Entropy of entangled states and SU(1,1) and SU(2) symmetries

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Entropy of entangled states and SU

„1,1… and SU„2… symmetries

A. E. Santana,1,2 F. C. Khanna,2,3and M. Revzen2,4

1

Instituto de Fı´sica, Universidade Federal da Bahia, Campus de Ondina, 40210-340 Salvador, Bahia, Brazil

2_{Physics Department, Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2J1} 3_{TRIUMF, 4004, Wesbrook Mall, Vancouver, BC, Canada V6T 2A3}

4_{Department of Physics, Technion, Haifa 32000, Israel} 共Received 25 April 2001; published 26 February 2002兲

Based on a recent definition of a measure for entanglement关Plenio and Vedral, Contemp. Phys. 39, 431

共1998兲兴, examples of maximum entangled states are presented. The construction of such states, which have

symmetry SU共1,1兲 and SU共2兲, follows the guidance of thermofield dynamics formalism.

DOI: 10.1103/PhysRevA.65.032119 PACS number共s兲: 03.65.Ud

I. INTRODUCTION

Due to the structure of the Hilbert space as well as the superposition principle, quantum mechanics gives rise to the notion of entangled states, which are states of two or more systems correlated with each other, but without a classical analog. In particular, this entanglement may have nonlocal features. Bell关1,2兴 was the first to present a systematic way to analyze such entangled states, by comparing such correla-tions to the classical correlated states, that are defined via classical probability distributions.

A renewed interest in entanglement arose because it can be used for teleporting quantum states, from one locus to another, which is a basic ingredient in quantum computers

关3,4兴. In order to progress with such a program of quantum

communication, the measure for entanglement is a crucial aspect of the theory that should be fully developed. And this aspect has been approached in several different methods

关5–7兴.

The conditions for teleporting require specific entangled states characterized by a maximum entanglement. In this sense, Barnett and Phoenix 关8兴 and Plenio and Vedral 关9兴, using the notion of thermal-like states with quantum correla-tions 共see also the paper by Ekert and Knight 关10兴 for a detailed account of two-mode squeezed states and thermal-like states兲, have suggested that a consistent measure of the entanglement of two subsystems, say A and B, described by a pure state兩␺(A,B)

典

, is the entropy of the reduced density operator␳A, where

␳A⫽TrB共兩␺共A,B兲

典具

␺共A,B兲兩兲, 共1兲

with TrB standing for the trace over the coordinates of the

subsystem B. In other words, a measure of entanglement of the state兩␺(A,B)

典

is given by

S共␳A兲⫽⫺Tr␳Aln␳A. 共2兲

Here our goal is to set forth examples of states that maximize the entanglement. We use as a guide the formalism of ther-mofield dynamics 共TFD兲关11–14兴, in order to build such

共maximum entangled兲 states that will be associated with the

SU共1,1兲 and SU共2兲 symmetries.

Let us go a bit further with the Barnett and Phoenix 关8兴 and Plenio and Vedral关9兴 ideas. Notice that S(␳A) is a

ho-mogeneous function of first degree in its dependency on EA,

the energy of the subsystem A. Then since we wish to maxi-mize S(␳A) we require␦S(EA)⫽0, under the constraints

EA⫽

具

HA

典

⫽Tr␳AHA, Tr␳A⫽1, 共3兲

where HA is the energy operator of systems A. Following

methods similar to those of statistical mechanics, we derive then a constraint equation for ␳_A, that is,

␣o⫺1⫹␣1HA⫹ln␳A⫽0, 共4兲

where␣oand␣1are the Lagrange multipliers attached to the given constraints. Using Eq.共4兲 we get a Gibbs-like density operator, that is,

␳A⫽

1

Zexp共␣1HA兲, 共5兲

where Z⫽exp(1⫺␣o). Multiplying Eq.共4兲 by␳A, taking the

trace and using Eqs. 共3兲 and 共5兲, we derive ln Z⫹␣1EA⫹S⫽0.

For the sake of convenience, let us write␣1⫽⫺1/␶, then we have␶ln Z⫽EA⫺␶S. The function F(␶)⫽␶ln Z describes

the Legendre transform of S since we assume that ␶

⫽⳵E/⳵S. Here ␶ is an intensive parameter describing the fact that the average EA⫽

具

HA

典

, given by Eq.共3兲, is constant

in the state described by ␳_A. Although the fluctuations can exist, these are not a result of any heat bath共or ensemble of states兲 but rather a consequence of the entanglement of the state A with B. Therefore, now we are in a position to look for an entangled state兩␺(A,B)

典

such that the corresponding reduced matrix as defined in Eq. 共1兲 is explicitly given by Eq. 共5兲, such that

Z⫽Tr exp共⫺␶HA兲. 共6兲

In this way we show in the following that, using the scheme of TFD for the SU共1,1兲 and SU共2兲 symmetries, we can ex-plicitly construct examples of maximum entanglement states, such that the measurement of the entanglement is given by Eq. 共2兲.

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II. MAXIMUM ENTANGLED STATES AND SU„1,1… SYMMETRY

Let us consider a two-mode bosonic system described by bosonic operators a,a†,b, and b†satisfying the algebraic re-lation

关b,b†_兴⫽关a,a†_兴⫽1, _共7兲

关a,b兴⫽关a,b†_兴⫽关a†_,b_兴⫽关a†_,b†_兴⫽0. _共8兲 Following the TFD approach 关11,12兴, with these two-boson operator, we can construct a two-mode linear canonical transformation presenting an SU共1,1兲 symmetry. First, con-sider the following operators:

S_⫹⫽a†b†, S_⫺⫽ab, So⫽ 1 2共a †_a_⫹bb†_兲,

which satisfy the su共1,1兲 algebra, namely, 关So,S⫾兴⫽S⫾,

and关S_⫹,S_⫺兴⫽⫺S_o. Introducing then a canonical transfor-mation by

U共␥兲⫽exp关␥共S_⫹⫺S_⫺兲兴,

典

⫽exp关␥共S⫹⫺S⫺兲兴兩0a,0b

典

, 共9兲

where ␥ is a parameter to be specified later. The canonical nature of U(␥) maintains the invariance of the algebra given by Eqs. 共7兲 and 共8兲 for the transformed operators given by

a共␥兲⫽U共␥兲aU共␥兲, a†共␥兲⫽U共␥兲a†U共␥兲, b共␥兲⫽U共␥兲bU共␥兲, b†共␥兲⫽U共␥兲b†U共␥兲.

.

Here we use the notation兩0(␥)

典

⫽兩␺(A,B)

典

in order to em-phasize that we have two bosonic systems, such that A de-scribes the degrees of freedom of the bosonic operators a and

a†, and B the operators b and b†. It is convenient to write Eq.

共9兲 by

兩␺共A,B兲

典

⫽exp关tanh␥a†b†兴exp关⫺ln cosh␥共bb†

⫹a†_a_{兲兴exp关tanh}_␥_{共⫺ba兲兴兩0}

a,0b

典

⫽exp共⫺ln cosh␥兲

兺

m 共⫺tanh␥兲 m ⫻_m!1 共a†_b†_兲m_兩0 a,0b

典

. 共10兲

Following the scheme delineated in the introduction, we show that the state兩␺(A,B)

典

is a maximum entangled state.

Consider␳(A,B)⫽兩␺(A,B)

典具

This expression can be written in the canonical Gibbs en-semble form by defining HA⫽wa⫹a and Z(␶)⫽Tre⫺␶HA ⫽(1⫺e⫺␶w)⫺1. Then we have

␳A⫽

1

Z共␶兲e ⫺␶HA_,

showing that the state兩␺(A,B)

典

⫽exp关␥(S_⫹⫺S_⫺)兴兩0a,0b

典

is a

maximum entangled state with symmetry SU共1,1兲. In the fol-lowing section we discuss a situation of an entangled state with SU共2兲 symmetry.

III. MAXIMUM ENTANGLED STATES AND SU„2… SYMMETRY

In order to construct a state of two systems A and B of maximum entanglement with SU共2兲 symmetry, we use the two-boson Schwinger representation for the su共2兲 Lie alge-bra, given 关14,15兴 S_⫹⫽a₁†a2, 共11兲 S_⫺⫽a₂†a1, 共12兲 So⫽ 1 2共a1 †_a 1⫺a2 †_a 2兲, 共13兲

where a1 and a2 are two bosonic operators. Then we have

关So,S_⫾兴⫽⫾S_⫾, 共14兲关S⫹,S⫺兴⫽2So. 共15兲

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With these two bosons关fulfilling the same algebra as that given in Eqs.共7兲 and 共8兲兴 the number operators

N₁⫽a₁†a₁, N₂⫽a₂†a₂, 共16兲 have the spectrum

_.

is the vacuum state for

S_⫺. The connection with the original su共2兲 algebra and the value of spin is obtained if we define

n1⫽s⫹m, 共21兲

n2⫽s⫺m, 共22兲

_, 共24兲

.

In order to construct the entangled state following the TFD procedure, we analyze another system of two bosons, denoted by b1 and b2, such that these operators commute among themselves and also with the boson operator a1 and

a2, giving rise to the doubling of the su共2兲 algebra. That is,

关So,S⫾兴⫽⫾S⫾, 共31兲关S⫹,S⫺兴⫽2So, 共32兲 关S˜o,S˜⫾兴⫽⫾S˜⫾, 共33兲 关S˜⫹,S˜⫺兴⫽2S˜o, 共34兲

such that the tilde operators S˜_⫺,S˜_⫹, and S˜o commute with

the nontilde operators and are now given by

S ˜_⫹_⫽b 1 † b2, 共35兲 S ˜_⫺_⫽b 2 † b1, 共36兲 S ˜ o⫽ 1 2共b1 † b1⫺b2 † b2兲. 共37兲

Consider the state兩␺(A,B)

典

given by

兩␺共A,B兲

典

⫽exp关␥共S⫹˜S⫹⫹S⫺˜S⫺兲兴兩0a,0b

典

, ⫽exp关␥共a1 † b₁†a2b2⫺a2 † b2†a1b1兲兴兩0a,0b

典

⫽共cos␥⫹sin␥a₁†b₁†a2b2兲兩0a,0b

典

, 共38兲

b₁兩m

典

b₂,

where the indices in the states as b₂ in兩1

典

_b

0兩a₂

具

1兩a₁. 共39兲

In the case of spin 1/2 we have 兩0

maximally entangled state. The generalization for any value of spin is straightforward.

IV. ENTANGLEMENT OF SYSTEM WITH FIXED SPIN: SCHWINGER METHOD REVISITED

In the preceding section 兩␺(A⫽S,B⫽S˜;␥)

典

was used to describe a maximally entangled state. If we consider an ar-bitrary spin共arbitrary values for the number operators n₁and

n2) value,兩␺(A⫽S,B⫽S˜;␥)

典

is a maximum entangled state of two systems, each one with two bosons. However, for the system of two 共defined兲 spin 1/2, for instance, we have the eigenvalues of the number operator as n1⫽0,1 and n2⫽0,1, which are no longer the spectrum of bosonic number opera-tors, but rather fermioniclike operators. In such a situation the bosonic algebra does not describe physical bosons but works as auxiliary variables to treat the entanglement of two

spin systems. Accordingly, since we define a fixed共not arbi-trary兲 value for the spin, we have to analyze more closely the consequences of that, over the Schwinger representation that is usually introduced for arbitrary spin.

Returning to the Schwinger bosonic representation as pre-sented in Sec. II, imposing the conditions on the spectrum of

n1 and n2 as is the case of Eqs.共25兲 and 共26兲共and so defin-ing s with a fixed value兲, then we are led to a situation of redefining the algebra of the auxiliary operators a1 and a2. Summarizing first our results, originally the operator a1 and

a₂ satisfy the bosonic algebra, say

关a1,a1 †_兴⫽关a 2,a2 †_兴⫽1, _共40兲关a1,a2兴⫽关a1,a2 †_兴⫽关a 1 † ,a2兴⫽关a1 † ,a₂†兴⫽0. 共41兲 In addition, for the spin 1/2, we have the subsidirary condi-tions 共allowing the fixed value for the spin兲,

关a1,a1 †_兴 ⫹⫽关a2,a2 †_兴 ⫹⫽1, 共42兲关a1,a2兴_⫹⫽关a1,a2 †_兴 ⫹⫽关a1 †_,a 2兴_⫹⫽关a1 †_,a 2 †_兴 ⫹⫽0, 共43兲

where 关,兴_⫹ stands for the anticommutator. A solution that fulfills all these conditions, Eqs. 共40兲–共43兲, is found by as-suming the algebra for the operators ai and ai

† (i⫽1,2) to be aiai †_⫽1, _共44兲 关Ni,ai †_兴⫽a i †_, _共45兲 关Ni,ai兴⫽⫺ai, 共46兲 with aiai⫽ai † a_i†⫽0 and ni⫽ai †

ai. Indeed it is a simple

mat-ter to show that in this case Ni⫽0,1, where i⫽1,2.

For the case of spin 1, we consider the basic algebra given by Eqs. 共44兲–共46兲 with Ni given by

Ni⫽ai

†

ai⫹ai

†

a_i†aiai,

such that a_i†a_i†a_i†⫽a_ia_ia_i⫽0, that is, three and higher

mono-mials of a_i† and ai are zero. In this case we derive ni ⫽0,1,2, and with Eqs. 共21兲 and 共22兲 we find s⫽1 and m ⫽⫺1,0,1.

The above procedure can be generalized for an arbitrary but fixed value of spin. That is, for a spin s, such that ni ⫽2s, we consider Eqs. 共44兲–共46兲 supplemented by a proper

definition of Ni, which reads

Ni⫽

兺

j⫽1 2s 共ai †_兲j_共a i兲j. 共47兲

For the particular situation in which s→⬁, we derive the approach of infinite statistics proposed by Greenberg 关16兴, and Chung关17兴. In the general situation presented here, it is worthy of noting that the algebra given in Eqs. 共44兲–共46兲 also satisfies a Hopf structure. Let us investigate this aspect closely.

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⌬r_共a兲⫽1 丢a, ⌬r_共a†_兲⫽1_丢_a†_, ⌬r_共N兲⫽N 丢1⫹1丢N, ⌬r_共I兲⫽I 丢I,

where⌬r provides a representation for the algebra given by Eqs. 共44兲–共46兲 for any value of s in Eq. 共47兲. Actually this fact has been shown in the particular situation in which s →⬁, in the context of the Greenberg operators 关16,17兴. A Hopf structure is introduced by using ⌬r as the coproduct and defining the counit ⑀and the antipode s by

⑀共I兲⫽1, ⑀共a兲⫽a, ⑀共a†_兲⫽a†_, _⑀_{共N兲⫽0, 共48兲}

s共I兲⫽1, s共a兲⫽a, s共a†兲⫽a†, s共N兲⫽⫺N. 共49兲

Another Hopf algebra is introduced, however, for the opera-tors a1⫽a丢I and a1

†_⫽a†

丢I, defining the coproduct, say⌬l, by ⌬l_共a兲⫽a_丢_1, ⌬l_共a†_兲⫽a† 丢1, ⌬l_{共N兲⫽⌬}r_共N兲, ⌬l_{共I兲⫽⌬}r_共I兲.

The counit⑀and the antipode s are the same as those given in Eqs.共48兲 and 共49兲. It is then a simple matter to verify all the Hopf-algebra axioms. Indeed, for the coproduct axiom we have (i丢⌬l₎_⌬l_(a)_⫽(i

丢⌬l_)(a

丢1)⫽a丢1丢1

⫽(⌬l_丢_i)(a_丢_{1), and so (i}_丢_⌬l₎_⌬l_⫽(⌬l_丢_i)_⌬l_{. Following}

the same procedure we can verify the counit axiom (⑀丢⌬l₎_⌬l_⫽(⌬l

丢⑀)⌬l as well as the antipode axiom (i丢s)⌬l⫽(s丢i)⌬l.

Closing this section let us emphasize, first, that a motiva-tion for studying the Hopf-algebra structure attached to the entangled states with SU共2兲 symmetry is for the sake of ex-periments. Actually, in the way as the algebra given in Eqs.

共44兲–共46兲 has emerged in our formalism, the Hopf-algebra

induced by that can be used to introduce a deformation pa-rameter, say q, associated with SU共2兲关18,19兴. This q param-eter in turn can be useful for fitting experimental results. Second any representation of the su共2兲 algebra can be de-composed in terms of operators a and a† satisfying Eqs.

共44兲–共46兲 with N given in Eq. 共47兲 for some fixed s, such a

formalism can be used to describe not only fixed spin but also, for instance, isospin. In the case of pion, isospin 1, it would be thought as being composed of 共at least, auxiliary兲 objects described by the operators a1,a1

†_,a

2, and a2

†_{. In this}

way we could find a physical interpretation for the Green-berg infinite-statistics approach 关18兴.

V. CONCLUDING REMARKS

In this paper we have improved the Barnett and Phoenix

关8兴 and Plenio and Vedral 关9兴 definition for a measure of

entanglement among states, based on a Liouville-von Neu-mann entropy. We then explore the similarity with the usual definition of entropy in statistical mechanics to construct states maximally entangled using the approach of ther-mofield dynamics. As TFD is a thermal formalism essentially founded on algebraic bases 共duplication of the usual Hilbert space and Bogoliubov transformations兲, it has been used as a compass to give the proper direction to build maximum en-tangled states with a well-defined symmetry. We have stud-ied the case of two bosons with SU共1,1兲 symmetry and four bosons 共actually two systems, each one with two bosons兲 with symmetry SU共2兲. Still in the case of SU共2兲 symmetry, we have studied the entanglement of two systems with fixed value of spin, using a modified version of the two-boson Schwinger representation for the su共2兲 Lie algebra.

The usual way to describe a fixed value of spin using boson operators was proposed by Holstein and Primakof

关20兴. However, such a method works if we are interested in

describing a system with spin via one bosonic operator. But this has not been the case here, since it has needed two operators associated to each spin to introduce the state of maximum entanglement through TFD. Obviously that for a finite spectrum, the couple of Schwinger operators loose the bosonic characteristic 共up to now, an aspect not investigated in the literature, to the best of our knowledge兲, giving rise to a new algebra. What we have shown were some properties of the new operators.

With these results we can set forth some final conclusions:

共i兲 the TDF states, seen as pure states, are naturally

maxi-mum entangled states; 共ii兲 the SU共1,1兲 symmetry could be written in terms of a doubled bosonic algebra, such that it can be derived from elements of a coproduct given by

⌬(a)⫽a丢1⫹1丢a⫽(⌬l⫹⌬r)a and ⌬(a†)⫽a†丢1⫹1 丢a†⫽(⌬l⫹⌬r)a† associated with the Weyl-Heisenberg al-gebra, induced from the bosonic algebra. This is similar to the case of the SU共2兲 symmetry. As a consequence, the struc-ture of Hopf algebra is at the bottom of all the construction, since the left (⌬l) and the right (⌬r) parts of coproducts have been used to find specific representations for the SU共2兲 and SU共1,1兲 symmetries and also to the TDF states 共which are here synonymous with the maximum entangled states兲.

ACKNOWLEDGMENTS

The authors wish to acknowledge the support by the Technion-Haifa University Joint Research Fund, by the Fund for Promotion of Sponsored Research at the Technion, CNPq

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