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44
UDC 004.38:530.145; MSC: 81P99; 81V80, 94A50
NOTE ON COMPLEXITY OF QUANTUM TRANSMISSION PROCESSES
N. Watanabe
Science University of Tokyo, Noda City, Chiba 278-8510, Japan.
E-mail: watanabe®is .noda. tus .ac.jp
In 1989, Ohya propose a new concept, so-called Information Dynamics (ID), to investigate complex systems according to two kinds of view points. One is the dynamics of state change and another is measure of complexity. In ID, two complexities Cs and Ts are introduced,. Cs is a measure for complexity of system itself, and Ts is a measure for dynamical change of states, which is called a transmitted complexity. An example of these complexities of ID is entropy for information transmission processes.
The study of complexity is strongly related to the study of entropy theory for classical and, quantum systems. The quantum entropy was introduced by von Neumann around 1932, which describes the amount of information of the quantum state itself. It was extended, by Ohya for C*-systems before CNT entropy. The quantum relative entropy was first defined by Umegaki for a-finite von Neumann algebras, which was extended, by Araki and, Uhlmann for general von Neumann algebras and, *-a,lgebra,s, respectively.
By introducing a new notion, the so-called compound state, in 1983 Ohya succeeded to formulate the mutual entropy in a complete quantum mechanical system (i.e., input state, output state and, channel are all quantum mechanical) describing the amount of information correctly transmitted through the quantum channel. In this paper, we briefly review the entropie complexities for classical and, quantum systems. We introduce some complexities by means of entropy functionals in order to treat the transmission processes consistently. We apply the general frames of quantum communication to the Gaussian communication processes. Finally, we discuss about a construction of compound states including quantum correlations.
Key words: quantum communication channel, von Neumann entropy, S-mixing entropy, Ohya mutual entropy, C*-system.
1. Introduction
In [1], Ohya introduced Information Dynamics (ID) synthesizing dynamics of state change and complexity of state. Based on ID, one can study various problems of physics and other fields. Channel and two complexities are key concepts of ID. Let us briefly review ID for quantum communication processes.
Let %k (k = 1)2) be complex separable Hilbert spaces. We denote the set of all bounded linear operators on Tik by B(%k) (k = 1,2) and we express the set of all density operators on Hk by &{%k) (k = 1,2). Let (B('Hk),&('Hk)) (k = 1,2) be input (k = 1) and output (k = 2) quantum systems, respectively.
1.1. Quantum Channels
A mapping from &(%{) to 6(^2) is called a quantum channel A*.
(1) A* is called a linear channel if A* satisfies the affine property such as
h*Œ2k^kpk) = Sfc^fc^*(Pfc) for any pk € S('Hi) and any nonnegative
number Ak € [0,1] with J2k = 1-
For the quantum channel A*, the dual map A of A* is defined by tr A*(p)5 = trpA(5), Vp € ©(fti),VB € B(H2).
Noboru Watanabe, Professor, Dept, of Information Sciences.
(2) A* is called a completely positive (CP) channel if A* is linear channel and its dual map A : B(fH2) —> B('Hi) of A* holds
(^x, ^2 A*k(A*Aj)Aj x^j ^ 0 (Væ € T-L\)
for any n € N, any {Ai} c B('%2) and any {Ai} c B('Hi).
One can describe almost all physical transform of states by using the CP channel [2-5].
1.2. Quantum Communication Channel
Here we explain the quantum communication channels as an example of the quantum channels.
In order to consider influence of the environment such as noise and loss, we suppose /Ci and K2 to be complex separable Hilbert spaces of noise and loss systems, respectively. Quantum channel of quantum communication process with noise and loss was discussed by [2,6].
1.3. Noisy quantum channel and Generalized Beam Splitter
For an input state p in &(%{) and a noise state £ € 6 (/Ci), Ohya and NW defined in [6] a generalized beam splitting n* by
H*(p®£)^F(p®£) V*,
where V is a linear mapping from T-L\ <g> Ki to %2 <S> K2 given by for the rii, m 1, j, (n\ + m 1 — j) photon number state vectors \n\) € Hi, \rrii) € /Ci, |j) € %2, \ni + mi ~j) € K.2
ni+mi
V(\ni) <g> \mi)) = C]'1’mi\i) ®\ni+mi- j)
j
and
= yv nn1+7-r Vnilmiljljni + mi ^jÿ. j JL r\(rii — j)\(j — r)\(mi — j + r)\ (1)
xami~j+2r (~fj)ni+j~2r
a and fi are complex numbers satisfying |ck|2 + |/?|2 = 1. K and L are constants given by K = min{ni,j}, L = max{mi — j, 0}. For the coherent input state p = 10) {0\ <g> |k) (k| € & (iii^Ki), the output state of n* is obtained by
n* (|0) {9\ <g> |k) (n\) = \a6 + fin) {ad + fin\
<g> | —fid + an) {—fid + an| .
By using n*, Ohya and NW introduced in [6] the noisy quantum channel A* with a fixed noise state £ € 6 (Ki) defined by
A* (p) = tr/c2 n* (p (8) 0 = tr^2 V(p®0V*. (2)
The generalized beam splitting II* with the vacuum noise state £o = |0) (0| is called the beam splitter IIq given by
Ho (|0> (0\ cx) Co) = \olQ) {ad| eg)\-po) {-pd\
for the coherent input state p ® £o = \0) (0\ <S> |0) (0| € & (Hi<S>lCi). The beam splitter IIq was described by means of the lifting ££ from (5 (H) to <8 (H<g>lC) in the sense of Accardi and Ohya [7] as follows
8*0(\9) (6\) = \ad) (ad\®\pd) </30|.
Based on the liftings, the beam splitting was studied by Accardi-Ohya and Ficht-ner-Freudenberg-Libsher [8]. Moreover, the noisy quantum channel Aq with the vacuum noise state £o = |0) (0| is called the attenuation channel given by Ohya [2] as
A*0(p) = tr/c2 IIo(p <g> Co) = tr/c2 V0(p <g> |0)<0|)Vb*, (3)
which plays an important role for investigating the quantum communication processes.
2. Complexities
Two kind of complexities Cs(p), Ts(p;A*) are used in ID. Cs(p) is a complexity of a state p measured from a subset S and Ts (p; A*) is a transmitted complexity according to the state change from p to A*p. These complexities should fulfill the following conditions: Let S, S, St be subsets of 6 ("Hi), 6 (^2), 6 (Hi ® H2), respectively.
(1) For any p € S, Cs(p) and T5(p; A*) are nonnegative ( Cs(p) ^ 0, Ts (p; A*) ^ 0 ).
(2) For a bijection j from ex&(Hi) to ex(5 (Hi),
Cs(p) = Cs(j(p))
is hold, where ex(5 (Hi) is the set of extremal point of <8 (Hi).
(3) For p <g) a € 6 (Hi <g> H2), p € 6 (Hi), a € 6 (H2),
CSt(p®a) = Cs(p)+C^(a).
It means that the complexity of the state p <g> a of totally independent systems are given by the sum of the complexities of the states p and a.
(4) Cs(p) and Ts (p;A*) satisfy the following inequality 0 ^ Ts (p;A*) ^
C'S(P)•
(5) If the channel A* is given by the identity map id, then Ts (p',id) = C (p) is hold.
One of the example of the above complexities are the Shannon entropy S (p) for Cs(p) and classical mutual entropy I (p; A*) for Ts (p; A*). Let us consider these complexities for quantum systems.
2.1. Example of Complexity Cs (p)
2.1.1. von Neumann Entropy and «S-mixing entropy
One of the example of the complexity Cs(p) of ID in quantum system is the von Neumann entropy S(p) [9] described by
Cs (p) S(p) = — tr p log p
for any density operators p Є 6 (Hi), which satisfies the above conditions (1), (2), (3).
Let (А, S(.A), Oi(G)) be a C*-dynamical system and S be a weak* compact and convex subset of 6(Л). For example, S is given by 6(Л) (the set of all states on A), 1(a) (the set of all invariant states for a), K(a) (the set of all KMS states), and so on. Every state p Є S has a maximal measure /л pseudosupported on exS such that
<p = J wdfji, (4)
where exS is the set of all extreme points of S. The measure /л giving the above decomposition is not unique unless S is a Choquet simplex. We denote the set of all such measures by M¥,(iS), and define
DV(S) = ^MV(S); Зцк C M+ and {<pk} C exS
s.t. = ц = ^fXkHVk)}, (5)
k k
where 5((p) is the Dirac measure concentrated on an initial state p. For a measure ц Є D^S), we put
H(p) = - (6)
k
The C*-entropy of a state (p Є S with respect to S (5-mixing entropy) is defined by
^-)«^-) = { (7)
It describes the amount of information of the state (p measured from the subsystem S. We denote Se(-A\ip) by S(ip) if S = 6(Л). It is an extension of von Neumann’s entropy.
This entropy (mixing 5-entropy) of a general state <p satisfies the following properties [10].
Theorem 2.1 .When A = B(W) and at = Ad(Ut) (i.e., at(A) = U£AUt for any А Є A) with a unitary operator Ut, for any state Lp given by ip( ■ ) = tr p ■ with a density operator p, the following facts hold:
(1) S(ip) = -trplogp.
(2) If Lp is an a-invariant faithful state and every eigenvalue of p is nondegenerate, then SI(-a\ip) = S(ip), where 1(a) is the set of all a -invariant faithful states.
(3) If ^р Є K(a), then Sk(№\lp) = 0, where K (a) is the set of all KMS states. Theorem 2.2.For any p Є K(a), we have
(1) SK{-a\ip) < SI{-a\ip).
(2) SK^(>p)^S(>p).
2.2. Example of Transmitted Complexity Ts (p\ Л*)
The classical mutual entropy I (p;A*) defined by using the joint probability distribution between the input state and the output state is an example of the transmitted complexity Ts (p;A*) of ID. In general, there does not exit the joint
states in the quantum system [11]. We need to introduce the compound state in quantum system instead of the joint probability distribution in classical system.
2.3. Compound state
The quantum mutual entropy I (p, A*) should satisfy the following three conditions:
1) If the channel is given by the identity channel id, then I (p; id) = S(p) (von Neumann entropy) is hold.
2) If the system is classical, then the quantum mutual equals to the classical mutual entropy.
3) The quantum mutual entropy should satisfy the Shannon’s type inequalities:
0< J(p,A*)<S(p).
Ohya introduced two compound states go and ge- go is the trivial compound state given by
do = p ® A*p.
ge is the compound state representing a certain correlation between the input state and the output state given by
ge
associated with the Schatten-von Neumann (one dimensional spectral) decomposition [12] p = J2n ^nEn of the input state p.
2.4. Ohya Mutual Entropy for density operator
An example of the transmitted complexity Ts (p; A*) of ID in quantum system is the Ohya mutual entropy with respect to the initial state p and the quantum channel A* defined by
Ts (p; A*) ^ I (p; A*) = sup i ^ S{ge, a0), p = ^ AnEn 1 ,
{ n n )
where S ( • , • ) is the Umegaki’s relative entropy [13] denoted by
n/ \ _ i tr p (log p — log g) (when rânp C rànâ)
’ y oo (otherwise) ^ '
which was extended to more general quantum systems by Araki and Uhlmann [1,3,4,14,16]. The Ohya mutual entropy holds the above conditions (1), (4), (5) such as
0</(p,A*) <5(p),
I (p, id) = S(p).
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 quantum capacity of quantum channel was studied in [17-20].
Theorem 2.3. Let <J>e be a compound state w.r.t. the initial state p, the quantum CP channel A* and aSchatten decomposition of p = AkEk defined by
^ (I ® Vn
n
under the condition
Y, \f^k'{xk/ \ ® (xy
k>
(i®v:
^2{i®v*){i®vn) = i®i
and A* is given by A*(p) = ’YlnVnpV*. By defining the compound state $e7 one can obtain the following theorem.
Theorem 2.4. For the compound state given above, one can obtain two
marginal states as follows
tr h2®e= S(p),
^h1^e= S (A*p).
The upper bound of the relative entropy S(&e, P ® A*p) is obtained as follows:
S($E,P® A*p) < 2S(p).
Let be a compoundstate defined by
'I'E,ti = p erg + (1 - p) (p € [0,1]).
We have the following theorem.
Theorem 2.5. For the compound state w.r.t. p € [0,1] given above, one
can obtain two marginal states as follows
tru2^E,n= S(p), tr-u^E,^ S (A*p).
The upper bound of the relative entropy S^e^, P ® A*p) is obtained as follows:
S(^E,fi, P ® A*p) ^ (2 - p) S(p) (p € [0,1]).
2.5. Ohya Mutual Entropy for general C*-system
Let (A, &(A),a (G)) be a unital C*-system and S be a weak* compact convex subset of 6 (.4). For an initial state <p € S and a channel A* : (5 („4.) —>• (5 (B), two compound states [3,10] are defined by
= / oj ® A*oj dp,, J s
$o = ip ® A*ip.
(9)
(10)
The compound state expresses the correlation between the input state <p and the output state A*(p. The mutual entropy with respect to S and /x is given by
/J(^;A*)=5(^,$o) (11)
and the mutual entropy with respect to S is defined by Ohya [3,10] as
TS (p', A*) Is (p; A*) = sup (<p; A*) ;/ieM,(5)}. (12)
2.6. Other Mutual Entropy Type Measures
Recently, several mutual entropy type measures were proposed by Shor [21] and Bennet et al [22,23], which defined by using the entropy exchange [24] given by
Se(p,A*) =-trWlogW, (13)
where W is a matrix W = (Wy)^ • with the elements
Wij = tr A*pA3 (14)
obtained by means of the input state p and the CP channel A* described by a Stinespring-Sudarshan-Kraus form
A*(-)^E jArAi- (15)
Based on the entropy exchange, the coherent entropy Ic{p', A*) [15] and the Lindblad-Nielson entropy II (p; A*) [23] were defined by
Ic(p]A*) = S(A*p)-Se(p,A*), (16)
IL (p; A*) = S(p) + S (A*p) - Se (p, A*). (17)
2.7. Comparison among these quantum mutual entropy type measures
In this section, we compare with these mutual types measures.
By comparing these mutual entropies for quantum information communication processes, we have the following theorem [25]:
Theorem 2.6. Let {Aj} be a projection valued measure with dim Aj = 1. For arbitrary state p and the quantum channel A* ( •) = ^ • Aj ■ A*, one has
(1) 0^1 (p; A*) ^ min{S,(p),5 (A*p)} (Ohya mutual entropy),
(2) Ic (p; A*) = 0 (coherent entropy),
(3) IL (p; A*) = S(p) (Lindblad entropy).
For the attenuation channel Aq, one can obtain the following theorems [25]: Lemma 2.1. For the attenuation channel Aq and the input state
p = A|0)(0| + (1 — A)|6,)(6I|,
there exists a unitary operator U such that
UWU* = A|0)(0| + (1 - A)|—/30)(—/301.
Theorem 2.7. For the attenuation channel Aq and the input state
P = A|0)<0| + (1 — A)|0)(0|, the entropy exchange is obtained by
l
Se (P, Aq) = —trW log W = - H lo§
3=0
where
N = ^ |1 + (—i)-7 y^1 — 4A(1 — A) (l-exp(— |/?|2|0|2))| (i = 0,1).
Theorem 2.8. For any state p = Ara |n) {n\ and the attenuation channel Aq with |ck| = |/3| = \, one has
(1) 0 ^ I (p; Aq) ^ min{S'(p),S' (Aqp)} (Ohya mutual entropy),
(2) Ic (p; Aq) = 0 (coherent entropy),
(3) IL (p; Aq) = S(p) (Lindblad entropy).
Theorem 2.9. For the attenuation channel Aq and the input state P = A |0) <0| + (1 — A) 19) {9\,
we have
(1) 0 ^ I (p; Aq) ^ min{S'(p),S' (A^p)} (Ohya mutual entropy),
(2) —S(p) ^ Ic (p] Aq) ^ S(p) (coherent entropy),
(3) 0 ^ II (p; Aq) ^ 2S (p) (Lindblad entropy).
It shows that the coherent entropy holds Ic (p] Aq) < 0 for |ck|2 < |/3|2 and the Lindblad entropy satisfies II (p; Aq) ^ S'(p) for |ck|2 > |/3|2. From the above theorems, we can conclude that the transmitted complexity in quantum system is the Ohya mutual entropy and it is most fitting measure for studying the efficiency of information transmission in quantum communication processes. It means that Ohya mutual entropy can be considered as the transmitted complexity for quantum communication processes.
Theorem 2.10. For the attenuation channel Aq and the input state
P = A|0)(0| + (1 — A)|6,)(6I|, if \ = \ and /3 = then there exists a compound state $ satisfying
IL(P]A*0) = S(<S>,P®A*0P).
Theorem 2.11. For the attenuation channel Aq and the input state
P = A|0)(0| + (1 — A)|6,)(6I|,
if X = ^ and a = 1, then there exists a compound state $ satisfying
S(<S>,p®A*0p) = S(p).
Acknowledgment. The author would like to thank Prof. I. Volovich for his many helpful sugestions during the preparation of the paper.
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Original article submitted 14/XI/2012; revision submitted 21/I/2013.
