数理解析研究所講究録
第 1565 巻 2007 年 169-178

169

On Entropies of Quantum Communication
Processes
Noboru Watanabe

Department of Information Sciences,
Tokyo University of Science
Noda City, Chiba, 278-8510, Japan
E-mail: watanabe@is.noda.tus.ac.jp
Abstract
The mutual entropy (information) denotes an amount of informationtransmitted correctly from the input system to the output system througha channel. The (semi-clas ical) mutual entropies for classical input andquantum output were defined by several researchers. The fully quantum
mutual entropy, which is called Ohya mutual entropy, for quantum input
and output by using the relative entropy was defined by Ohya in 1983.
In this paper, we compare with mutual entropy-type measures and show
some resuls for quantum capacity$s$1

Introduct ion

The development of communication
is closely connected with study ofentropy theory. The signal of the input system is carried through a physical device, which is called a channel. The mathematical representation of the channel
is a mapping ffom the input state space to the output state space. In classicalcommunication theory, the mutual entropy was formulated by using the joint
probability distribution between the input system and the output system. The(seni-classical) mutual entropies for classical input and quantum output were
defined by several researchers $[9, 10]$ . In fUlly quantum system, there does notexist the joint probability distribution in goneral. Instead of the joint probability distribution, Ohya [14] invented the quantum (Ohya) compound state,and he introduced the fully quantum mutual entropy (information), which is
caUed Ohya mutual entropy, for quantum input and output systerns, describes
the amount of information correctly sent from the quantum input system to thequantum output system through the quantum channel.
In this paper, we compare with mutual entropy-type measures and show
some resuls for quantum capacity for the attenuation channel.
$th\infty ry$$\mathcal{H}$

170

2

Quantum Channels

The concept of channel has been carried out an important role in the progressof the quantum communication. In particular, an attenuation channel
introduced in [14] is one of the most inportaint model for discussing the information transmission in quantum optical communication. Here we review thedefinition of the quantum channels.
be the complex separable Hilbert spaces of an input and anLetoutput systems, respectively, and let
be the set of all bounded linear
operators on. We denote the set of all density operators on$(k=1,2)$by$th\infty ry$$\mathcal{H}_{1},\mathcal{H}_{2}$$B(\mathcal{H}_{k})$$\mathcal{H}_{k}$$\mathcal{H}_{k}$$\mathfrak{S}(\mathcal{H}_{k})\equiv\{\rho\in B(\mathcal{H}_{k});\rho\geq 0,tr\rho=1\}$

.

(1)

A map
from the quantum input system to the quantum output system is$cag_{ed}$a (fully) quantum channel.
$\Lambda^{*}$

1.

$\Lambda^{*}$

is $caUd$ a linear channel if it atisfies the affine property, i.e.,
$s$

1

$=$
$\sum_{k}\lambda_{k}$$\Lambda^{*}(\sum_{k}\lambda_{k}\rho_{k})=\sum_{k}\lambda_{k}\Lambda^{*}(\rho_{k}),\forall\rho_{k}\in \mathfrak{S}(\mathcal{H}_{1})$$\Rightarrow$2.

$(\forall\lambda_{k}\geq 0)$.

:
is called a completely positive (CP) channel
if its dual map A satisfies$\Lambda^{*}$$\mathfrak{S}(\mathcal{H}_{1})arrow \mathfrak{S}(\mathcal{H}_{2})$(2)$\sum_{j,k=1}^{n}B_{j}^{*}\Lambda(A_{j}^{*}A_{k})B_{k}\geq 0$

for any $n\in N$ , any
map:

and any, where the dualofsatisfies $tr\rho\Lambda(A)=tr\Lambda^{*}(\rho)$ $A$ for any$B_{j}\in B(\mathcal{H}_{1})$$A_{k}\in B(\mathcal{H}_{2})$$\Lambda:B(\mathcal{H}_{2})arrow B(\mathcal{H}_{1})$$\Lambda^{*}$$\mathfrak{S}(\mathcal{H}_{1})arrow \mathfrak{S}(\mathcal{H}_{2})$and any

2.1$A\in B(\mathcal{H}_{2})$.

$\rho\in \mathfrak{S}(\mathcal{H}_{1})$Quantum Communication ProcessLetandbe two Hilbert spaces expressing noise and loss systems, respectively. Quantum communication process including the influence of noise andloss is described by the quantum channel [14]:$\mathcal{K}_{1}$$\mathcal{K}_{2}$$\Lambda^{*}(\rho)\equiv tr_{\mathcal{K}_{2}}\pi^{*}(\rho\otimes\xi)$for any input state in
and a noise stateis a CP channel fromtoproperties of the communication device.
$\rho$$\pi^{*}$$\mathfrak{S}(\mathcal{H}_{1})$$\mathfrak{S}(\mathcal{H}_{1}\otimes \mathcal{K}_{1})$

$\xi$

in

$\mathfrak{S}(\mathcal{H}_{2}\otimes \mathcal{K}_{2})$, where the map
determined by physical$\mathfrak{S}(\mathcal{K}_{1})$171

3Ohya Mutual Entropy and Capacity
The quantum entropy was introduced by von Neumann around 1932 [13], which
is defined by$S(\rho)\equiv-trp$log$\rho$. It denotes the amount of information of
for any density operators in
the quantum state .
In order to define such a quantum mutual entropy, we need the quantum
relative entropy and the joint state, which is called a compound state, describing
through a
the correlation between an input state and the output state,. For a statechannel$\rho$$S(\mathcal{H}_{1})$$\rho$$\Lambda^{*}\rho$$\rho$$\Lambda^{*}$$\rho\in \mathfrak{S}(\mathcal{H}_{1})$$\rho=\Sigma_{k}\lambda_{k}E_{k}$,(3)ensional
is theis called a Schatten decomposition [24] of , where
decomposition
is not.
projection
orthogonal
associated withThe Schatten
unique usually depending on a degeneracy of the eigenvalue of p.For, the compound states are define by:
and$on\triangleright d\dot{m}$$E_{k}$$\rho$$\lambda_{k}$$\rho\in \mathfrak{S}(\mathcal{H}_{1})$$\Lambda^{*}$$\mathfrak{S}(\mathcal{H}_{\sim}\backslash )arrow \mathfrak{S}(\mathcal{H}_{2})$$\sigma_{E}=\sum_{n}\lambda_{n}E_{n}\otimes\Lambda^{*}E_{n}$,$\sigma_{0}=\varphi\otimes\Lambda^{*}\varphi$.

(4)

The first compound state is called a Ohya compund state associating with the, which generalizes the joint probability in
Schatten docomposition
classical dynamical system and it shows a certain correlation between the initial.
state and the final state
Ohya mutual entropy with respect to andis defined by$\rho=\Sigma_{k}\lambda_{k}E_{k}$$\Lambda^{*}\rho$$\rho$$\Lambda^{*}$$\rho$$I( \rho;\Lambda^{*})\equiv\sup\{S(\sigma_{E},\sigma_{0});E=\{E_{n}\}\}$

whereis Umegaki’s relative entropy [25].
non’s type inequality :$S(\sigma_{E}, \sigma_{0})$$I(\rho;\Lambda^{*})$$0 \leq I(\rho, \Lambda^{*})\leq\min\{S(\rho),S(\Lambda^{*}\rho)\}$

3.1

,

(5)

satisfies the Shan-.

Quantum Capacity

The capacity means the ability of the information transmission of the channel,
which is used as a measure for construction of channels. The quantum capacity
is formulated by taking the supremum of the Ohya mutual entropy with respect
to a certain subset of the initial state space. The capacity of quantum channel
was studied in [17, 18, 19, 20].
Let $S$ be the set of all input states $satis\theta ing$ some physical conditions. Let
us consider the ability of information transmission for the quantum channel .
The answer of this question is the capacity of quantum channel
for a certain$\Lambda^{*}$$\Lambda^{*}$

set$S\subset S(\mathcal{H}_{1})$defined by$C_{q}^{S}( \Lambda^{*})\equiv\sup\{I(p;\Lambda^{*});p\in S\}$When

$S=S(\mathcal{H}_{1})$, the capacity of quantum channel$\Lambda^{*}$.

(6)
is denoted by$C_{q}(\Lambda$

“ $)$

.

172

4

Quantum Entropy for Quantum Communication Channels

4.1

Attenuation channel

One of the example of the quantum communication channels is the attenuation
introduced by Ohya [14], which is defined bychannel$\Lambda_{0}^{*}$$\Lambda_{0}^{*}(p)$where$|0\rangle$$\mathcal{H}_{2}\otimes \mathcal{K}_{2}$iii$tr_{\mathcal{K}_{2}}\pi_{0}^{*}(p\otimes\xi_{0})$is vacuum state ingiven by$\langle 0|$,$\mathcal{H}_{1}$$\xi_{0}\equiv|0\rangle$

and$V_{0}$\langleand$0|$$\pi_{0}^{*}(\cdot)\equiv V_{0}(\cdot)V_{0}^{l}$is a linear mapping &om,(7)$\mathcal{H}_{1}\otimes \mathcal{K}_{1}$

to

(8)

$V_{0}(|n \rangle\otimes|0\rangle)\equiv\sum_{j=0}^{n}\sqrt{\frac{n!}{j!(n-j)!}}\alpha^{j}\overline{\beta}^{n-j}|j\rangle\otimes|n-j\rangle$

for anyinandare complex numbers $satis\theta ing|\alpha|^{2}+|\beta|^{2}=1$ .
, which is the transmission rate of the channel.
is called a beam splittings,
which means that one beam comes and two beams appear after passing through. The attenuation channel is generalized by the noisy optical channel $[20, 21]$ ,which is also reformulated by Accardi and Ohya [1] using the liftinge. The
noisy optical channel consists of the generaliezed beam splittings , which was
extended on generalized Fock space by Fichtner, Freudenberg and Libsher [6]by means of the concept of compound Hida-Malliavin derivative [8] and so on.
For the attenuation channel , one can obtain the following theorem provedin [22].
$|n\rangle$$\mathcal{H}_{1}$$\alpha,\beta$$\eta=$$|\alpha|^{2}$$\pi_{0}^{*}$$\pi_{0}^{*}$$\pi^{*}$$\Lambda_{0}^{*}$

Theorem 1 For a subset{
auenmtion channel
satisfies$S_{n}\equiv$$\rho\in S(\mathcal{H}_{1})$; dim 8 ($\rho)=n$ }, the capa city of the$\Lambda_{0}^{*}$$C_{q}^{S_{n}}(\Lambda_{0}^{*})=\log n$

where$s(\rho)$is the support rojection of$p$$\rho$,.

When the mean energy of the input state vectors
can be taken$th\inftyrem$, the above
infinite, i.e.,tells that the quantum
capacity for the attenuation channel
with respect tobecomes log . It isa natural result, however it is impossible to take the mean energy of input statevector infinite.
$\{|\tau\theta_{k}\rangle\}$$\lim_{\tauarrow\infty}|\tau\theta_{k}|^{2}=\infty$$S_{n}$$\Lambda_{0}^{*}$

4.2

$n$

Quantum Teleportation Channel

In usual quantum communication processes, quantum states are transmitted
from input system to an output system through a channel. Since the channel
representing a physical apparatus is affected by outside system, the final state
sent through the channel is different from the itial state. On the contrary, in$\dot{i}$173
telePortation scheme, a particle is not transmitted from Alice’s system to Bob’s

system, but one can reconstruct the initial state by means of the entangledstate located at Bob’s system. Bennett et al. $[2, 3]$ proposed a state change,
so-called a quantum teleportation, in terms of EPR entangled state denotedby Bell’s base. It is difficult to realize such telePortation scheme because the
EPR entangled state dissipates easily. In order to avoid this demerit, Ohya td
Fichtner $[4, 5]$ introduced a new teleportation scheme on boson-Fock space bymeans of the entangled coherent states and the general beam splitting.
In perfect quantum telePortation scheme, the total Process of the quantum
teleportation is consist of three systems denoted by the complex 正駈 lbert spaces. Alice controls. Alice has an unknown
and Bob treats
teleports
initial stateand sheit to Bob by using an entangled
belonging to Alice and Bob. At first, Alice measures
state
by means of an observablefor the Part ofin the state, whereis a set of orthogonal projections$\mathcal{H}_{1},$$\mathcal{H}_{2},$$\mathcal{H}_{3}$$\mathcal{H}_{1},\mathcal{H}_{2}$$\mathcal{H}_{3}$$\rho^{(1)}\in \mathfrak{S}(\mathcal{H}_{1})$$\sigma^{(23)}\in \mathfrak{S}(\mathcal{H}_{2}\otimes \mathcal{H}_{3})$$\{p_{lm}^{(12)}\}$$\sum_{l,m}z_{lm}F_{lm}^{(12)}\in \mathbb{B}(\mathcal{H}_{1}\otimes \mathcal{H}_{2})$. If Alice obtains a vaIue
is changed to$F_{lm}^{(12)}\in \mathbb{B}(\mathcal{H}_{1}\otimes \mathcal{H}_{2})$state$\rho^{(1)}\otimes\sigma^{(23)}$$F^{(12)}\equiv$$\rho^{(1)}\otimes\sigma^{(23)}$$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$$z_{lm}$after measurement, then the$\rho_{lm}^{(123)}=\frac{(F_{lm}^{(12)}\otimes I_{3})(p^{(1)}\otimes\sigma^{(23)})(F_{lm}^{(12)}\otimes I_{\partial})}{tr_{12}s(F_{lm}^{(12)}\otimes I_{3})(\rho^{(1)}\otimes\sigma^{(23)})(F_{lm}^{(12)}\copyright I_{3})}$.

(9)

Alice send the resultof the measurement to Bob through a classical communication channel. Bob reconstructs the unknown initial state
fromby applying a unitary key
created by thevaluereceived ffom Alice. The total process of the perfect teleportation
scheme is described by$z_{lm}$$\rho^{(1)}$$\Lambda_{lm}^{*}(\rho^{(1)})\equiv tr_{12}p_{lm}^{(123)}\in \mathfrak{S}(\mathcal{H}_{3})$$U_{lm}$$z_{lm}$$\rho^{(1)}\cong U_{lm}(\Lambda_{lm}^{*}(\rho^{(1)}))U_{lm}^{*}$

.

(10)

In the perfect teleportation scheme, Bob can reconstruct the unknown initialstate
by applying the unitary key once. On the contrary, in the non-perfect
teleportation scheme, Bob can obtain the unknown initial state
by applying
the unitary operator
more than twice. Applying the unitary operator
once to the state, the state change of the non-perfect teleportation
scheme is denoted by$\rho^{(1)}$$\rho^{(1)}$$V_{nm}$$V_{nm}$$\Lambda_{nm}^{l}(p^{(1)})$$—*nm(\rho^{(1)})\equiv V_{nm}\Lambda_{nm}^{*}(\rho^{(1)})V_{nm}^{*}$

.

(11)

For the perfect teeportation, we have the following theorem [11]:Theorem 2 If the perfect teleportation channel is lnear and the input stateis finite rank operator, then one can obtain$I(\rho^{(1)},\Lambda_{T}^{*})=S(\rho^{(1)})$for$\rho^{(1)}$

174
4.3

Quantum channel for $Ekedkin-Toffoli$-Milburn gate

In usual computer, we could not determine two inputs for the logical gates AND
and OR after we know the output for these gates. This property is called anirreversibility of logical gate. This property leads to the loss of information andthe heat generation. Thus there exists an upper bound of computational speed.
Fredkin and Toffoli proposed a conservative gate, by which any logical gate
is realized and it is shown to be a reversible gate in the sense that there is no lossof information. This gate was developed by Milburn as a quantum gate withquantum input and output. We call this gate Redkin-Toffoli-Milburn (FTM)
gate. Recently, we reformulate a quantum channel for the FTM gate and werigorously study the conservation of information for FTM gate [23].
The FTM gate is composed of two input gatesand one control gate
C. Two inputs come to the first beam splitter and one spliting input passes$throug_{1}$ the control gate made from an optical Kerr device, then two spliting
inputs come in the second beam splitter and appear as two outputs (Fig.2.1).
Two beam splitters and the optical Kerr medium are needed to describe thegate.
$I_{1},$$I_{l}$$I_{2}$$0$$F$h) Beam splitters: (a) Based on [19], let$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$with transmission rate$\eta_{1}$given by$V_{1}$

be a mapping from

$V_{1}(|n_{1} \rangle\otimes|n_{2}\rangle)\equiv\sum_{j=0}^{n_{1}+n_{3}}C_{j}^{n_{1},n_{2}}|j\rangle\otimes|n_{1}+n_{2}-j\rangle$$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$

to

(12)

for any photon number state vectors. The quantum
expressing the first beam splitter (beam splitter 1) is defined bychannel$|n_{1}\rangle$$\Pi_{BS1}^{*}$$\otimes|n_{2}\rangle$$\in \mathcal{H}_{1}\otimes \mathcal{H}_{2}$

175
(13)

$\Pi_{BS1}^{*}(\rho_{1}\otimes p_{2})\equiv V_{1}(\rho_{1}\otimes p_{2})V_{1}^{*}$

for any states
gates

and

$I_{1}$$\rho_{1}\otimes\rho_{2}\in \mathfrak{S}(\mathcal{H}_{1}\otimes \mathcal{H}_{2})$$I_{2}$. In particular, for an input state in two

given by the tensor product of two coherent statesis written as$|\theta_{1}\rangle\langle\theta_{1}|\otimes|\theta_{2}\rangle\langle\theta_{2}|,$$\rho_{1}\otimes p_{2}=$$\Pi_{BS1}^{*}(p_{1}\otimes\rho_{2})$$\Pi_{BS1}^{*}(p_{1}\otimes\rho_{2})$$|\sqrt{\eta_{1}}\theta_{1}+\sqrt{1-\eta_{1}}\theta_{2}\rangle\langle\sqrt{\eta_{1}}\theta_{1}+\sqrt{1-\eta_{1}}\theta_{2}|$$\otimes|-\sqrt{1-\eta_{1}}\theta_{1}+\sqrt{\eta_{1}}\theta_{2}\rangle\langle-\sqrt{1-\eta_{1}}\theta_{1}+\sqrt{\eta_{1}}\theta_{2}|$(b) Let$\dot{g}ven$$V_{2}$

be a mapping$hom\mathcal{H}_{1}\otimes \mathcal{H}_{2}$to$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$

by

.

(14)

with transmission rate$\eta_{2}$(15)$V_{2}(|n_{1} \rangle\otimes|n_{2}\rangle)\equiv\sum_{j=0}^{n_{1}+n_{2}}C_{j}^{n_{2},n\iota}|n_{1}+n_{2}-j\rangle\otimes|j\rangle$

for any photon number state vectors. The quantum
channelexpressing the second beam splitter (beam splitter 2) is defined$|n_{1}\rangle$$\otimes|n_{2}\rangle$$\in \mathcal{H}_{1}\otimes \mathcal{H}_{2}$$\Pi_{BS2}^{*}$

by

(16)

$\Pi_{BS2}^{*}(p_{1}\otimes p_{2})\equiv V_{2}(p_{1}\otimes\rho_{2})V_{2}^{*}$

for any states$\rho_{1}\otimes\rho_{2}\in \mathfrak{S}(\mathcal{H}_{1}\otimes \mathcal{H}_{2})$$p_{1}\otimes\rho_{2}=|\theta_{1}\rangle\langle\theta_{1}|\otimes|\theta_{2}\rangle\langle\theta_{2}|,$$\Pi_{BS2}^{*}(\rho_{1}\otimes p_{2})$$=$. In particular, for coherent input statesis written as$\Pi_{BS2}^{*}(\rho_{1}\otimes\rho_{2})$$|\sqrt{\eta_{2}}\theta_{1}-\sqrt{1-\eta_{2}}\theta_{2}\rangle\langle\sqrt{\eta_{2}}\theta_{1}-\sqrt{1-\eta_{2}}\theta_{2}|$$\otimes|\sqrt{1-\eta_{2}}\theta_{1}+\sqrt{\eta_{2}}\theta_{2}\rangle\langle\sqrt{1-\eta_{2}}\theta_{1}+\sqrt{\eta_{2}}\theta_{2}|$. (17)(2) Optical Kerr medium: The interaction Hamiltonian in the optical Kerr
medium is given by the number operators
andfor the input system 1
and the Kerr medium, respectively, such as$N_{1}$$H_{int}=\hslash\chi(N_{1}\otimes I_{2}\otimes N_{a})$$N_{c}$,(18)

where is the Plank constant divided by
is a constant proportional to thesusceptibility of the Kerr medium and. Let $T$is the identity operator on
be the passing time of a beam through the Kerr medium and put $\sqrt{F}=\hslash\chi T$,
a parameter exhibiting the power of the Kerr effect. Then the unitary operator
describing the evolution for time $T$ in the Kerr medium is given by$\hslash$$2\pi,$$\chi$$I_{2}$$\mathcal{H}_{2}$$U_{K}$$U_{K}=\exp(-i\sqrt{F}(N_{1}\otimes I_{2}\otimes N_{c}))$

.

(19)

176
We assume that an initial (input) state of the control gate is thephoton\langle , a quantum channel
number state
representing the optical Kerr
effect is given by$n$$\xi=|n\rangle$$n|$$\Lambda_{K}^{*}$(20)$\Lambda_{K}^{*}(p_{1}\otimes p_{2}\otimes\xi)\equiv U_{K}(p_{1}\otimes p_{2}\otimes\xi)U_{K}^{l}$for any state$p_{1}\otimes\rho_{2}\otimes\xi\in \mathfrak{S}(\mathcal{H}_{1}\otimes \mathcal{H}_{2}\otimes \mathcal{K})$$p_{1}\otimes p_{2}\otimes\xi=|\theta_{1}\rangle\langle\theta_{1}|\otimes|\theta_{2}\rangle\langle\theta_{2}|\otimes|n\rangle\langle n|,$. In particular, for an initial state$\Lambda_{K}^{*}(\rho_{1}\otimes p_{2}\otimes\xi)$is denoted by$\Lambda_{K}^{*}(p_{1}\otimes\rho_{2}\otimes\xi)$

$|$

exp

$(-i\sqrt{F}n)\theta 1\rangle\langle$$\otimes|\theta_{2}\rangle\langle\theta_{2}|\otimes|n\rangle\langle n|$

exp $(-i\sqrt{F}n)\theta 1|$

,

(21)

Using the above channels, the quantum channel for the whole FTM gate isconstructed as follows: Let both one input and output gates be described by, another input and output gatos be described byand the control gate
, all of which are Fock spaces. For a total state
be done byof
two input states and a control state, the quantum channels
fromare written byto$\mathcal{H}_{1}$$\mathcal{H}_{2}$$\mathcal{K}$$\rho_{1}\otimes\rho_{2}\otimes\xi$$\Lambda_{BS1}^{*},$$\Lambda_{BS2}^{*}$$\mathfrak{S}(\mathcal{H}_{1}\otimes \mathcal{H}_{2}\otimes \mathcal{K})$$\mathfrak{S}(\mathcal{H}_{1}\otimes \mathcal{H}_{2}\otimes \mathcal{K})$$\Lambda_{BSk}^{*}(p_{1}\otimes\rho_{2}\otimes\xi)=\Pi_{BSk}^{*}(\rho_{1}\otimes\rho_{2})\otimes\xi$

Therefore, the whole quantum channel$\Lambda_{FTM}^{*}$of the FTM gate is defined by

.

(23)

$p_{1}\otimes\rho_{2}\otimes\xi=|\theta_{1}\rangle$$\langle\theta_{1}|\otimes|\theta_{2}\rangle\langle\theta_{2}|\otimes|n\rangle\langle n|$$\Lambda_{FTM}^{*}\equiv\Lambda_{BS2}^{l}0\Lambda_{K}^{*}0\Lambda_{BS1}^{*}$

In particular, for an initial state
is obtained by(22)$(k=1,2)$,$\Lambda_{FTM}^{*}(\rho_{1}\otimes p_{2}\otimes\xi)$$\Lambda_{FTM}^{*}(\rho_{1}\otimes\rho_{2}\otimes\xi)$$=$$|\mu_{n}\theta_{1}+\nu_{n}\theta_{2}\rangle$$\langle\mu_{n}\theta_{1}+\nu_{n}\theta_{2}|$$\otimes|\nu_{n}\theta_{1}+\mu_{n}\theta_{2}\rangle(\nu_{n}\theta_{1}+\mu_{n}\theta_{2}|\otimes|n\rangle$\langle

(24)

$n|$

where
$\mu_{n}$$=$$\frac{1}{2}\{\exp(-i\sqrt{F}n)+1\}$,$\nu_{n}$$=$$\frac{1}{2}\{\exp(-i\sqrt{F}n)-1\}$,(25)$(k=0,1,2, \cdots)$

.

(26)

Then we obtain the following $th\infty rem$ :$Th\infty rem3$

If$\sqrt{F}$satisfies the conditions $\sqrt{F}n=0$ or $\sqrt{F}n=(2m+1)\pi$$(m=0,1,2, \cdots)$ , then one can obtain$I(\rho_{1}\otimes\rho_{2},\tilde{\Lambda}_{FTM}^{*})=S(\rho_{1})+S(\rho_{2})$177
for input states$p_{1}\otimes\rho_{2}$given by$\rho_{i}=\lambda_{i}|0\rangle\langle 0|+(1-\lambda_{i})|x_{i}\rangle\langle x_{i}|\in \mathfrak{S}(\mathcal{H}_{i})$where$|x_{i}\rangle$,$\lambda_{i}\in[0,1]$,$(i=1,2)$ ,is defined by.
$|x_{i} \rangle=\frac{|\theta_{i}\rangle-|-\theta_{i}\rangle}{\sqrt{2(1+\exp(-\frac{1}{2}|\theta_{i}|^{2}))}}$

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