Scholarly article on topic 'A hierarchy of Hamilton operators and entanglement'

A hierarchy of Hamilton operators and entanglement Academic research paper on "Physical sciences"

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Academic research paper on topic "A hierarchy of Hamilton operators and entanglement"

Cent. Eur. J. Phys. • 7(4) • 2009 • 854-859 DOI: 10.2478/s11534-009-0075-z

Central European Journal of Physics

A hierarchy of Hamilton operators and entanglement

Short Communication

Willi-Hans Steeb*, Yorick Hardy

International School for Scientific Computing, University of Johannesburg, Auckland Park 2006, South Africa

Received 11 February 200S; accepted 10 April 200S

VERS ITA

Abstract: We consider a hierarchy of Hamilton operators HN in finite dimensional Hilbert spaces C2 . We show

that the eigenstates of HN are fully entangled for N even. We also calculate the unitary operator UN(t) = exp(-iHNt/h) for the time evolution and show that unentangled states can be transformed into entangled states using this operator. We also investigate energy level crossing for this hierarchy of Hamilton operators.

PACS C200B): 03.65.Ud, 03.67Bg, 31.15xh

Keywords: entanglement • energy level crossing • eigenvalue problem

© Versita Warsaw and Springer-Verlag Berlin Heidelberg.

1. Introduction

with the corresponding normalized eigenvectors

Two-level quantum systems and their physical realizations have been studied by many authors [1-11]. The Hamilton operator H is given by

H = hwoz + Aox =

hw A A -hw

where w is the frequency and A is a real parameter (dimension energy). Throughout the paper we use the abbreviation

E =y/ h2w2 + A2 .

VA2 + (E - hw)2 \ E - hwj '

_1_ ( A ).

VA2 + (E + hw)2 \-E - hw j

We can use the Cayley-Hamilton theorem to calculate U|(t) = exp(-iH1tlh) and obtain

cos (Et IK) - í sin (Et/K)Kw/E -í sln(Et/K)A/E

-í sln(Et/K)A/E cos (Et/K) + í sln (Et/K)Kw/E

The eigenvalues of H1 are

E± = +V h2w2 + A2 .

*E-mail: steebwLllL@gmail.com

Obviously, exp(-iHit/h) is a unitary matrix. Here we study a higher dimensional extension, we consider the hierarchy of Hamilton operators

N-factors

N-factors

H N = Kw(oz ® oz oz ) + A(ox ® ox %■■■% ox ),

Springer

with N > 1. For N > 2 we can also study entanglement. Here ® denotes the Kronecker product [11—13], ox, az are the Pauli spin matrices, w> 0 and A > 0. Thus the Hamilton operator Hn acts

in the Hilbert space H = C2 . Entanglement and energy level crossing are studied for N > 2.

2. Properties of the Hamilton operators

The properties of the Hamilton operators depend strongly whether N is even or odd. Since trHn = 0 for all N we obtain

E Ej = 0 ■

where Ej are the eigenvalues of Hn. Consider the operators

Zzin = oz » az

ZXiN = Ox » Ox

We have to distinguish between the case N even and the case N odd. If N is even then the commutator vanishes, i.e.

[Ez.N, tx.N 1 = 0 .

If N Is odd then the anti-commutator vanishes, i.e

E1 = A + ho, E2 = -(A + ho), E3 = A - ho, E4 = -(A - ho). with the corresponding normalized eigenvectors

|0+ ) =

|Y + ) =

|T-) =

|*-) =

0 0 -i

Note that the states do not depend on the parameters o and A. These states are the Bell states (see [10, 11] and references therein). The Bell states are fully entangled. Entanglement of states in finite-dimensional Hilbert spaces (dimH > 4) has been investigated by many authors (see [10,11] and references therein). The measure of entanglement for bipartite states are the von Neumann entropy, concurrence and the 2-tangle. As a measure of entanglement we apply the tangle which is the squared concurrence. The concurrence C for a pure state |m) in H = C4 is given by

[ez.n, ex.n 1+ =0.

coo coi

Cio c11

Note that Zx,n and Zz,n are elements of the Pauli group. Thus setting Hn = HN0 + Hni with

with the state m) written in the form

H No = ho(Oz » Oz

1 Oz), HN1 = A(Ox » Ox

we find that for N even [H N0, H ni] = 0. Then the unitary operator Un(t) = exp(-iHNt/h) can easily be calculated since

Un(t) = exp(-iHN0t/h) exp(-iHni t/h) ■

m = E cJk n )»k )

and \J) (J = 0,1) denotes the standard basis in the Hilbert space C2. Next we calculate exp(-iH'2t/h). Since

If N is odd we have [HN0, Hni]+ = 0. Here too the time evolution Un (t) = exp(-iH Nt/h) can easily be calculated.

U2(t) = exp(-lH2t/h) = e-lot(°z »Oz ' e

= e-lot(Oz »Oz ) e-ltA(Ox »Ox )/h

3. Special cases e-M = ^m - ^ ® ^m

e-lAt(0xmx>№ = /4cos(tA/h) - i(ax ® ax)sin(tA/h)

We consider now the cases N = 2 and N = 3. Then we generalize to arbitrary N. Consider the case N = 2. The four eigenvalues are given by we obtain

-íH2t/h

= I4 cos(wf) cos(fA/h) — í(az 0 oz) sln(wf) cos(f A/h) — í(ax 0 ax) cos(wf) sln(fA/h) — ) 0 (ozox) sln(wf) sln(fA/h).

The Hamilton operator H2 shows energy level crossing (when keeping hu fixed and varying A). The unitary operator U2(t) = exp(-iH'2t/h) can generate entangled states from unentangled states. However note that applying the unitary operator U2(t) to one of the Bell states given above cannot disentangle these states since they are eigenstates. For example, we have U2(t)|$+) = e-iEit№|$+).

If we start with the unentangled state (product state)

= (1000)7

under the evolution U2(t)depending on t, hu and A we can find entangled states using the concurrence as measure. For the case hu = A (level crossing) the state reduces to

j cos2(ut) — i sin(ut) cos(ut) \ 0 0

- sin2(ut) - i cos(ut) sin(ut)

Consider now the case N = 3. For the case N = 3 we find the eigenvalue E = Vh2u2 + A2 (four-times degenerate) with the normalized eigenvectors

va2 + (e — m2

va2 + (e + m2

A 0 0 0 0 0 0

\e — huj

A 0 0 0 0

E + hw 0

va2 + (e — hw)2

va2 + (e + hw)2

E — hw 0 0 0

0 0 A 0 0

E + hw 0 0

and the eigenvalue —E (four times degenerate) with the normalized eigenvectors

va2 + (e + hw)2

A 0 0 0 0 0 0

\—E — hw/

va2 + (e + hw)2

— E — hw 0 0 0

va2 + (e — hw)2

0 A 0 0 0 0

—E + hw 0

Va2 + (e — hw)2

0 0 A 0 0

— E + hw 0 0

If A = E — hu then the first and second eigenstates are fully entangled. If A = E + hu the third and fourth states are fully

entangled. As measure we can use the 3-tangle [14].

Since we have the eigenvalues and eigenvectors of H3 the unitary operator U'3(t) can easily be calculated. We find

The eigenvalues are 2N 2 times degenerate. The corresponding 2n normalized eigenvectors for the case N even are

U3(t) = /8 cos(Et/K)

+ ^ (Kwaz 0 az 0 az + Aax 0 ax 0 ax) sin (Et/K)

If we start with the unentangled state (product state)

p) = (10000000)T

under the evolution U(t)\p) depending on t, Kw and A we can find entangled states.

4. General case

Consider now the general cases. If N is odd the Hamilton operator has only two eigenvalues, namely E and -E. Both are 2N-1 times degenerate. The eigenvectors for +E before normalization are given by

E — hw 0

0 E + hw

\E — hw)

The eigenvectors for -E before normalization are given by

0 —E + hw

—E — hw

—E — hw 0

For N odd the time evolution is given by

UN (t) = e(-iwtZzN-<-At^xN I1»

= I2N cos(ti/h) + I---sin(fci/h).

For N even the four eigenvalues are given by

E1 = Kw + A' E2 = -Kw - A' E3 = -Kw + A' E4 = Kw - A.

/ 1 \ 0\ 0

0 0 0

1 1 1

0 ' 72 0 .....71 ±1 0

0 ±1

\±V 0 Vo/

They do not depend on A and Kw. The first vector corresponds to the GHZ-states. These 2N eigenvectors form an orthonormal

basis in the Hilbert space C2 . Wong and Christensen [15] introduced an n-tangle for all even n and n = 3. Using this measure of entanglement the eigenvectors given above are fully entangled. For N even the unitary operator Un(t) for the time evolution is given by

e(-iwt"Lz'NxN)№ = /2n cos(wt) cos(At|K)

- iLzin sin(wt) cos(At/K) - iZXiN cos(wt) sin(At/K)

- Zzn sin(wt) sin(At/K).

5. Energy level crossing

A basic problem in quantum mechanics is the calculation of the energy spectrum of a given (hermitian) Hamilton operator H. It is assumed that the hermitian Hamilton operator acts in a Hilbert space "H. Here we assume we have a finite dimensional Hilbert space. Thus the spectrum is discrete. In many cases the Hamilton operator depends on a real parameter. The question whether or not energy levels can cross by changing the parameter was first discussed by Hund [16]. He studied examples only and conjectured that, in general no crossing of energy levels can occur. In 1929 von Neumann and Wigner [17] investigated this question more rigorously and found the following theorem: Real symmetric matrices (respectively the hermitian matrices) with a multiple eigenvalue form a real algebraic variety of codimension 2 (respectively 3) in the space of all real symmetric matrices (respectively all hermitian matrices). This implies the famous "non-crossing rule" which asserts that a "generic" one parameter family of real symmetric matrices (or two-parameter family of hermitian matrices) contains no matrix with multiple eigenvalue. "Generic" means that if the Hamilton operator HH admits symmetries the underlying Hilbert space has to be decomposed into invariant Hilbert subspaces using group theory [18]. A large number of papers have published studying energy level crossing (see [19] and references therein).

For the case N even we can study energy level crossing. We restrict ourselves to the case N = 2. Energy level crossing occurs if A = Kw. Then we have the eigenvalues E1 = 2Kw, E3 = 0, E4 = 0, E2 = -2Kw. For the degenerate eigenvalue 0 we have

the eigenvectors

|Y+> =

/0\ i i

|0-> =

i1 \ 0 0

The projection operator ni projects into a two-dimensional Hilbert space spanned by elements of the standard basis

Since we have energy level crossing the Hamilton operator H2 with the corresponding matrix for the Hamilton operator admits a discrete symmetry. We have

[H2,OX 0 Ox ] = 0, [H2,Oz 0 Oz ] = 0 ■ (a hu

Now both { /2 0 /2, ox 0 ox } and { /2 0 /2, oz 0 oz } form a finite abelian group under matrix multiplication, where /2 is the 2 x 2 identity matrix. Both can be used to find the reduction of the Hilbert space C4 to Hilbert subspaces. Consider first the group { /2 0 /2, ox 0 ox }. The character table provides the projection operators

n = -(/2 ® I2 + ox. ® ax), n2 = — (/2 ® /2 - ax ® ax)

The projection operator n2 projects into a two-dimensional Hilbert space spanned by the elements of the standard basis

with the corresponding matrix for the Hamilton operator

The projection operator ni projects into a two-dimensional Hilbert space spanned by the Bell states

-hu A A -hu

1 0 1 1

72 0 , 72 1

Note that the elements of these two groups are elements of the Pauli group V2 which is defined by

V„ = { /2,ox,av,az }®n ®{±1, +i },

n = 1,2,

with the corresponding matrix for the Hamilton operator

hu - A 0 \ 0 -hu - A/ '

where /2 is the 2 x 2 identity matrix. For N > 2 and N even the Hamilton operators admit more symmetries since the eigenvalues are degenerate 2N—2 times. Again we can reduce the Hilbert

space C2 to Hilbert subspaces.

The projection operator n2 projects into a two-dimensional Hilbert space spanned by the Bell states

I 1 \ I 0 \

1 0 1 1

72 0 , 12 -1

—1 0

with the corresponding matrix for the Hamilton operator

hu + A 0 \ 0 —hu + a) ■

Consider now the group { /2 0 /2, oz 0 oz }. The character table provides the projection operators

ni = ~ (/2 0 /2 + Oz 0 Oz), n = ^(/2 0 /2 — Oz 0 Oz)■

6. Conclusion

We have studied a hierarchy of Hamilton operators in the Hilbert

space C2 . The behaviour depends strongly on whether N is even or odd. For the case N odd the Hamilton operator admits only two eigenvalues and the eigenvectors depend on hu and A. If N is even the eigenvectors do not depend on the parameters hu

and A and form an entangled basis in the Hilbert space C2 . The unitary operator Un = exp(—iHNt/h) (N > 2) can convert unentangled states into entangled ones and vice versa.

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