that $V$ is block diagonal, $V=\mathrm {diag}[V_1,\ldots ,V_k]$, with blocks $V_i$ of the same order as $I_i$ in (8.80).

(4):

Since $V$ is PSD, each block $V_i$ is PSD. Then, we perform the EID of the blocks $V_i=X_i\varSigma _i^2X_i^*$ and we get the diagonalization $V=X\varSigma ^2 X^*$ with $\varSigma ^2=\mathrm {diag}[\varSigma _1^2\ldots ,\varSigma _k^2]$ and $X=\mathrm {diag}[X_1,\ldots ,X_k]$.

(5):

By reversing the previous unitary transformations we get the diagonalization $T=U_SVU_S^*=U_SX\varSigma ^2X^*U_S^*$ and, remembering that $X_iX_i^*=I_i$, the simultaneous diagonalization $S=U_S\varLambda U_S^*=U_SX\varLambda X^*U_S^*\;.$ Thus, we get (8.72) with $U=U_S\,X$.

Note that $SU_S=U_S\varLambda$ and $U_S^*S=\varLambda U_S^*$ (see point 1). Then

$$ V=U_S^*TU_S=\sum _{k=0}^{m-1}U_S^*S^k\rho _0(S^*)^kU_S =\sum _{k=0}^{m-1}\varLambda ^kU_S^*\rho _0U_S\varLambda ^{-k}\;. $$

Since $V$ and $\varLambda$ are block–diagonal

$$\begin{aligned} V_i=\sum _{k=0}^{m-1}\lambda _k^iI_i(U_S^*\rho _0U_S)_{ii} \lambda _k^{-i}I_i=(U_S^*\rho _0U_S)_{ii} \end{aligned}$$

(8.81)

and the block $V_i$ coincides with the $i$th diagonal block of $U_S^*\rho _0U_S$.