# QuYBE - An Algebraic Compiler for Quantum Circuit Compression

Sahil Gulania† Mathematics and Computer Science Division
Argonne National Laboratory
Lemont, Illinois, 60439, United States
<EMAIL_ADDRESS>Zichang He† Department of Electrical and Computer
Engineering
University of California, Santa Barbara
Santa Barbara, California, 93106, United States
<EMAIL_ADDRESS>Bo Peng† Physical and Computational Sciences Directorate
Pacific Northwest National Laboratory
Richland, Washington, 99352, United States
<EMAIL_ADDRESS>Niranjan Govind  Physical and Computational Sciences
Directorate
Pacific Northwest National Laboratory
Richland, Washington, 99352, United States
<EMAIL_ADDRESS>Yuri Alexeev † Equal contributions  Computational
Science Division

Argonne National Laboratory
Lemont, Illinois 60439, United States
<EMAIL_ADDRESS>
###### Abstract

QuYBE is an open-source algebraic compiler for the compression of quantum
circuits. It has been applied for the efficient simulation of the Heisenberg
Hamiltonian on quantum computers. Currently, it can simulate the time dynamics
of one-dimensional chains. It includes modules to generate the quantum
circuits for the above as well as produce the compressed circuits, which are
independent of the time step. It utilizes the Yang-Baxter equation (YBE) to
perform the compression. QuYBE enables users to seamlessly design, execute,
and analyze the time dynamics of the Heisenberg Hamiltonian on quantum
computers. QuYBE is the first step toward making the YBE technique available
to a broader community of scientists from multiple domains. The QuYBE compiler
is available at https://github.com/ZichangHe/QuYBE.

##  I Introduction

Current major efforts towards demonstrating a real quantum advantage are
focused on designing practical large-scale quantum computer architectures. To
accommodate these efforts with the supporting software, robust quantum
compilation processes are much needed. A typical quantum compilation includes
three-level circuit generation or optimization, namely, the circuit generation
from a reduced set of universal quantum gates, the fault-tolerant design of
the original circuit, and the circuit compilation to hardware-specific
instructions. There is extensive literature devoted to employing classical
approaches for quantum compiling (see Refs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
for past examples using temporal planning, machine learning, and other
techniques). Recent exciting ideas include utilizing algebraic relations to
compress and optimize the original circuit for more fault-resilient
performance on NISQ devices[12, 13, 14, 15, 16, 17, 18, 19]. The Yang-Baxter
equation (YBE) or star-triangle relation has recently attracted much attention
in quantum information science. This equation, which was originally introduced
in theoretical physics[20] and statistical mechanics[21], is a consistency
condition which arises in systems where the dynamics are two-body
factorizable. The YBE link to quantum computing arises from the connection to
topological entanglement, quantum entanglement, and quantum computational
universality[22, 23, 24, 25, 26, 27, 28].

Briefly, the quantum YBE can be written as

$(\mathcal{R}\otimes\mathds{1})(\mathds{1}\otimes\mathcal{R})(\mathcal{R}\otimes\mathds{1})=(\mathds{1}\otimes\mathcal{R})(\mathcal{R}\otimes\mathds{1})(\mathds{1}\otimes\mathcal{R})$
(1)

where the $\mathcal{R}$ operator is a linear mapping $\mathcal{R}:V\otimes
V\rightarrow V\otimes V$ of vector space $V$. In the present discussion,
$\mathcal{R}$ represents a parameterized unitary gate and is limited as a two-
qubit gate parameterized by a phase factor and a rotation. Since the YBE
connects topological concepts like knots and links to entangled quantum
states, the CNOT gate can be replaced by some unitary gate $R$ via the YBE to
maintain the universality of quantum computation.[21] Here, the unitary $R$
gate serves as the solution for the condition of topological braiding, as well
as the unitary solution to the YBE.

In our previous work[28], we have proved that for some model systems it is
feasible to compress their time evolution circuit based on the YBE to a depth
that scales linearly with respect to the number of qubits $N$. Furthermore,
the YBE compilation in the classical pre-processing step scales as
$\mathcal{O}(N^{3})$. Following our previous conceptual work, and with an eye
toward large-scale applications, here we report the design and execution of a
quantum compiler (QuYBE) based on our previous conceptual YBE analysis. We
demonstrate that our QuYBE compiler can successfully perform efficient YBE
circuit compression for certain classes of the Heisenberg model.

##  II Methodology

In this section, as an illustration, we introduce how our QuYBE compiler
compresses the quantum circuit corresponding to the time dynamics of the
1D-Heisenberg XY model Hamiltonian. The same method can also be applied to
other Heisenberg model Hamiltonians with or without external field. For an
$N$-spin 1D-Heisenberg XY model Hamiltonian, to simulate its time dynamics
with $T$ Trotterization steps, the corresponding circuit has $M=2T$ layers.
For such a circuit with $M>N$, the QuYBE compiler is able to compress the full
circuit to $N$ layers analytically without loss of accuracy. The workflow is
summarized in Fig. 1. In the following, we elaborate our algorithms that have
been used in the QuYBE compiler. We will first be focused on a sequential
compression scheme for a demonstration purpose, and then we will briefly
discuss how it can be made parallel.

Figure 1: Worflow of QuYBE compiler.

###  II-A Sequential compression scheme

The overall algorithm is described in Algorithm. 1. It first transforms the
first $N$-layer circuit to the reflection symmetric form. Then the $(N+1)$-th
layer can be merged into the reflected circuit, and the circuit depth is
decreased by one. By repeating these two steps, we can compress the original
circuit to a compact form with only $N$ layers, i.e. circuit depth is a linear
function of number of qubits $N$. As an example how YBE fascilitate the
reflection symmetry, Fig 2 shows how to achieve reflection symmetry for four
qubits where YBE is true.

Figure 2: Reflection symmetry can be achieved by using the YBE four times on
four qubits. For each step, the triplets of qubits on which YBE act is shown
by green color. Algorithm 1 A sequential circuit compression employing QuYBE

1:A $N$-qubit time evolution circuit with $M(M>N)$ layers

2:Functionally equivalent circuit with $N$ layers

3:while $M>N$ do

4: Execute the function CircuitReflection on the first $N$ layers.

5: Merge the adjacent gates on the same qubits

6: $M\leftarrow M-1$

7:end while

Algorithm 1 simply describes how a sequential compression works for a given
time evolution circuit of the Heisenberg XY model. Inside the compression
algorithm, the key function CircuitReflection is the multi-qubit reflection
operation, of which the detailed workflow is given in Algorithm 2. As can be
seen, CircuitReflection is a recursive function, which also reflects how we
proved the existence of a general reflection symmetry of an $N$-layer
$N$-qubit time evolution circuit for the studied models in our previous
analysis. [28] Essentially, what CircuitReflection does is to gradually
offload the two-qubit gates in a given time evolution circuit from top left to
lower right and restore the untouched two-qubit gate from lower right to top
left to get the reflection done. To better illustrate how exactly the key
function CircuitReflection works, take a five-qubit time evolution circuit as
an example, Fig. 3 shows how we can offload the restore the two-qubit gates to
vertically reflect the first five layers of the circuite to enable the merge
between the fifth and sixth layers.

Algorithm 2 Multi-qubit circuit reflection operation

1:function CircuitReflection(circuit of $N$-qubit)

2: Offload the gate on the first two qubits to the side.

3: Perform CircuitReflection for the circuit of ($N-1$)-qubit.

4: Restore the untouched two-qubit gates in Step 1 back to the top row.

5:end function

Another key component in Algorithm 2, as can be seen from Fig. 3, is the
three-qubit YBE (or reflection) operation which we have taken in our QuYBE
compiler as a basic operation. As demonstrated in Fig. 4, the three-qubit YBE
operation includes transforming an A-shape circuit to a V-shape circuit
(denoted as the A2V transformation), and its reverse operation (denoted as the
V2A transformation). We have coded these two transformations, as well as a
gate merge function, as core functions in our QuYBE compiler. The detailed
analytic transformation rules are based on the Theorem I in Ref.28. Also,
regarding the scaling of CircuitReflection, the reflection of an $N$-qubit
time evolution circuit with $N$ layers requires $\mathcal{O}(N^{3})$ number of
3-qubit YBE transformations (see Fig. 7 (red), also see the derivation in
Appendix A of Ref.28).

Figure 3: Schematic demonstration of compressing one layer from the five-qubit
time evolution circuit. (a) shows the original circuit. (b-g) converts the
first five-layer circuit into its reflection symmetric form, where (c-e)
reflects the four-layer four-qubit time evolution circuit to its reflection
symmetric form. (g) shows the circuit that enables the merge of the fifth and
six-th layers. (f) shows the compressed circuit. Figure 4: Three-qubit
reflection operation is treated as a basic operation in our QuYBE compiler.

###  II-B Parallel compression scheme

In comparison with some heuristic approaches, the QuYBE compiler offers a
deterministic approach for compressing the time evolution circuits for some
model systems. Nevertheless, a sequential version would become a very time
consuming pre-processing step given only one layer is compress after
$\mathcal{O}(N^{3})$ three-qubit YBE operations. To reduce the time-to-
solution, we also explore the feasibility of having a parallel scheme.

####  II-B1 Starting point of the circuit compression

From the above demonstration, one important observation is that the
compression in Fig. 3 starts from the very beginning of the circuit structure
where there are no gates prior to the circuit fragment that is about to get
compressed. Therefore, we only achieve an ($N+1$)-to-$N$ compression.
Alternatively, if we choose to start the compression from anywhere other than
the beginning, we will get an ($N+2$)-to-$N$ two-layer compression, i.e., one
layer gets compressed at the beginning of the circuit fragment and the other
gets compressed at the end of the circuit fragment, after performing one
reflection operation (see the inset in Fig. 5).

####  II-B2 Parallel compression

Another important observation is that the ($N+2$)-to-$N$ compression in the
$(N+2)$-layer is independent of other $(N+2)$-layer circuit fragments. The
independence between the circuit fragments then triggers a straightforward
parallel scheme to reduce the time-to-solution. Specifically, as shown in Fig.
5, we only need to partition the original circuit to multiple circuit
fragments, each of which is an $(N+2)$-layer circuit. The compression of these
circuit fragments can straightforwardly be distributed to classical computing
resources. Suppose the entire time evolution circuit include $M$ layers, the
simple parallel scheme would require $\lceil\frac{M}{N+2}\rceil$ processors at
the maximum to do the compression.

Figure 5: A parallel execution of QuYBE compiler. p denotes the maximal number
of processes. The inset at the lower right demonstrate a two-layer compression
for a seven-layer five-qubit time evolution circuit if the reflection takes
place at the layers #2$-$#6.

##  III Results

In this section, we discuss the impact of circuit compression by showing the
reduction in two-qubits gate, i.e., CNOTs, and the growth of CNOTs with
increasing system size for the time dynamics of 1D-XY Hamiltonian, a subclass
of Heisenberg Hamiltonian. The Heisenberg Hamiltonian [29, 30, 31] is widely
used to study magnetic systems, where the magnetic spins are treated quantum
mechanically. The Hamiltonian (with nearest neighbor interaction) can be
written as

$\hat{H}=-\sum_{\alpha}\\{J_{\alpha}\sum_{i=1}^{N-1}\sigma_{i}^{\alpha}\otimes\sigma_{i+1}^{\alpha}\\},$
(2)

where $\alpha$ sums over $\\{x,y,z\\}$, the coupling parameter $J_{\alpha}$
denotes the exchange interaction between nearest-neighbour spins along the
$\alpha-$direction, and $\sigma^{\alpha}_{i}$ is the $\alpha$-Pauli operator
on the $i$th spin. For XY Hamiltonian, $J_{z}=0$ in Eq. (2). We also show a
real device (IBM-Manila) simulation using the same compression scheme.

Fig. 6 shows the growth of CNOTs between the canonical and compressed circuits
with increasing steps in the quantum time dynamics circuit.

Figure 6: Comparison between the canonical and compressed quantum circuits for
the growth of CNOTs with Trotter steps. The dashed line represents the growth
of CNOTs in the canonical quantum circuit, while the filled line represents
the growth of CNOTs in the compressed quantum circuit.

The dashed line represents the growth of CNOTs with Trotter steps in the
canonical circuit. On the other hand, after the compression, CNOTs grow until
a step number and plateau to a constant after that, as shown by the filled
line. Fig 7 (blue) shows the scaling of the maximum CNOTs required for
arbitrary time dynamics in compressed circuits with the number of qubits.

Figure 7: Scaling of number of 3-qubit YBE transformations to finish one
reflection operation (red). Scaling of maximum CNOTs is required in compressed
quantum circuits for arbitrary time evolution with the number of qubits
(blue).

The growth of CNOTs is quadratic with the number of qubits. Fig 8, shows the
evolution of the staggered magnetization for 5-spins for the Heisenberg XY
Hamiltonian ($J_{x}=-0.8$, $J_{y}=-0.2$). Each Trotter step is 0.05 units of
time. In order to confirm the validity of compression, we plot results from
the canonical circuit and compressed circuit using the Qiskit simulator for
every tenth step. It shows that the compressed circuit (composed of only 20
CNOTs) provides the same level of accuracy as the canonical circuit on the
simulator.

Figure 8: Comparison of the staggered magnetization evolution for 5 spins with
the XY Hamiltonian ($J_{x}=-0.8$, $J_{y}=-0.2$) between the canonical circuit
and compressed circuit via YBE. Each simulation on IBM-Manila used 20000
shots.

The impact of the compression is clearly seen when the same simulation is run
on a real quantum device. Fig. 8 shows the results of the canonical circuit
and compressed circuit on the IBM-Manila quantum device. We computed the time-
dependent staggered magnetization, $m_{s}(t)$, which can be connected to the
antiferromagnetism and ferrimagnetism in materials as follows:

$m_{s}(t)=\frac{1}{N}\sum_{i}(-1)^{i}\braket{\sigma_{z}(t)}.$ (3)

The initial state is the ground state (Neél state) of the XY Hamiltonian,
defined as
$\Psi_{0}=\ket{\uparrow\downarrow\uparrow\downarrow...\uparrow\downarrow}$.
The staggered magnetization of the Neél state is one. We performed the time
evolution for 5 units of time with a Trotter step size of 0.05 units.

The staggered magnetization using the canonical circuit quickly drops to zero
and remains zero for future time steps. The increasing circuit depth and noise
results in a maximally mixed state corresponding to zero magnetization. This
does not happen in the compressed circuit, which only requires 20 CNOTs for
arbitrary time evolution. In other words, the maximum error is capped, and we
observe better qualitative and quantitative results on real quantum devices.

##  IV Conclusions

In this paper, we have presented the QuYBE, which is an open-source algebraic
compiler for the compression of quantum circuits. It has been applied for the
efficient simulation of the time dynamics of the Heisenberg Hamiltonian on
quantum computers. The compiler is based on a circuit compression algorithm
based on the quantum Yang-Baxter equation. We demonstrate the efficacy of this
approach by comparing time dynamics simulations with canonical and compressed
circuits of the 1D-Heisenberg XY model on a real quantum device.

The QuYBE compiler is available at https://github.com/ZichangHe/QuYBE. One of
our goals is to make the QuYBE compiler easy to use with existing popular
quantum compilers. It is achieved by using QuYBE as an intermediate step in
the compilation pipeline, where YBE takes a QASM circuit, compresses it, and
generates a QASM circuit in the target format.

The proposed YBE technique is a general approach that can be applied to any
quantum circuit compliant partially or fully with YBE. For example, other
types of Ising Hamiltonians are obvious targets. This work is important
because it paves the way for large-scale applications and investigations of
broader classes of models.

## Acknowledgment

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, and National Quantum Information Science Research Centers.
Y.A. acknowledges support from the U.S. Department of Energy, Office of
Science, under contract DE-AC02-06CH11357 at Argonne National Laboratory.

## References

  * [1] A. Oddi and R. Rasconi, “Greedy randomized search for scalable compilation of quantum circuits,” in _Integration of Constraint Programming, Artificial Intelligence, and Operations Research_ , W.-J. van Hoeve, Ed. Cham: Springer International Publishing, 2018, pp. 446–461.
  * [2] T. Häner, D. S. Steiger, K. Svore, and M. Troyer, “A software methodology for compiling quantum programs,” _Quantum Science and Technology_ , vol. 3, no. 2, p. 020501, feb 2018.
  * [3] L. E. Heyfron and E. T. Campbell, “An efficient quantum compiler that reduces t count,” _Quantum Science and Technology_ , vol. 4, no. 1, p. 015004, sep 2018.
  * [4] L. Cincio, Y. Subaşı, A. T. Sornborger, and P. J. Coles, “Learning the quantum algorithm for state overlap,” _New Journal of Physics_ , vol. 20, no. 11, p. 113022, nov 2018.
  * [5] D. Venturelli, M. Do, E. Rieffel, and J. Frank, “Compiling quantum circuits to realistic hardware architectures using temporal planners,” _Quantum Science and Technology_ , vol. 3, no. 2, p. 025004, feb 2018.
  * [6] F. Chong, D. Franklin, and M. Martonosi, “Programming languages and compiler design for realistic quantum hardware,” _Nature_ , vol. 549, p. 180–187, 2017.
  * [7] Y. Nam, N. Ross, Y. Su, A. Childs, and D. Maslov, “Automated optimization of large quantum circuits with continuous parameters,” _npj Quantum Inf._ , vol. 4, p. 23, 2018.
  * [8] J. Booth, “Quantum compiler optimizations,” 2012.
  * [9] A. G. Fowler, “Constructing arbitrary steane code single logical qubit fault-tolerant gates,” _Quantum Info. Comput._ , vol. 11, no. 9–10, p. 867–873, sep 2011.
  * [10] D. Maslov, G. W. Dueck, D. M. Miller, and C. Negrevergne, “Quantum circuit simplification and level compaction,” _IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems_ , vol. 27, no. 3, pp. 436–444, 2008.
  * [11] K. E. C. Booth, M. Do, J. C. Beck, E. Rieffel, D. Venturelli, and J. Frank, “Comparing and integrating constraint programming and temporal planning for quantum circuit compilation,” 2018.
  * [12] L. Bassman, R. V. Beeumen, E. Younis, E. Smith, C. Iancu, and W. A. de Jong, “Constant-depth circuits for dynamic simulations of materials on quantum computers,” 2021.
  * [13] E. Kökcü, D. Camps, L. Bassman, J. K. Freericks, W. A. de Jong, R. V. Beeumen, and A. F. Kemper, “Algebraic compression of quantum circuits for Hamiltonian evolution,” _Physical Review A_ , vol. 105, no. 3, mar 2022\. [Online]. Available: https://doi.org/10.1103%2Fphysreva.105.032420
  * [14] D. Camps, E. Kökcü, L. Bassman, W. A. de Jong, A. F. Kemper, and R. Van Beeumen, “An algebraic quantum circuit compression algorithm for hamiltonian simulation,” 2021. [Online]. Available: https://arxiv.org/abs/2108.03283
  * [15] S.-H. Lin, R. Dilip, A. G. Green, A. Smith, and F. Pollmann, “Real-and imaginary-time evolution with compressed quantum circuits,” _PRX Quantum_ , vol. 2, no. 1, p. 010342, 2021.
  * [16] C. Cirstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, and A. Sornborger, “Variational fast forwarding for quantum simulation beyond the coherence time,” _npj Quantum Information_ , vol. 6, no. 1, pp. 1–10, 2020.
  * [17] Y. Atia and D. Aharonov, “Fast-forwarding of hamiltonians and exponentially precise measurements,” _Nature Communications_ , vol. 8, no. 1, pp. 1–9, 2017.
  * [18] N. F. Berthusen, T. V. Trevisan, T. Iadecola, and P. P. Orth, “Quantum dynamics simulations beyond the coherence time on NISQ hardware by variational Trotter compression,” 2021. [Online]. Available: https://arxiv.org/abs/2112.12654
  * [19] F. Barratt, J. Dborin, M. Bal, V. Stojevic, F. Pollmann, and A. G. Green, “Parallel quantum simulation of large systems on small nisq computers,” _npj Quantum Information_ , vol. 7, no. 1, pp. 1–7, 2021.
  * [20] C.-N. Yang, “Some exact results for the many-body problem in one dimension with repulsive delta-function interaction,” _Physical Review Letters_ , vol. 19, no. 23, p. 1312, 1967.
  * [21] R. J. Baxter, _Exactly solved models in statistical mechanics_. Elsevier, 2016.
  * [22] M.-L. Ge, K. Xue, R.-Y. Zhang, and Q. Zhao, “Yang–Baxter equations and quantum entanglements,” _Quantum Information Processing_ , vol. 15, no. 12, pp. 5211–5242, 2016.
  * [23] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” _Rev. Mod. Phys._ , vol. 80, pp. 1083–1159, Sep 2008. [Online]. Available: https://link.aps.org/doi/10.1103/RevModPhys.80.1083
  * [24] L. H. Kauffman and S. J. J. Lomonaco, “Topological quantum information theory,” in _Proceedings of Symposia in Applied Mathematics_ , ser. Vol. 68, S. J. Lomonaco, Ed. Washington, DC: AMS, 2010.
  * [25] Y. Zhang, “Integrable quantum computation,” _Quantum Inf. Process._ , vol. 12, pp. 631–639, 2013.
  * [26] F. A. Vind, A. Foerster, I. S. Oliveira, R. S. Sarthour, D. d. O. Soares-Pinto, A. M. d. Souza, and I. Roditi, “Experimental realization of the Yang–Baxter equation via NMR interferometry,” _Scientific reports_ , vol. 6, no. 1, pp. 1–8, 2016.
  * [27] M. T. Batchelor and A. Foerster, “Yang–Baxter integrable models in experiments: from condensed matter to ultracold atoms,” _Journal of Physics A: Mathematical and Theoretical_ , vol. 49, no. 17, p. 173001, 2016.
  * [28] B. Peng, S. Gulania, Y. Alexeev, and N. Govind, “Quantum time dynamics employing the yang-baxter equation for circuit compression,” _Phys. Rev. A_ , vol. 106, p. 012412, Jul 2022.
  * [29] P. Fazekas, _Lecture notes on electron correlation and magnetism_. World scientific, 1999, vol. 5.
  * [30] R. Skomski, _Simple models of magnetism_. Oxford University Press, 2008.
  * [31] A. S. T. Pires, _Theoretical Tools for Spin Models in Magnetic Systems_. IOP Publishing, 2021.


# On capacity of quantum channels generated by irreducible projective unitary
representations of finite groups

G.G.Amosov This work was funded by the Ministry of Science and Higher
Education of the Russian Federation (grant number 075-15-2020-788) and
performed at the Steklov Mathematical Institute of the Russian Academy of
Sciences. Steklov Mathematical Institute of Russian Academy of Sciences

###### Abstract

We study mixed unitary quantum channels generated by irreducible projective
unitary representations of finite groups. Under some assumptions on the
probability distribution determining a mixture the classical capacity of the
channel is found. Examples illustrating the techniques are given.

##  1 Introduction

The quantum coding theorem [1, 2] gives rise to the estimation from above for
a number of states $N$ that can be asymptotically accurately transmitted
through $m$ copies of the quantum channel $\Phi$ of the form

$N\sim e^{mC({\Phi})},\ m\to+\infty,$

where $C(\Phi)$ is said to be a classical capacity of $\Phi$. The capacity
$C(\Phi)$ can be calculated by the formula

$C(\Phi)=\lim\limits_{m\to\infty}\frac{C_{1}(\Phi^{\otimes m})}{m},$

where $C_{1}(\Phi)$ is a one-shot capacity determined by the Holevo upper
bound as follows

$C_{1}(\Phi)=\sup\limits_{\pi_{j},\rho_{j}\in{\mathfrak{S}}(H)}(S(\sum\limits_{j}\pi_{j}\Phi(\rho_{j}))-\sum\limits_{j}\pi_{j}S(\Phi(\rho_{j}))).$

Here $S(\rho)=-Tr(\rho\log\rho)$ is the von Neumann entropy of quantum states
$\rho_{j}$ from the set of all states $\mathfrak{S}(H)$ in a Hilbert space $H$
and the supremum is taken over all probability distributions $(\pi_{j})$.

The main feature of quantum theory is the presence of entangled states in
$\mathfrak{S}(H^{\otimes m})$, that is, such states that are not factorized
with respect to the tensor product $H^{\otimes m}$. The presence of entangled
states leads to the fact that taking the supremum in the definition of
$C_{1}(\Phi^{\otimes m})$ becomes a very difficult mathematical problem. If
the equality of the form

$C_{1}(\Phi^{\otimes m})=mC_{1}(\Phi)$ (1)

were true, it would mean that the use of entangled states does not give an
advantage in encoding. In the last case, the capacity of $\Phi$ coincides with
the one-shot capacity

$C(\Phi)=C_{1}(\Phi).$ (2)

Nevertheless it is known that (2) doesn’t take place in general [3].

It is natural to start calculating the capacity of a channel with the case of
mixed unitary channels having the form

$\Phi(\rho)=\sum\limits_{j}\pi_{j}U_{j}\rho U_{j}^{*},$ (3)

where $\rho\in{\mathfrak{S}}(H)$, $(\pi_{j})$ is a probability distribution
and $(U_{j})$ are unitary operators in $H$. Operators $U_{j}$ simulate errors
that may occur during the transmission of information. Since the content part
of the problem concerns the study of possible encodings in the tensor product
$H^{\otimes m}$, it turns out that only error operators acting locally on the
qudit $H$ forming a composite system $H^{\otimes m}$ are allowed. For the
qubit case $dimH=2$ quantum channels of the form (3) were studied in [4]. The
property (2) was proved and the capacity of channels were found. Here unitary
operators in (3) may mean bit flip or sign (phase) flip errors. In [5] the
same problem was solved for the quantum depolarizing channel having the form
(3) in any dimension. Both the cases considered relate to channels of a
narrower type than (3). More precisely, these channels have the form

$\Phi(\rho)=\sum\limits_{g\in G}\pi_{g}U_{g}\rho U_{g}^{*},$ (4)

where $G\ni g\to U_{g}$ is an irreducible projective unitary representation of
a finite group $G$ and $(\pi_{g})$ is a probability distribution on $G$.
Moreover, as we will show and discuss it appeared that $(U_{g},\ g\in G)$
generate the group ${\mathcal{U}}_{G}$ containing a maximum Abelian normal
subgroup. It should be noted that the hypothetical counterexample of [3] has
the form (3) but we suppose that it doesn’t belong to the class (4).

Since the unital channel (4) is covariant with respect to the irreducible
projective unitary representation $g\to U_{g}$ we obtain that [6]

$C_{1}(\Phi)=\log n-\inf\limits_{\rho\in\mathfrak{S}(H)}S(\Phi(\rho)).$ (5)

If the property

$\inf\limits_{\rho\in\mathfrak{S}(H^{\otimes^{m}})}S(\Phi^{\otimes^{m}}(\rho))=m\inf\limits_{\rho\in\mathfrak{S}(H)}S(\Phi(\rho))$
(6)

known as the weak additivity for output entropy holds true the classical
capacity of $\Phi$ can be found by means of a simple formula

$C(\Phi)=C_{1}(\Phi)=\log
n-\inf\limits_{\rho\in\mathfrak{S}(H)}S(\Phi(\rho)).$ (7)

At the moment, (7) is proved for only few partial cases [4, 5, 7].

Recently, there has been renewed interest in computing the capacity of quantum
channels, related to the application of the majorization procedure [8, 9, 10,
11, 7]. The idea of applying the majorization procedure [12] for channels in a
finite dimensional Hilbert space $H,\ dimH=n<+\infty,$ is proposed in [10] and
based upon the division of elements of the probability distribution
$(\pi_{g})$ from (4) into subsets, the sums of the elements of which are
placed in descending order. This allows us to obtain estimates of eigenvalues
for $\Phi(\rho)$. However, in order for the estimates to be achievable, some
additional conditions must be met. In [11] it was shown that if the conditions
of achievability for [10] are fulfilled for $\Phi$ it would be not true for
$\Phi^{\otimes m}$. We will show that this complexity can be overcome.

This work is a continuation of [7], where the classical capacity was
calculated for the Weyl channels that are perturbations of quantum-classical
(q-c) channels. Here we justify the results of [7] and develope the techniques
of [7] to the channels generated by irreducible projective unitary
representations of discrete groups. We use the ideas that originate from [13,
14, 15, 16]. Applying the techniques of the group theory was inspired by [17,
18]. These papers in turn were encouraged by [19, 20, 21].

This paper is organized as follows. We start with Preliminaries containing
basic notations used in the text. The next two section are devoted to basic
results from the theory of majorization and the theory of representations for
finite groups. Our main result concerning the calculation of one-shot and
classical capacities for mixed unitary channels is placed in Section 5. The
proof is given in Section 6. Section 7 is devoted to the construction of
examples. The paper is completed by concluding remarks.

##  2 Preliminaries

Throughout this paper we use the following standard notations of quantum
information theory [6] and the group theory [24, 25].

$H$ is the Hilbert space with the dimension $dimH=n$ and the fixed orthonormal
basis $\ket{j},\ j\in{\mathbb{Z}}_{n}$;

$B(H)$ is the algebra of all bounded operators in $H$;

${\rm I}={\rm I}_{H}$ is the identity operator in $H$;

$\mathfrak{S}(H)$ the convex set of all states (positive unit trace operators)
in $H$;

$\mathcal{U}(H)$ the group of all unitary operators in $H$;

Given $\rho\in\mathfrak{S}(H)$ we denote $S(\rho)=-Tr(\rho\log\rho)$ the von
Neumann entropy of $\rho$;

The linear map $\Phi:B(H)\to B(H)$ is said to be a quantum channel if it is
completely positive and trace preserving such that
$\Phi:\mathfrak{S}(H)\to\mathfrak{S}(H)$;

The channel $\Phi$ is said to be a quantum-classical (q-c) channel if
$\Phi(\rho)$ belongs to the maximum Abelian algebra ${\mathcal{A}}\subset
B(H)$ for all $\rho\in{\mathfrak{S}}(H)$;

${\mathbb{Z}}_{n}$ is the cyclic group with elements $\\{0,\dots,n-1\\}$;

$S_{n}$ is the symmetric group consisting of all permutations on the set of
indexes $\\{0,\dots,n-1\\}$;

$S_{n}$ is the set of all permutations of indexes $j\in\\{0,1,2,\dots,n-1\\}$
identifying with elements $j\in{\mathbb{Z}}_{n}$;

A subgroup $T$ of the group $G$ is said to be normal if $gtg^{-1}\in T$ for
all $g\in G,\ t\in T$. We denote elements of the quotient $G/T$ by $[g]$.

##  3 Majorization

Here we give the basic concept of the majorization theory [12] following to
[22].

Given a probability distribution $\mu=\\{\mu_{l},\ l\in{\mathbb{Z}}_{n}\\}$
denote $\mu^{\downarrow}$ the same distribution rearranged in the descending
order. For two probability distributions $\mu=\\{\mu_{l},\
l\in{\mathbb{Z}}_{n}\\}$ and $\nu=\\{\nu_{l},\ l\in{\mathbb{Z}}_{n}\\}$ we say
that $\mu$ majorizes $\nu$ and write [22]

$\nu^{\downarrow}\prec\mu^{\downarrow}$

iff

$\sum\limits_{l=0}^{r}\nu_{l}^{\downarrow}\leq\sum\limits_{l=0}^{r}\mu_{l}^{\downarrow},\
0\leq r<n.$

The famous characteristic of majorization is placed below.

Theorem II.3.1 [12, 22]. Suppose that $\mu=\mu^{\downarrow}$ and
$\nu=\nu^{\downarrow}$. The following two conditions are equivalent:

(i) $\nu\prec\mu$.

(ii) $\sum\limits_{j}\phi(\nu_{j})\leq\sum\limits_{j}\phi(\mu_{j})$ for all
convex functions $\phi$ from $\mathbb{R}$ to $\mathbb{R}$.

Applying Theorem given above we obtain that the following inequalities hold
true

$\sum\limits_{l\in{\mathbb{Z}}_{n}}\nu_{l}^{p}\leq\sum\limits_{l\in{\mathbb{Z}}_{n}}\mu_{l}^{p},\
p\geq 1,$ (8)

and

$-\sum\limits_{l\in{\mathbb{Z}}_{n}}\nu_{l}\log\nu_{l}\geq-\sum\limits_{l\in{\mathbb{Z}}_{n}}\mu_{l}\log\mu_{l}.$
(9)

The inequality (8) concerns the multiplicativity property for $l_{p}$-norms
[23]. We especially interested in the inequality (9) allowing to calculate a
classical capacity of the channel under the conditions we consider.

##  4 Projective unitary representations of finite groups

Here we put the standard facts of the representation theory [24, 25].

Let $G$ be a finite group. The map $G\ni g\to U_{g}\in{\mathcal{U}}(H)$ is
said to be a projective unitary representation of $G$ if

$U_{g}U_{h}=\omega(g,h)U_{gh},$ (10)

where $\omega(g,h)$ is an unimodular function on $G\times G$. In order for
(10) to be correct, the following condition must be met

$\omega(g,h)\omega(gh,r)=\omega(g,hr)\omega(h,r),\ g,h,r\in G.$ (11)

Function $\omega(g,h)$ satisfying the property (11) is said to be $2$-cocycle
of $G$. Two representation $g\to U_{g}$ and $g\to V_{g}$ are said to be
equivalent if there exists the unimodular function $c(g)$ such that

$V_{g}=c(g)U_{g},\ g\in G.$ (12)

It is straightforward to check that

$V_{g}V_{h}=\tilde{\omega}(g,h)V_{gh},$

with

$\tilde{\omega}(g,h)=\frac{c(g)c(h)}{c(gh)}\omega(g,h).$

Given a representation $g\to U_{g}$ of the group $G,\ |G|=N$ there exists the
equivalent representation with the function $\omega(g,h)$ satisfying [25]

$\omega(g,h)^{N}=1,\ g,h\in G.$ (13)

The $2$-cocycles $\omega(g,h)$ form the Abelian group with respect to a
pointwise multiplication. This group is said to be the Schur multiplier.
Adding to the unitary operators $\\{U_{g},\ g\in G\\}$ the Schur multiplier we
obtain the unitary group acting in the Hilbert space $H$ [25]. We denote
${\mathcal{U}}_{G}$ this group.

##  5 The main result

Let $T\triangleleft G$ be a normal Abelian subgroup of $G$. Consider a unitary
representation of $t\to U_{t}$ of the group $T$ in the Hilbert space $H,\
dimH=n$. Without a sake of generality we can suppose that the operators
$U_{t},\ t\in T,$ are diagonal in the basis $(\ket{j})$. Let $g\to U_{g}$ be a
projective unitary representation of $G$ in a Hilbert space $H$ extending the
representation $t\to U_{t}$ of $T$.

Let us define a mixed unitary quantum channel by the formula

$\Phi(\rho)=\sum\limits_{g\in G}\pi_{g}U_{g}\rho U_{g}^{*},\
\rho\in\mathfrak{S}(H),$ (14)

where $\pi=(\pi_{g},\ g\in G)$ is a probability distribution on $G$.

Together with the probability distribution $(\pi_{g},\ g\in G)$ determining
the channel (14) let us consider the probability distribution $p_{[g]}$ on the
quotient $G/T$ defined by the formula

$p_{[g]}=\sum\limits_{t\in T}\pi_{gt}.$ (15)

Denote $p=(p_{j},\ j\in{\mathbb{Z}}_{n})$ values (15) placed in descending
order such that

$p_{0}\geq p_{1}\geq\dots\geq p_{n-1}$

Let $[g_{j}]$ be the element of $G/T$ such that $p_{[g_{j}]}=p_{j}$. Let us
define the following condition

$j>k\Rightarrow\pi_{g_{j}t}\leq\pi_{g_{k}s}$ (16)

for all $s,t\in T$ and $j,k\in{\mathbb{Z}}_{n}$.

Theorem. Let $G$ and $T$ be a finite group and its normal Abelian subgroup
with the property $|G/T|=n$. Suppose that $g\to U_{g}$ is an irreducible
projective unitary representation of $G$ in a Hilbert space $H,\ dimH=n$, and
the restriction $t\to U_{t},\ t\in T$ is unitary representation. Then, if the
condition (16) is satisfied the one-shot and classical capacities of (14) are
given by the formula

$C(\Phi)=C_{1}(\Phi)=\log n+\sum\limits_{j\in{\mathbb{Z}}_{n}}p_{j}\log
p_{j}.$

The rest of the text is devoted to clarifying the condition (16), proving
Theorem, and giving several examples of how to apply Theorem.

##  6 Quantum channels generated by projective unitary representations

The following proposition shows how the projective unitary representation of
the group satisfying the conditions of Theorem is arranged.

Proposition 1. Let $T$ be a normal Abelian subgroup of $G$ and $g\to U_{g}$ is
an irreducible projective unitary representation of $G$ in $H$ extending the
unitary representation $t\to U_{t}$ of $T$. Suppose that $|G/T|=n$, then there
is the action $\alpha_{[g]}$ of $G/T$ consisting of permutations on
${\mathbb{Z}}_{n}$ considered as a set such that

$U_{g}\ket{j}\bra{j}U_{g}^{*}=\ket{\alpha_{[g]}(j)}\bra{\alpha_{[g]}(j)},$
(17)

$g\in G,\ j\in{\mathbb{Z}}_{n}$.

Proof.

Since the representation is irreducible we get

$\frac{1}{|G|}\sum\limits_{g\in G}U_{g}\ket{f}\bra{f}U_{g}^{*}=\frac{1}{n}{\rm
I}$ (18)

for any unit vector $f\in H$. Projections $\ket{j}\bra{j}$ belong to the
algebra of fixed elements with respect to the actions of $B(H)\ni x\to
U_{t}xU_{t}^{*},\ t\in T$. Substituting $\ket{f}=\ket{j}$ to (18) we obtain

$\sum\limits_{[g]\in G/T}U_{g}\ket{j}\bra{j}U_{g}^{*}={\rm I}$ (19)

because $|G/T|=n$ by the condition. The sum of projections in (19) is equal to
the identity operator only if they are pairwise orthogonal.

$\Box$

The next statement explains how the majorization works.

Proposition 2. Suppose that the conditions of Proposition 1 are satisfied and
the condition (16) holds true. Then, given a unit vector $f\in H$ the
eigenvalues $\lambda=(\lambda_{m})$ of the operator $\Phi(\ket{f}\bra{f})$
satisfy the relation

$\lambda^{\downarrow}\prec p.$

Proof.

Since the representation $g\to U_{g}$ is irreducible (18) is fulfilled.
Because $|G/T|=n$ we can rewrite (18) as follows

$\sum\limits_{[g_{j}]\in G/T,t\in T}|\braket{e,U_{g_{j}t}f}|^{2}=|T|$ (20)

for any unit vector $e\in H$. Let $(e_{m})$ be the eigenvectors corresponding
to the eigenvalues $\lambda^{\downarrow}=(\lambda_{m}^{\downarrow})$. It
follows from (20) that

$\sum\limits_{m=0}^{k}\lambda_{m}^{\downarrow}=\sum\limits_{m=0}^{k}\braket{e_{m},\Phi(\ket{f}\bra{f})e_{m}}=$
$\sum\limits_{m=0}^{k}\sum\limits_{[g_{j}]\in G/T,t\in
T}\pi_{g_{j}t}|\braket{e_{m},U_{[g_{j}]t}f}|^{2}\leq\sum\limits_{m=0}^{k}p_{m},\
0\leq k\leq n-1,$ (21)

due to each sum in (15) consists of $|T|$ elements and (16).

$\Box$

Consider the partial case of probability distribution satisfying (16)

$\pi_{g_{j}t}=\frac{1}{n}p_{j},\ j\in{\mathbb{Z}}_{n}$

and the corresponding channel

$\Phi_{0}(\rho)=\frac{1}{n}\sum\limits_{j\in{\mathbb{Z}}_{n},t\in
T}p_{j}U_{g_{j}t}\rho U_{g_{j}t}^{*},\ \rho\in{\mathfrak{S}}(H).$ (22)

Proposition 3. The channel (22) is the q-c channel with the image belonging to
the maximum Abelian subalgebra $\mathcal{A}$ generated by $\ket{j}\bra{j},\
j\in{\mathbb{Z}}_{n}$.

Proof.

The conditional expectation $\Theta$ on the algebra $\mathcal{A}$ is given by
the formula

$\Theta(x)=\frac{1}{|T|}\sum\limits_{t\in T}U_{t}xU_{t}^{*},\ x\in B(H).$

It is straightforward to check that

$\Theta\circ\Phi_{0}=\Phi_{0}.$

$\Box$

The quantum channel $\Phi$ possessing the property (16) is naturally to call a
perturbation of q-c channel (22).

Now we are ready to proceed to the study of the properties of channels for
tensor products. Given a probability distribution $q$ on ${\mathbb{Z}}_{n}$
denote $q^{\oplus N}$ the probability distribution
$(q_{0},q_{1},\dots,q_{n-1},0,\dots,0)$, where $(n-1)N$ zeros are added to the
distribution $q$.

Proposition 4. Fix the probability distribution $q$ on ${\mathbb{Z}}_{n}$.
Suppose that the conditions of Proposition 1 are satisfied and

$\lambda^{\downarrow}\prec q,$

for the eigenvalues $\lambda$ of the state $\Phi(\ket{e}\bra{e})$ under any
choice of a unit vector $e\in H$. Consider
$Id\otimes\Phi:\mathfrak{S}(K\otimes H)\to\mathfrak{S}(K\otimes H)$. Then,
given a unit vector $f\in K\otimes H$ the eigenvalues $\mu=(\mu_{j})$ of the
state $(Id\otimes\Phi)(\ket{f}\bra{f})$ satisfy the relation

$\mu^{\downarrow}\prec q^{\oplus N},$

where $N=dimK$.

Proof.

Consider the subspace ${\mathcal{L}}\subset K\otimes H$ generated by vectors
$(I\otimes U_{g})f,\ g\in G$. Since the representation $g\to U_{g}$ is
irreducible and $|G|=n^{2}$ the unitary operators $(U_{g},\ g\in G)$ are
linearly independent. Hence $dim{\mathcal{L}}=n$ and $G\ni g\to({\rm I}\otimes
U_{g})|_{\mathcal{L}}$ is the irreducible projective unitary representation of
$G$ in $\mathcal{L}$. Moreover, $U_{g}\to(I\otimes U_{g})|_{\mathcal{L}}$
defines a unitary representation of the group ${\mathcal{U}}_{G}$. Since the
representation $U_{g}\to I\otimes U_{g}$ of the group ${\mathcal{U}}_{G}$ can
be represented as the orthogonal sum of $n$ irreducible representations each
of which is equivalent to ${\mathcal{U}}_{G}$ and the resolution of a
reducible representation in the sum of orthogonal irreducible representations
is unique up to the permutation of terms [8] we get that the group generated
by $(I\otimes U_{g})|_{\mathcal{L}},\ g\in G,$ is unitarily equivalent to
${\mathcal{U}}_{G}$. Hence, the channel
$Id\otimes\Phi:\mathfrak{S}({\mathcal{L}})\to\mathfrak{S}({\mathcal{L}})$ has
the same characteristics as $\Phi$.

$\Box$

Proposition 5. Let $\Phi:\mathfrak{S}(H)\to\mathfrak{S}(H)$ and
$\Omega:\mathfrak{S}(K)\to\mathfrak{S}(K)$ be two channels of the form (14),
$\Phi$ satisfies the conditions of Propositions 1, 2 and there exists the
probability distribution $q=(q_{j})$ such that the eigenvalues
$\lambda=(\lambda_{j})$ of $\Omega(\ket{e}\bra{e})$ obeys the condition

$\lambda^{\downarrow}\prec q$

for any choice of unit vector $e\in K$. Then, the eigenvalues $\mu=(\mu_{j})$
of $(\Phi\otimes\Omega)(\ket{f}\bra{f})$ satisfies

$\mu^{\downarrow}\prec(pq)^{\downarrow}$

for any choice of unit vector $f\in H\otimes K$.

Proof.

Given a unit vector $f\in H\otimes K$ denote $\lambda^{f}=(\lambda_{j}^{f})$
the eigenvalues of $(Id\otimes\Omega)(\ket{f}\bra{f})$ arranged in the
descending order. It follows from Proposition 4 that

$\lambda^{f}\prec q^{\oplus dimH}.$

Denote $\Pi=(\Pi_{j})$ the probability distribution $(p_{k}q_{m})$ arranged in
the descending order,

$\Pi_{0}\geq\Pi_{1}\geq\dots\geq\Pi_{ndimK-1}.$

Given a unit vector $f\in H\otimes K$ let us consider

$(\Phi\otimes\Omega)(\ket{f}\bra{f})=\sum\limits_{[g_{j}]\in G/T,t\in
T}\pi_{g_{j}t}(U_{g_{j}t}\otimes
I_{K})((Id\otimes\Omega)(\ket{f}\bra{f}))(U_{g_{j}t}^{*}\otimes I_{K}).$

Notice that

$\sum\limits_{[g_{j}]\in G/T,t\in T}(U_{g_{j}t}\otimes
I_{K})((Id\otimes\Omega)(\ket{f}\bra{f}))(U_{g_{j}t}^{*}\otimes I_{K})=n{\rm
I}_{H}\otimes\Omega(Tr_{H}(\ket{f}\bra{f})).$ (23)

The identity (23) implies that

$\sum\limits_{[g_{j}]\in G/T,t\in T}\braket{h_{m},(U_{g_{j}t}\otimes
I_{K})(Id\otimes\Omega)(\ket{f}\bra{f})(U_{g_{j}t}^{*}\otimes I_{K})h_{m}}\leq
n.$ (24)

It follows from Proposition 4 that

$\sum\limits_{m=0}^{k-1}\braket{h_{m},(U_{g_{j}t}\otimes
I_{K})((Id\otimes\Omega)(\ket{f}\bra{f}))(U_{g_{j}t}^{*}\otimes
I_{K})h_{m}}\leq\sum\limits_{m=0}^{k-1}q_{m}^{\oplus dimH}$ (25)

for any choice of $k$ pairwise orthogonal unit vectors $h_{m}\in H\otimes K$.
Thus, we obtain

$\sum\limits_{m=0}^{k-1}\braket{h_{m},(\Phi\otimes\Omega)(\ket{f}\bra{f})h_{m}}=$
$\sum\limits_{m=0}^{k-1}\sum\limits_{[g_{j}]\in G/T,t\in
T}\pi_{g_{j}t}(U_{g_{j}t}\otimes
I_{K})((Id\otimes\Omega)(\ket{f}\bra{f}))(U_{g_{j}t}^{*}\otimes I_{K})\leq$
$\sum\limits_{m=0}^{k-1}\Pi_{m},\ 0\leq k\leq ndimK,$

where we have combined (24), the majorization condition (16) and (25).

$\Box$

Proof of Theorem.

The value for one-shot capacity is immediately follows from Propositions 1, 2
and (9). Suppose that the conditions of Proposition 5 are satisfied for some
channel $\Omega$. Hence, given a unit vector $f\in H\otimes K$ for the
eigenvalues $\mu=(\mu_{j})$ of $(\Phi\otimes\Omega)(\ket{f}\bra{f})$ we obtain

$\mu^{\downarrow}\prec(pq)^{\downarrow}$ (26)

due to Proposition 5. Put $\Omega=\Phi$, then the channel $\Phi\otimes\Phi$
satisfies (26). Thus, we can apply the same procedure for $\Phi$ and
$\Omega=\Phi\otimes\Phi$. Denote $\Pi$ the probability distribution
$(\prod_{j=1}^{N}p_{s_{j}})$. Acting consistently, we get for the eigenvalues
$\lambda=(\lambda_{j})$ of $\Phi^{\otimes N}(\ket{f}\bra{f})$ with an
arbitrary unit vector $f\in H^{\otimes N}$

$\mu^{\downarrow}\prec\Pi^{\downarrow}.$

Hence, the minimal output entropy

$\inf\limits_{\rho\in{\mathfrak{S}}(H^{\otimes
N})}(\Phi(\rho))=-N\sum\limits_{j\in{\mathbb{Z}}_{n}}p_{j}\log p_{j}$

due to (9). Now the result follows from (5).

$\Box$

##  7 Construction of representations.

In this section we give several examples of irreducible projective unitary
representations of Abelian and non-Abelian groups for which Theorem can be
applied. Surely there are many other examples, see e.g. [26].

Let $T={\mathbb{Z}}_{n}$ and $\mathcal{S}$ be a subgroup of the symmetric
group $S_{n}$. Notice that any finite group is isomorphic to a subgroup of
$S_{n}$ for some $n$ by the Caley theorem. There is a natural action of
$\mathcal{S}$ on $T$ considered as a set of $n$ elements which is determined
by permutations as

$\alpha_{s}(t)=s(t),\ s\in{\mathcal{S}},\ t\in T.$

Fix the set $E=(\chi_{0},\dots,\chi_{n-1})$ consisting of $n$ characters of
$T$,

$\chi_{j}(tt^{\prime})=\chi_{j}(t)\chi_{j}(t^{\prime}),\ t,t^{\prime}\in T,$

giving rise to $\chi_{j}(t)^{n}=1,\ j\in{\mathbb{Z}}_{n}$. Characters from $E$
can coincide. Let us define unitary operators $W_{t}\in{\mathcal{U}}(H)$ by
the formula

$W_{t}\ket{j}=\chi_{j}(t)\ket{j},\ j\in{\mathbb{Z}}_{n},\ t\in T.$ (27)

Consider the representation of $\mathcal{S}$ in $H$ determined as follows

$V_{s}\ket{j}=\ket{s(j)},\ j\in{\mathbb{Z}}_{n},\ s\in{\mathcal{S}}.$ (28)

The next statement immediately follows from Theorem.

Corollary. Suppose that (27) and (28) determine the irreducible projective
unitary representation of the group $G={\mathcal{S}}\times{\mathbb{Z}}_{n}$.
Then, if the condition (16) is satisfied the classical capacity of (14) is
given by the formula

$C(\Phi)=C_{1}(\Phi)=\log n+\sum\limits_{j\in{\mathbb{Z}}_{n}}p_{j}\log
p_{j}.$

Example 1. Qubit channels.

Here $G={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ and ${\mathcal{U}}_{G}$ is the
group generated by the Pauli matrices. Let us define the characters as follows
$\chi_{0}(0)=\chi_{0}(1)=\chi_{1}(0)=1,\ \chi_{1}(1)=-1$ and the action
$s(0)=1$. Any unital qubit channel $\Phi$ is determined by the parameters
$(\lambda_{x},\lambda_{y},\lambda_{z})$ giving the action on the Pauli
matrices [4]

$\Phi(\sigma_{x})=\lambda_{x}\sigma_{x},\
\Phi(\sigma_{y})=\lambda_{y}\sigma_{y},\
\Phi(\sigma_{z})=\lambda_{z}\sigma_{z}.$ (29)

Four numbers including in the probability distribution $\pi$ determining a
channel $\Phi$ of the form (14) can be arranged in the descending order. If
maximal one is related to the action $\sigma_{a}\cdot\sigma_{a}$,
$a\in\\{x,y,z\\}$ we can consider
$\tilde{\Phi}(\cdot)=\sigma_{a}\Phi(\cdot)\sigma_{a}$. Now the maximal value
in the distribution $\pi$ corresponds to the identical action. There are three
possibility for determining a unitary representation of the Abelian group
${\mathbb{Z}}_{2}$. These are $0\to{\rm I},\ 1\to\sigma_{a},\
a\in\\{x,y,z\\}$. For any of them we get the probabilities in the mixture
corresponding to (29) as follows

$\pi_{00}+\pi_{01}-\pi_{10}-\pi_{11}=\lambda_{x},$
$\pi_{00}+\pi_{10}-\pi_{01}-\pi_{11}=\lambda_{y},$
$\pi_{00}+\pi_{11}-\pi_{01}-\pi_{10}=\lambda_{z}.$

Taking the majorization procedure over the set
$\\{\pi_{00},\pi_{01},\pi_{10},\pi_{11}\\}$ we obtain

$C(\Phi)=C_{1}(\Phi)=\log
2+\frac{1+\lambda_{max}}{2}\log\frac{1+\lambda_{max}}{2}+\frac{1-\lambda_{max}}{2}\log\frac{1-\lambda_{max}}{2},$

where

$\lambda_{max}=\max(|\lambda_{x}|,|\lambda_{y}|,|\lambda_{z}|).$

Example 2. $G={\mathbb{Z}}_{n}\times{\mathbb{Z}}_{n}$ and ${\mathcal{U}}_{G}$
is the Heisenberg-Weyl group.

Let ${\mathcal{S}}=T={\mathbb{Z}}_{n}$ and the set $E$ consists of pairwise
different characters $\chi_{k},\ k\in{\mathbb{Z}}_{n}$

$\chi_{k}(j)=e^{\frac{2\pi kj}{n}},\ k,j\in{\mathbb{Z}}_{n}.$

Suppose that ${\mathbb{Z}}_{n}$ acts on ${\mathbb{Z}}_{n}$ as shifts modulo
$n$. Then, the unitary operators (27) and (28) generates the Heisenberg-Weyl
group. Here we give the example constructed in [7] for $n=3$. Put

$\pi_{00}=\frac{1}{4},\ \pi_{10}=\frac{1}{8},\ \pi_{20}=\frac{1}{8},$
$\pi_{01}=\frac{1}{8},\ \pi_{11}=\frac{1}{8},\ \pi_{21}=\frac{1}{12},$ (30)
$\pi_{02}=\frac{1}{12},\ \pi_{12}=\frac{1}{24},\ \pi_{22}=\frac{1}{24}$

and consider the channel

$\Phi(\rho)=\sum\limits_{k,m\in{\mathbb{Z}}_{3}}\pi_{km}V^{k}W^{m}\rho
W^{*m}V^{*k},\ \rho\in\mathfrak{S}(H).$

Denote

$p_{k}=\sum\limits_{m\in{\mathbb{Z}}_{3}}\pi_{km},\ k\in{\mathbb{Z}}_{3},$

then

$p_{0}=\frac{1}{2},\ p_{1}=\frac{1}{3},\ p_{2}=\frac{1}{6}.$

It follows from Theorem

$C(\Phi)=C_{1}(\Phi)=\log(3)-\frac{1}{2}\log(2)-\frac{1}{3}\log(3)-\frac{1}{6}\log(6).$
(31)

Example 3. $G={\mathbb{K}}_{4}\times{\mathbb{Z}}_{4}$.

Let ${\mathcal{S}}={\mathbb{K}}_{4}$ be the Klein group generated by a unit
$e$ and three elements $x,y,z$ satisfying the relations

$x^{2}=y^{2}=z^{2}=e,\ xy=yx=z.$

Let us define the action $s$ of ${\mathbb{K}}_{4}$ on ${\mathbb{Z}}_{4}$ by
the rule

$s_{x}(0)=1,\ s_{x}(1)=0,\ s_{x}(2)=3,\ s_{x}(3)=2,$ $s_{y}(0)=2,\
s_{y}(2)=0,\ s_{y}(1)=3,\ s_{y}(3)=1,$ $s_{z}(0)=3,\ s_{z}(3)=0,\ s_{z}(1)=2,\
s_{z}(2)=1.$

Put $\chi_{0}(1)=\chi_{1}(1)=\chi_{0}(2)=\chi_{2}(2)=1,\
\chi_{2}(1)=\chi_{3}(1)=\chi_{1}(2)=\chi_{3}(2)=-1$. Now we can define a
projective unitary representation of ${\mathbb{K}}_{4}\times{\mathbb{Z}}_{4}$
by the formula

$(h,k)\to V_{s_{h}}W_{k},\ h\in{\mathbb{K}}_{4},\ k\in{\mathbb{Z}}_{4}$ (32)

with $V_{s_{h}}$ and $W_{k}$ determined by (28) and (27).

Proposition 6. The representation (32) is irreducible.

Proof.

The commutant of $V_{s_{h}},\ h\in{\mathbb{K}}_{4}$ consists of matrices

$\begin{pmatrix}a&b&0&0\\\ b&a&0&0\\\ 0&0&a&b\\\ 0&0&b&a\end{pmatrix},$ (33)

where $a,b\in{\mathbb{C}}$. The matrices of the form (33) commuting with
$W_{k},\ k\in{\mathbb{Z}}_{4}$ are multiplies of identity.

$\Box$

Proposition 6 implies that the mixed unitary channel (14) with
$G={\mathbb{K}}_{4}\times{\mathbb{Z}}_{4}$ for which the majorization
condition is satisfied has the capacities given by Corollary.

Example 4. $G={\mathbb{D}}_{n}\times{\mathbb{Z}}_{2n}$.

Dihedral group ${\mathbb{D}}_{n}$ is generated by unit $e$ and two elements
$x,y$ satisfying the relation

$x^{n}=y^{2}=e,\ xy=yx^{-1}.$

While the groups ${\mathbb{D}}_{1}={\mathbb{Z}}_{2}$ and
${\mathbb{D}}_{2}={\mathbb{K}}_{4}$ are Abelian, all other groups
${\mathbb{D}}_{n},\ n>2,$ are non-Abelian. Let us define the action of
${\mathbb{D}}_{n}$ on ${\mathbb{Z}}_{2n}$ as follows

$s_{x}(j)=j+2\ mod\ 2n,\ 0\leq j\leq 2n,\ s_{y}(2j)=2j+1,\ 0\leq j\leq n-1.$
(34)

Put $\chi_{2j}(1)=e^{i\frac{2\pi j}{n}},\ \chi_{2j+1}(1)=-e^{i\frac{2\pi
j}{n}},\ j\in{\mathbb{Z}}_{n}$. Define the irreducible projective unitary
representation of $G={\mathbb{D}}_{n}\times{\mathbb{Z}}_{2n}$ by the formula

$(h,k)\to V_{s_{h}}W_{k},\ h\in{\mathbb{D}}_{n},\ k\in{\mathbb{Z}}_{2n}$ (35)

with $V_{s_{h}}$ and $W_{k}=W_{1}^{k}$ determined by (28) and (27).

Proposition 7. The representation $G$ defined by (35) is irreducible.

Proof.

The Abelian group $W_{k},\ k\in{\mathbb{Z}}_{2n}$ generates the maximal
Abelian algebra $\mathcal{A}$ consisting of all operators that are diagonal in
the basis $\ket{j}$ because all eigenvalues of $W_{1}$ are pairwise different.
Applying to $x\in\mathcal{A}$ the unitary operators $V_{s_{h}}$ we see that
$V_{s_{h}}xV_{s_{h}}^{*}=x$ for all $h\in{\mathbb{D}}_{n}$ only if $x=c{\rm
I}$ for some $c\in{\mathbb{C}}$.

$\Box$

It follows from Proposition 7 that if the majorization condition (16) is
satisfied the classical capacity of the channel (14) is given by Corollary.

##  8 Conclusion

We have studied the mixed unitary channels generated by irreducible projective
unitary representations of finite groups. We have calculated the capacity of
such channels under some additional assumptions about the structure of the
group and the probability distribution generating a mixture. It is shown that
our assumptions mean that the channel can be considered as a perturbation of
quantum-classical channel of the same class. The proof is based upon the
majorization theory and the theory of projective unitary representations for
finite groups. We provide the text with examples illustrating the techniques
for representations of Abelian as well as non-Abelian groups.

## Acknowledgments

The author is grateful to A.S. Holevo, A.V. Vasilev and I.Yu. Zhdanovskii for
many fruitful discussions.

## References

  * [1] Holevo A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Inform. Theory 44(1), 269–273 (1998)
  * [2] Schumacher B., Westmoreland M.D.: Sending classical information via noisy quantum channels. Physical Review A 56, 131 (1997)
  * [3] Hastings M.B.: A counterexample to additivity of minimum output entropy. Nature Physics 5, 255–257 (2009)
  * [4] King C.: Additivity for unital qubit channels, J. Math. Phys. 43, 4641-4653 (2002)
  * [5] King C.: The capacity of the quantum depolarizing channel. IEEE Trans. Inform. Theory 49, 221-229 (2003)
  * [6] Holevo A.S.: Quantum Systems, Channels, Information. De Gruyter (2019)
  * [7] Amosov G.: On classical capacity of Weyl channels. Quantum Information Processing 19, 401 (2020)
  * [8] Mari A., Giovannetti V., Holevo A.S.: Quantum state majorization at the output of bosonic Gaussian channels. Nature Communications 5, 3826 (2014)
  * [9] Holevo A.S.: On the proof of the majorization theorem for quantum Gaussian channels. Russian Math. Surveys 71(3), 585–587 (2016)
  * [10] ur Rehman J., Jeong Y, Kim J.S., Shin H.: Holevo capacity of discrete Weyl channels. Scientific Reports 8, 17457 (2018)
  * [11] Siudzińska K.: Classical capacity of generalized Pauli channels. J. Phys. A: Math. Theor. 53(44), 445301 (2020)
  * [12] Karamata J.: Sur une inegalite relative aux fonctions convexes. Publications Math. de l’Universite de Belgrade 1 145–148 (1932)
  * [13] Amosov G.G.: Remark on the additivity conjecture for a quantum depolarizing channel. Problems Inform. Transmission 42(2), 69–76 (2006)
  * [14] Amosov G.G.: On Weyl channels being covariant with respect to the maximum commutative group of unitaries. J. Math. Phys. 48(1), 012104 (2007)
  * [15] Amosov G.G.: Strong superadditivity conjecture holds for the quantum depolarizing channel in any dimension. Phys. Rev. A 75(6), 060304(R) (2007)
  * [16] Amosov G.G., Mancini S.: The decreasing property of relative entropy and the strong superadditivity of quantum channels. Quantum Inf. Comput. 9(7-8), 594–609 (2009)
  * [17] Amosov G.G., Zhdanovskii I.Yu.: Structure of the algebra generated by a noncommutative operator graph which demonstrates the superactivation phenomenon for zero-error capacity. Math. Notes 99(6), 924–927 (2016)
  * [18] Amosov G.G., Zhdanovskii I.Yu.: On the noncommutative deformation of the operator graph corresponding to the Klein group. J. Math. Sci. (N. Y.) 215(6), 659–676 (2016)
  * [19] Shirokov M.E.: On multipartite superactivation of quantum channel capacities. Problems Inform. Transmission 51(2), 87–102 (2015)
  * [20] Shirokov M.E.: On channels with positive quantum zero-error capacity having vanishing n-shot capacity. Quantum Inf. Process. 14(8), 3057–3074 (2015)
  * [21] Shirokov M.E.: On quantum zero-error capacity. Russian Math. Surveys 70(1), 176–178 (2015)
  * [22] Bhatia R.: Matrix analysis. Springer (1996)
  * [23] Amosov G.G., Holevo A.S.: On the multiplicativity hypothesis for quantum communication channels. Theory Probab. Appl. 47(1), 123–127 (2003)
  * [24] Hall M.: The Theory of Groups. MacMillan, New York (1957)
  * [25] Hamermesh M.: Group Theory and its applications to physical problems. Addison Wesley Publishing Company, Inc., Reading, Massachusetts, USA, London (1962)
  * [26] Janssen T.: On projective unitary-antiunitary representations of finite groups. J. Math. Phys. 13, 342 (1972)