数理解析研究所講究録 1354 巻 2004 年 68-73

88translation-invariant quantum Markov states
東北大学大学院情報科学研究科
大野 博道 (Hiromichi Ohno)

Graduate School of Information Sciences,Tohoku University

1Introduction

The notion of quantum Markov states was first introduced by Accardi andPrigerio ([1], [3]), and was further discussed ffom a somewhat different viewpoint by Fannes, Nachtergaele and Werner ([5]). A Markov state by Accardi
and Prigerio is defined on a UHF algebra and is determined by an initialstate and a family of completely positive quasi-conditional expectations However, thanks toand also [6], conditional expectations can be used inplace of quasi-conditional expectations. Although Accardi and Prigerio defined their Markov states without translation-invariance, we restrict ourselves
to translation-invariant ones and clarify their fine structure.
In ] it was implicitly stated that any translation-invariant Markov statein the sense of ] is determined by a single conditional expectation ( that
it is a C’-finitely correlated state in [and an explicit form of translationMarkov states was given. In Section 2 we make the relation between
two notions of quantum Markov states more precise and consider the question
concerning the commutativity of local density matrices of a Markov state. In
Section 3 we see explicit form of quantum Markov states due to ] .
$[2\rfloor, [3]$$\underline{|}4\lceil$$\downarrow 3\lceil$$\mathrm{s}\mathrm{o}\backslash$$51)\lrcorner$’$\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{a}\mathrm{r}\dot{\mathrm{a}}\mathrm{n}\mathrm{t}$$\lfloor\Gamma 4$

2

Characterization of translation-invariant Markovstates, thecomplex matrix algebra, forLetand. We denotefor
21 be the infinite C’-tensor productarbitrary subset A N. The translation ) is the right shift on 21. We write, and in particular
for the localization
for $n=1.$ The following
definition is from [3] with a slight modification.
$d\mathrm{x}d$$\mathfrak{U}_{i}=M_{d}=M_{d}(\mathbb{C})$$i\in \mathrm{N}$$\otimes_{=1}^{\infty}\dot{.}\mathfrak{U}_{i}$$\mathfrak{U}_{\Lambda}=\otimes_{n\in\Lambda}\mathfrak{U}_{n}$$\subset$$\phi|\mathfrak{U}[1,n]$$\mathrm{A}[1,n]$$\phi_{1}$Definition 2.1 A state on is called a (quantum) Markov state if for each
there exists a conditional expectationfrom $2[[1,n11]$ intosuch. A Markov state is said tothat
andbe translation-invariant if.
$\mathfrak{U}$$\phi$$n\in \mathrm{N}$$E_{n}$$E_{n}(\mathfrak{U}_{[1,n+1]})\supset \mathit{2}L_{[1,n-1]}$$\#[\mathrm{r},*+\mathrm{i}]$$\phi\circ\gamma=\phi$$=$$\#[1,n]$$\circ E_{n}$$\mathfrak{U}[1,n]$asAlthough the above definition is a bit different from the original one ofAccardi and Prigerio in [3], it is known that both definition are equivalent ([2],[3] and also [6] .
is faithful for all $n\in$ N. The
We assume that is locally faithful, i.e.
next theorem was implicitly stated in [4]; we here give a proof.
$)$$\phi_{[1,n]}$$\phi$

Theorem 2.2 Let be a state on 1. Then the following are equivalent(i) is a translation-invariant Markov state,suchinto(ii) There eists a conditional expectation $E$ from
$A\inM_{d}$$\phi_{1}\circ
E(I\A)=\phi_{1}(A)$andthatfor all$\phi$$\phi$$M_{d}$$M_{d}\otimes M_{d}$6 $(A_{1}\otimes\cdots\otimes A_{n})=\phi_{1}(E(A_{1}\otimes E(A_{2}\otimes\cdots\otimes E(A_{n-1}\otimes A_{n})\cdots)))$for all, $\ldots,A_{n}\in M_{d}$ .

$A_{1}$

Proof.

(ii).

$arrow$$2\mathrm{L}_{(1,n+1]}$$\Rightarrow(\mathrm{i})$$\mathfrak{U}_{[1,n]}$,. Assume (ii), and define conditional expectation$n\in$$E_{n}$:

N, by$E_{n}(X\otimes A)=X\otimes E(A)$for $X\in$$\mathit{2}_{(1,n-1]}$

and $A\in$$\phi(A_{1}\otimes\cdots \otimes A_{n})$$2\mathrm{L}[n,n+1]$. Then for$A_{1}$,$\ldots$, $A_{n}\in M_{d}$ ,$\phi_{1}\circ E(A_{1}\otimes E(A_{2}\otimes\cdots\otimes E(A_{n-1}\otimes A_{n})\cdots)$$=$$=\phi(A_{1}\otimes \cdots\otimes A_{n-2}\otimes E(A_{n-}\mathrm{J} A_{n}))$$\phi \mathrm{o}E_{n-1}(A_{1}\otimes\cdots\otimes A_{n})$

$=$

and
$\phi(I\otimes A_{1}\otimes\cdots\otimes A_{n})$$=$$6_{1}\circ E(I\otimes E(A_{1}$(& $E(A_{2}\otimes\cdots\otimes E(A_{n-1}\otimes A_{n})\cdots)$$/\mathrm{N}(A1\otimes E(A_{2}\otimes\cdots \otimes E(A_{n-1} \ \wedge 4_{n})\cdot\cdot.)$$=$$\phi_{1}\circ$$=$$\phi(A_{1}\otimes\cdots\otimes 4)$.

is a translation-invariant Markov state.
(i). We fix $n\in$ N, and define
defined. Indeed, for any $A\in$ $2[[1,n -1]$ andSo$\phi$$F_{n-1}=\gamma^{-1}\circ E_{n}\circ\gamma$$\Rightarrow(\mathrm{i}\mathrm{i})$$B\in \mathit{2}i_{1}$. This is well,$E_{n}(I\otimes A)B\otimes I^{\theta n-1}=E_{n}(B\otimes A)=B\otimes I^{\otimes n-1}\cdot$ $E_{n}(I\otimes A)$.

. Similarly, we defineHence, $E_{n}(I\otimes 4)$$(1\leq i\leq n-1)$ . Then for $1\leq i\leq n$ and, $A_{i+1}\in M_{d}$ , we have,
$F_{\dot{\iota}}=\mathrm{y}$$\in \mathfrak{U}[2,n_{d}\uparrow$$A_{1}$$F_{\dot{\iota}}(A_{1}\otimes\cdots\otimes A_{:+1})$$-(n-i)\mathrm{o}E_{n}\circ\gamma^{n-:}$$\ldots$$=$$(A_{1}\otimes\cdots\otimes A_{:-1}\otimes I^{Q2})F_{i}$$=$$4_{1}\otimes\cdots\otimes A_{\dot{\iota}-1}3$($I^{\otimes i-1}\otimes$$F_{1}(A_{\dot{1}}\otimes A_{i+1})$A.

$\otimes A_{i+1}$).

Now, letdenote the set of all conditional expectations $F:M_{d}\otimes M_{d}arrow M_{d}$$=A_{1}$,such that if we define& $\cdots\$$1\leqi\leqn.$theThenfor each, $A_{i+1}\in M_{d}$ , thenfor ,$\mathrm{f}\mathrm{f}_{n}$$F_{i}(A_{1} \ \cdots\otimes \mathrm{A}_{+1}.)$$A_{1}$$\ldots$$\mathrm{E}_{[1,\mathrm{i}]}$$\circ F\dot{.}=\phi(1,\mathrm{i}+1)$$A_{*-1}.\otimes F(A_{i} \ \mathrm{A}_{+1}.)$

70
above argument guarantees the non-emptyness of . Since ’s are compact)
and, it follows thatis not empty. Choose
and define$S_{n}$$\mathrm{f}\mathrm{f}_{2}\cdots$$\mathrm{f}\mathrm{f}_{1}$$\bigcap_{n\in \mathrm{N}}$$E_{n}(A_{1}\otimes\cdots\otimes A_{n+1})=A_{1}$for$A_{1}$$S_{n}$$J_{n}$&$E \in\bigcap_{n\in \mathrm{N}}\mathrm{f}\mathrm{f}_{n}$$\cdots\$ $A_{n-1}$$\otimes$$E(A_{n} \ 4 +1)$

, . . . , $A_{n+1}\in M_{d}$ . Then$\phi(A_{1}\otimes\cdots\otimes A_{n})$$=$$\phi 1$$\mathrm{o}E_{1}\mathrm{o}\cdots \mathrm{o}E_{n}(A_{1}\otimes\cdots\otimes A_{n})$$=\phi_{1}\circ E(A_{1}\otimes E(A_{2}\otimes\cdots\otimes E(A_{n-1} \ A_{n})\cdots))$,and$\phi_{1}\circ E(I\otimes A)=\phi(I\otimes A)=\phi_{1}(A)$for $A\in M_{d}$ .

$\square$

The following definition is from [5].

Definition 2.3 A state on is called a C’-finitely correlated state if there
exist a finite dimensional -algebra , a completely positive map $E:M_{d}$ @$@arrow@$ and a stateon such that$\mathfrak{U}$$\phi$$C^{*}$$\mathfrak{B}$$\mathfrak{B}$$\rho$$\rho$( $E$ ($I_{d}$cg $B)$ ) $=\rho(B)$for all $B\in B$ and$\phi(A_{1}\otimes\cdots\otimes A_{n})--,(E(A1\otimes E(A_{2} (\ \cdots\otimes E(A_{n}\otimes I_{\mathfrak{B}})\cdots )))$

for all$A_{1}$,$\ldots$, $A_{n}\in M_{d}$ .

Let be a translation-invariant Markov state, and be as in (ii) of Theorem
2.2. We set $1=E(M_{d}\otimes M_{d})$ and. Then is a (7’-finite
correlated state with a triple. Hence any translation-invariant
Markov state becomes a C’-finitely correlated state.
Now, let ,, be minimal central projections of 1, so thatMd.
for some $4\in$ N. Let
be the multiplicity of. Then: in$E$$\phi$$\hat{E}=E|M_{d}\otimes \mathfrak{B}$$\phi$$1\mathrm{y}$$(\mathfrak{B},\hat{E}, \phi|\mathfrak{B})$$q_{1}$$\ldots$$f\mathit{3}q_{i}\cong$$q_{k}$$M_{d}$$m_{i}$$\mathfrak{B}=\oplus^{k}\mathfrak{B}q_{i}=\oplus^{k}M_{d_{i}}$$i=1$$M_{d}$.

$\dot{.}=1$

Moreover we set$C$ $=\oplus^{k}M_{d_{i}}\otimes M_{m:}i=1$

and letbe the pinching
The next proposition is included in [4].
$E_{\mathrm{C}}:\cdot M_{d}arrow C$$A \vdasharrow\sum$

k

$=1$

$q_{i}Aq_{i}$.

71
Proposition 2.4 There exist positive linear functional$i,j$such that$s\rho_{ij}$on$M_{m:}\otimes M_{d_{j}}(1\leq$$\leq k)$$\hat{E}=$$(\oplus^{k}\mathrm{i}\mathrm{d}_{M_{d}}i,j=1^{\cdot}.

\otimes\rho_{ij})(E_{\mathrm{C}}\otimes \mathrm{i}\mathrm{d}_{\mathfrak{B}})$.

We remark that the unitality ofi equivalent to the condition thatis a state on
for each $1\leq i\leq k.$ Furthermore, the condition
thatfor all
is equivalent to the condition that,for$\hat{E}$$\mathrm{s}$$M_{m_{i}}\otimes@$$\oplus_{j=1}^{k}\rho_{\dot{0}j}$$6_{1}\circ\hat{E}(I\otimes B)=\phi_{1}(B)$$B\in \mathfrak{B}$$B_{j}\in M_{d_{\mathrm{j}}}(1\leq j\leq k)$$\sum_{i=1}^{k}\psi_{\mathrm{i}}(q_{i})\rho_{\dot{*}j}(I_{m}\dot{.}\otimes B_{\mathrm{j}})=\psi_{j}(B_{j})$where$6|"=\oplus_{1=1}^{k}\cdot$.

$\mathit{7}_{\iota}$

equation (1) saysSet$\pi_{j}\dot{.}=\rho_{ij}(I_{m}$.

$[\alpha_{1} \alpha_{k}][_{\pi_{k1}}\pi_{11}..$..

:&$q_{j})$

(1)

,

and

$\alpha_{i}=\psi_{\dot{l}}(q_{i})$; then the$\pi\pi_{\mathrm{I}}kk]=[\alpha_{1}\wedge \alpha_{k}]$The unitality ofmeans that the matrix
is a stochastic matrix. The
faithfullness ofguarantees $\pi_{ij}>0$ for all $1\leq i,j\leq k.$ Hence,
isuniquely determined byfrom the Perron-Probenius theorem. So, by (1),is also uniquely determined by.
By Proposition 2.4, we get the next corolary.
$\hat{E}$$[\pi_{ij}]$$\hat{E}$$\{\alpha_{i}\}$$\{\pi_{ij}\}$$\{\psi_{i}\}$$\{\rho_{ij}\}$Corollary 2.5 Letandbe the density matrices oftively. Then the density matrix
isof$S_{i}$$T_{i_{J}}$$\hat{D}_{n}$$\phi|\mathfrak{U}_{[1,n-1]}\otimes$$\psi_{\dot{l}}$and$\rho_{\dot{\iota}j}$, respec-$f\mathit{3}$$i_{1},\ldots,i_{n}\oplus 5_{i_{1}}\otimes T_{i_{1}i_{2}}\otimes\cdots\otimes T_{i_{n-1}i_{n}}$.

In the above the density matrix
is taken with respect to the usualon, i.e. thehaving the value 1 for each rank one projection.
If all summands of 3 are of multiplicity one, then the density in the above
corollary is actually the density matrixof. Hence, the densities
are all commuting in this case (see [6]).
$\hat{D}_{n}$$2\mathrm{L}[1,n-1]\otimes$$95$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$D_{n}$$D_{n}$$\phi_{[1,n]}$Consider the case $d=2.$ Subalgebras ofor C. Iforareisor , any translation-invariant Markov state relative to 3 is a product, all summands ofstate. If fl is
are of multiplicity one. Hence, thedensity matrices’s are commuting in this case (see [3]).
are nonThe following is the simplest example where the densities$\mathfrak{B}$$M_{2}$$M_{2}$$M_{2}$$\mathbb{C}\oplus \mathbb{C}$$\mathbb{C}$$\mathbb{C}\oplus \mathbb{C}$$\mathrm{B}$$D_{n}$$\{D_{n}\}$commuting$\mathfrak{B}$72
Example 2.6 We set $d=3$ and$(1\leq i, j\leq 3)$ are the matrix unit ofiswhere , $A_{2}\in M_{2}$ , , $A_{2}\geq 0$ ,, where. Assume that the density matrix of,, $c_{1}+c_{2}=1.$and ,$\mathit{1}\mathit{3}--\mathbb{C}\oplus \mathbb{C}=\mathbb{C}(e_{11}+e_{22})+\mathbb{C}e_{33}$$=A_{1}\oplus A_{2}\oplus c_{1}\oplus c_{2}\in$$T_{11}\oplus T_{12}\oplus T_{21}\oplus 722$$\oplus_{i,j=1}^{2}\rho_{ij}$$A_{1}$$M_{3}$$\mathrm{T}\mathrm{r}(A_{1}+A_{2})=1,$$A_{1}$$c_{1}$$e_{ij}$$\#_{2}\oplus M_{2}\oplus \mathbb{C}\oplus \mathbb{C}$$c_{2}\in \mathbb{R}^{+}$We define$\mathrm{A}_{1}(e_{11}+e_{22})=\frac{c_{1}}{c_{1}+7\mathrm{k}(A_{2})}$’,$2(e_{33})= \frac{\mathrm{b}(A_{2})}{c_{1}+\mathrm{R}(A_{2})}$,then it is easily seen that the condition (1) is satisfied. In this case, the densityofismatrix$D_{n}$$2\mathrm{t}[\mathrm{i},\mathrm{n}]$$(T_{i_{n-1}1}\otimes(A_{1}+A_{2})\oplus T_{n-1}.\cdot 2\otimes(c_{1}+c_{2}))$$\cdots\bigoplus_{:_{1\prime\prime n-1}}.\cdot S_{i_{1}}\otimes T_{i_{1}i_{2}}\otimes\cdots\otimes T_{i_{n-2}i_{n-1}}\otimes$So,3$D_{n}$’s are non-commuting if so are$A_{1}$

and$A_{2}$.

.

Disintegration of quantum Markov states

In this section, we survey the explicit form of Markov states due to [4]. Let 6 bea translation-invariant Markov state as in Section 2. We put$\Omega_{n}=\{1, \ldots k\}$,$\Omega=\prod_{n\in \mathrm{N}}\Omega_{n}$and$(x_{1},x_{2}, \ldots,x_{n})=\{(y_{1},y_{2},\ldots)\in\Omega|y_{i}=x:,1 \leq i\leq n\}$We define the mesure

$\nu$

on

$\Omega$.

by$\nu((x_{1},x_{2},\ldots,x_{n}))=7(q_{x_{1}} \ q_{x_{2}} \ \cdot..

q_{x_{\hslash}})$$= \alpha_{x_{1}}.\cdot\prod_{=1}^{n-1}\pi_{x\dot{.}xx}:+1^{\cdot}$

Then is a probability mesure on 0. For an arbitraly element, we set$\nu$$\omega$$=$$(\omega_{1},\omega_{2},$$)\in$\ldots$\Omega$$\mathfrak{B}_{\omega}=M_{d_{\omega_{1}}}\otimes M_{m.1}\otimes M_{d.2}$

and the state$\psi_{\omega}$

on fi, by$I_{\omega}= \tilde{\psi}_{\omega_{1}}\otimes.\cdot\bigotimes_{=1}^{\infty}\tilde{\rho}_{\omega}$

andwheremap defined by$\tilde{\psi}_{i}=\psi_{i}/\alpha_{i}$$\tilde{\rho}_{ij}=\rho_{\mathrm{i}j}/\pi_{j}.\cdot$. Let$E_{\omega}(A_{1}\otimes A_{2}\otimes\cdots\otimes’)$for any$A_{1}$,\ldots,..

$A_{n}\in M_{d}$, and$E_{\omega}$:

jCA j $+1$

$21arrow \mathfrak{B}_{\omega}$$=q_{\omega_{1}}A_{1}q_{\omega_{1}}$$\phi_{\omega}=\psi_{\omega}\circ E_{\{d}$,.

$\otimes\cdot..$

be a completely positive$\otimes q_{\omega_{n}}A_{n}q_{\omega_{n}}$73Theorem 3.1 Define$\Omega$,

and$\nu$$\phi_{\omega}$

as above. Then$\phi=f_{\Omega}\phi_{\omega}\nu(M)$Proof.

If$\phi$$\omega,\omega’\in$,(t$(x_{1}$,.. ,$1\otimes\cdots\otimes A_{n-1}$$x_{n})$.

, then)$=\#_{\omega}(q_{x_{1}}A_{1}q_{x_{1}} \ \cdots\otimes q_{x_{\mathfrak{n}}}A_{n-1}q_{x_{n}})$n-l$= \frac{1}{\nu((x_{1},\ldots,x_{n}))}\cdot(\psi_{x_{1}}\otimes\dot{.}\bigotimes_{=1}\rho_{x\dot{.}x:+1})(q_{x_{1}}A_{1}q_{x_{1}}\otimes\cdots\otimes q_{x_{n- 1}}A_{n-1}q_{x_{n- 1}})$$=\phi_{\omega’}$for any($A_{1}\otimes\cdots$$A_{n-}1$), $A_{n-1}\in M_{d}$ . Therefore,$A_{1},..\tau$6cg$(A_{1}\otimes\cdots\otimes A_{n-1})$$=\emptyset\circ E_{\mathbb{C}}(A_{1}\otimes\cdots\otimes A_{n-1})$$= \sum_{x_{1,\ldots\prime}x_{n}}(\psi_{x_{1}}\otimes\bigotimes_{i=1}\rho_{x\dot{.}x_{j+1}})(q_{x_{1}}A_{1}q_{x_{1}}\otimes\cdots\otimes q_{x_{n-1}}A_{n-1}q_{x_{n-1}})$$\sum\nu((x_{1},\ldots,x_{n}))\cdot\phi_{\omega_{(x_{1},\ldots,x_{\hslash})}}(A_{1}\otimes\cdots\otimes A_{n-1})$$=$$x_{1}$,... ,$x_{n}$$= \int_{\Omega}\phi_{\mathrm{t}d}(A_{1}\otimes\cdots\otimes A_{n-1})\nu(d\omega)$

where$\omega_{(x_{1}}$,...

,$x_{n}$) is,

an arbitraly element in $(x_{1}, .., x_{n})$ .

口References[1] L. Accardi, Topics in quantum probability, Phys. Rep., 77, (1981) 169-192.
[2] L. Accardi and C. Cecchini, Conditional expectations in von Neumann
algebras and a theorem of Takesaki, J. Fund. Anal, 45, (1982) 245-273.

[3] L. Accardi and A. Prigerio, Markovian cocycles, Proc. Roy. Irish. Acad.,, (1983) 251-263.
$83\mathrm{A}(2)$[4] L. Accardi and V. Liebscher, Markovian KMS-states for one-dimensional
spin chains, Infin. Dimens. Anal Quantum Probab. Relat Top., 2, (1999)

645-661.

[5] M. Fannes, B. Nachtergaele and R. F. Werner, Finitely correlated stateson quantum spin chains, Commun. Math. Phys., 144, (1992) 443-490.
[6] V. Y. Golodets and G. N. Zholtkevich, Markovian KMS states, Teoret
Mat Fiz., 56(1), (1983) 80-86.