Quantum Teleportation for Nonmaximal Entangled States in the Generalized Bell Measurement with Latin Square

Quantum Teleportation for Nonmaximal Entangled States in the Generalized Bell Measurement with Latin Square

REPORTS ON MATHEMATICAL PHYSICS Vol. 69 (2012) No. 1 QUANTUM TELEPORTATION FOR NONMAXIMAL ENTANGLED STATES IN THE GENERALIZED BELL MEASUREMENT WITH...

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REPORTS ON MATHEMATICAL PHYSICS

Vol. 69 (2012)

No. 1

QUANTUM TELEPORTATION FOR NONMAXIMAL ENTANGLED STATES IN THE GENERALIZED BELL MEASUREMENT WITH LATIN SQUARE YOSHIHARU TANAKA , M ASANARI A SANO and M ASANORI O HYA Department of Information Sciences, Tokyo University of Science, Noda, Chiba 278-8510, Japan (e-mails: [email protected], [email protected], [email protected]) (Received January 29, 2011) In this paper, we construct a teleportation model with nonmaximal entangled state. This model, called the m-level teleportation, is discussed on the basis of the Kossakowski and Ohya teleportation scheme. For this study, we define a generalized Bell state in terms of Latin square, by which we derive a general form of appropriate nonmaximal entangled state for a perfect m-level teleportation. Keywords: teleportation, entanglement, Bell state.

1.

Introduction

The idea of quantum teleportation was proposed by Bennett et al. [2]. In their model, an EPR pair of qubits is shared by Alice and Bob. Alice’s purpose is to send a quantum state of input qubit a |0 + b |1 to Bob. She performs a quantum measurement on the input qubit and a qubit of EPR pair at her side. The quantum measurement is a joint measurement called the Bell measurement. Alice informs Bob about the result of her measurement via a classical channel, then Bob can recover the original input state by applying a unitary transformation to the qubit at his side. Recently A. Kossakowski and M. Ohya discussed a general formalism of teleportation process [7]. In their representation, an input state is written as an N-dimensional state vector in a Hilbert space H1 , an entangled state between Alice and Bob is given by a state vector in N 2 -dimensional Hilbert space H2 ⊗ H3 . The measured states are written by a set of N 2 orthogonal state vectors defined in H1 ⊗ H2 . The process of teleportation is mathematically represented as a map of an input state to an output state which Bob obtains after Alice’s measurement. Such a map is called the teleportation map. If this map is linear, then there exists a unitary key to recover the input state from the output state. When an entangled state on H2 ⊗ H3 is maximal, the teleportation map is always linear, and the teleportation succeeds perfectly. Many experiments for realization of teleportation tried to make and preserve the maximal entangled state by various physical resources [3–5]. However, it is difficult [57]

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Y. TANAKA, M. ASANO, and M. OHYA

to generate such ideal entangled state, and most of states realized in the experiments are nonmaximal entangled states. Therefore it is important to discuss how to succeed in teleportation with high fidelity by a realizable nonmaximal entangled state. For such problem, Kossakowski and Ohya defined an ideal teleportation map such that its linearity is preserved even if nonmaximal entangled state is postulated. In [1], we proposed a model of perfect teleportation, in which the ideal teleportation in the Kosakowski–Ohya scheme might be realized. In this model, called the two-level teleportation, an input state is represented as | = a |x + b |y, n where |x and |y are mutually orthogonal state vectors in H1 = C2 , and the complex numbers a and b satisfy |a|2 + |b|2 = 1. We showed that such a two-level state | can be transmitted perfectly with a nonmaximal entangled state defined n n on H2 ⊗ H3 = C2 ⊗ C2 (n ≥ 2). Moreover, in [14], we designed a physical model of two-level teleportation in quantum optics. The input state is described as a superposition of Schr¨odinger’s cat state [9, 12]. The shared entangled state is generated by a photon number state and beam-splitter [6], and it is nonmaximal in general. Alice measures the sum of photon numbers and the phase difference of photons [8, 10]. Bob can recover the input state by shifting the phase [13]. In the present paper, we discuss the scheme of two-level teleportation in detail, and we propose an extended model, called the m-level teleportation. An input state  |x a of the m-level teleportation is an m-level state vector | = m k=1 k k , where ak mn 2 m are the complex numbers satisfying k=1 |ak | = 1, and the state vectors {|xk }k=1 are mutually orthogonal. The m-level teleportation succeeds perfectly when the input state, the entangled state and the measurable states satisfy a certain condition shown in Section 5. The previous study of [1] only showed an example of such condition. We discuss the condition for the perfect m-level teleportation generally. We define the general form of the Bell state by Latin square. The generalization of the Bell state was showed in [2], and it is useful for quantum dense coding [15]. Our generalization is different from the one in [2]. In Section 2, we briefly explain the Kossakowski and Ohya scheme. In Section 3, we explain the details of two-level teleportation. It is shown that there exists a set of unitary keys even if nonmaximal entangled state is used. In Section 4, we discuss the measurement in two-level teleportation, and we define the generalized Bell state. In Section 5, we extend the two-level teleportation to m-level teleportation, where the generalized Bell state plays important role. 2.

The Kossakowski and Ohya scheme for perfect teleportation

In this section, we briefly explain the scheme of teleportation proposed by Kossakowski and Ohya. Let us take the conditions that all Hilbert spaces H1 , H2 and H3 be CN . Alice has an unknown quantum state ρ on H1 and she transfers it to Bob. For this purpose, an entangled state σ is prepared on H2 ⊗ H3 , and H2 is attached to Alice’s side and H3 is to Bob’s side. Any entangled state σ can be represented as

QUANTUM TELEPORTATION FOR NONMAXIMAL ENTANGLED STATES

σ =

N 

|i j | ⊗

N2 

μs fs |i j | fs∗

59

 (1)

s=1

i,j =1

N 2 N with s=1 μs = 1 and μs ≥ 0. Here {|i}i=1 is the fixed orthonormal basis (ONB) 2 N in CN , and {fs }N s=1 is an ONB in the set of all bounded operators on C , which is simply denoted by MN . Alice performs a joint measurement of the observable F on H1 ⊗ H2 , which is defined by N N2 N2    F = λk Pk ≡ λk gk∗ |i j | gk ⊗ |i j | , k=1

k=1

i,j =1

2

where {gk }N k=1 is another ONB in MN . Kossakowski and Ohya defined a not normalized teleportation map for an input state ρ and the measured value λk by Tk (ρ) ≡ tr12 [(Pk ⊗ I ) ρ ⊗ σ (Pk ⊗ I )] =

N2 

μs fs gk ρgk∗ fs∗ ,

s=1

where I is a unity of MN , and tr12 is the partial trace over the space H1 ⊗ H2 . It is easily seen that Tk (ρ) is completely positive but not trace preserving. In order to make a trace preserving map from Tk (ρ), they considered the dual map T˜k (ρ) of Tk (ρ) such that trATk (ρ) =trT˜k (A)ρ is satisfied. It is expressed as T˜k (A) =

N2 

μs gk∗ fs∗ Afs gk ,

A ∈ MN .

s=1

The map T˜k is normalizable if and only if the operator T˜k (I ) is invertible, i.e. rankT˜k (I ) = N . In this case, the dual teleportation map T˜k is normalized as 1

1

− − ϒ˜ k = κk 2 T˜k κk 2 ,

where κk ≡ T˜k (I ) =

N2 

μs gk∗ fs∗ fs gk .

s=1

The dual map of ϒ˜ k , ϒk (ρ) =

N2 

−1

−1

μs fs gk κk 2 ρκk 2 gk∗ fs∗

s=1

is trace preserving and linear. In the K-O scheme, ϒk is the teleportation map, that is, the state ϒk (ρ) means an output state which Bob obtains after the measurement of λk .

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Let us consider a case when an entangled state σ is pure, σ =

N 

|i j | ⊗ f |i j | f ∗ ≡ |ξ  ξ | ,

(2)

i,j =1

where |ξ  =

N 

|i ⊗ f |i .

i=1

Note that the operator f satisfies trf ∗ f = 1. The state σ is a maximal entangled state if and only if f ∗ f = ff ∗ = I /N . If rankf = rankgk = N, then the output state of the teleportation map ϒk is −1

−1

ϒk (ρ) = f gk κk 2 ρκk 2 gk∗ f ∗ for the input state ρ. Here the operator κk is defined as κk = gk∗ f ∗ f gk . Bob can recover the input state ρ from ϒk (ρ) with the following unitary key, −1

Uk = κk 2 gk∗ f ∗ . Remark that there are a lot of nonmaximal entangled states σ satisfying the condition of rankf = rankgk = N. Thus a teleportation realizing the process of the map ϒk succeeds perfectly even if a nonmaximal entangled state is used. 3. 3.1.

Two-level teleportation The concept of two-level teleportation

The two-level teleportation proposed in [1] is a realization for the idea of teleportation map ϒk . In that model, the input state on H1 is a pure state ρ = | | with the state vector | = a |x + b |y , where |x and |y are mutually orthogonal. An entangled state σ on H2 ⊗ H3 is a pure state defined by σ = |   | ,

|  =

N 

|ei  ⊗ f |ei  ,

i=1 N ∗ where the {|ei }N i=1 is an ONB of C , and the operator f satisfies trf f = 1. The observable which Alice measures is

F =

N2  k=1

λk Pk .

QUANTUM TELEPORTATION FOR NONMAXIMAL ENTANGLED STATES

61

Here λk is the value obtained by Alice, and Pk is the projection operator corresponding to λk defined by Pk = | k   k | ,

| k  =

N 

gk∗ |ei  ⊗ |ei  ,

(3)

i=1 2

N where the {gk }N and they satisfy trgk∗ gk = δkk . When k=1 are operators on C  Alice   measures λk , the output state which Bob obtains on H3 is written as ρk = k  k  with    = f gk κk − 12 | k

1

1

= a · f gk κk − 2 |x + b · f gk κk − 2 |y ., where κk = gk∗ f ∗ f gk and κk  = | κk |. Here, let us introduce the map k from ρ to ρk : k (ρ) = ρk . When the equations x| κk |x = y| κk |y , x| κk |y = y| κk |x = 0

(4) (5)

are satisfied, there exists a unitary operator Wk such that Wk k Wk∗ is equivalent to the ideal teleportation map ϒk . In this sense, ϒk can be realized under the conditions of (4) and (5). Indeed, when these conditions are satisfied, the output state vector is rewritten as        = a x + b y , k k k       where xk and yk are given by     y = √f gk |y . x = √f gk |x , (6) k k y| κk |y x| κk |x     Note that the state vectors xk and yk are mutually orthogonal, so that there exists a unitary operator Uk such that     Uk x = |x , Uk y = |y , (7) k

k

and Uk plays the role of key to recover the input ρ, Uk ρk Uk∗ = ρ. We rewrite Eqs. (4) and (5) as N 

 

fi | κk fj x¯i xj − y¯i yj = 0,

(8)

i,j =1

by means of Eqs. (9)–(11) below, N  i,j =1

N       fi | κk fj x¯i yj = fi | κk fj y¯i xj = 0, i,j =1

(9)

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with the orthogonal basis {|fi }N i=1 , which are derived from the Schatten decomposition of f ∗ f . The Schatten decomposition of f ∗ f is explained as follows: There  exists a unique spectral decomposition f ∗ f = N i=1 ωi Ei , where ωi is an eigenvalue of f ∗ f and Ei is the corresponding projection to each ωi . The projection Ei is not one-dimensional when ωi is degenerate, so that the spectral decomposition can be further decomposed into one-dimensional projections. Such a decomposition is called the Schatten decomposition [11], namely, f ∗f =

N 

ωi |fi  fi | ,

(10)

i=1

where |fi  fi | is the one-dimensional projection associated with ωi and the degenerate eigenvalue ωi repeats dim Ei times; for instance, if the eigenvalue ωi has the degeneracy 3, then ω1 = ω2 = ω3 < ω4 . The values of xi and yi in Eqs. (8) and (9) are coefficients in the form of |x =

N 

xi |fi  ,

i=1

|y =

N 

yi |fi  .

(11)

i=1

From Eqs. (8) and (9) we show that a perfect teleportation is realized even if σ is not a maximal entangled state.

3.2.

An example of a perfect two-level teleportation

We show an example of a perfect teleportation in the case of N = 4. We take  an entangled state σ = |   | , |  = 4i=1 |ei  ⊗ f |ei  , with rank(f ) = 4. As discussed in the previous section, for a perfect teleportation, the conditions of Eqs. (8) and (9) should be satisfied for the given f . That is, we should find an appropriate  in κk = gk∗ f ∗ f gk and an appropriate basis of input |x = 4i=1 xi |fi , set {gk }16 k=1  |y = 4i=1 yi |fi  . Let us consider the following operators {gk }16 k=1 : g1 g5 g9 g13

= X 1 Y1 , = X 1 Y2 , = X 1 Y3 , = X 1 Y4 ,

g 2 = X 2 Y1 , g 6 = X 2 Y2 , g10 = X2 Y3 , g14 = X2 Y4 ,

g 3 = X 3 Y1 , g 4 = X 4 Y1 , g 7 = X 3 Y2 , g 8 = X 4 Y2 , g11 = X3 Y3 , g12 = X4 Y3 , g15 = X3 Y4 , g16 = X4 Y4 ,

(12)

where Xk and Yk are operators represented by the following matrices in the basis

QUANTUM TELEPORTATION FOR NONMAXIMAL ENTANGLED STATES

of {|fi }4i=1 ,

1 1 1 1 1 X1 = √ I, X2 = diag √ , − √ , − √ , √ , 4 4 4 4 4 1 1 1 1 X3 = diag √ , − √ , √ , − √ , 4 4 4 4 1 1 1 1 X4 = diag √ , √ , − √ , − √ , 4 4 4 4 ⎛

63

(13)

⎞ 0 1 0 0

Y1 = I, ⎛ 0

⎜ ⎜0 ⎜ Y3 = ⎜ ⎜1 ⎝ 0

⎜ ⎟ ⎜ 1 0 0 0⎟ ⎜ ⎟ Y2 = ⎜ ⎟, ⎜ 0 0 0 1⎟ ⎝ ⎠ 0 0 1 0 ⎛ ⎞ 0 1 0 0 ⎜ ⎟ ⎜0 0 0 1⎟ ⎟ ⎜ Y4 = ⎜ ⎟, ⎜0 0 0 0⎟ ⎠ ⎝ 1 0 0 1

Then the operators κk = gk∗ f ∗ f gk are written as ⎧ ω ω ω ω  1 2 3 4 ⎪ diag , , , (k ⎪ ⎪ ⎪ 4 4 4 4 ⎪ ⎪   ⎪ ω2 ω1 ω4 ω3 ⎪ ⎪ , , , (k ⎨diag 4 4 4 4 κk = ω ω ω ω  3 4 1 2 ⎪ ⎪ , , , (k ⎪diag ⎪ ⎪ 4 4 4 4 ⎪ ⎪   ⎪ ⎪ ⎩diag ω4 , ω3 , ω2 , ω1 (k 4 4 4 4

⎞ 0

0

0

1

1

0

0

0

1

⎟ 0⎟ ⎟ ⎟. 0⎟ ⎠ 0

(14)

= 1, 2, 3, 4) = 5, 6, 7, 8) (15) = 9, 10, 11, 12) = 13, 14, 15, 16)

Here {ωi }4i=1 are eigenvalues of f ∗ f in the Schatten decomposition of Eq. (10). By means of Eq. (15), we rewrite Eqs. (8) and (9) as ⎛ ⎞ |x1 |2 − |y1 |2 ⎜ ⎟  ⎜|x |2 − |y |2 ⎟  2 ⎟ ⎜ 2 (16) ⎟ = 0, f1 | κk |f1  f2 | κk |f2  f3 | κk |f3  f4 | κk |f4  ⎜ 2 ⎜|x3 | − |y3 |2 ⎟ ⎝ ⎠ |x4 |2 − |y4 |2

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x¯1 y1



⎜ ⎟   ⎜x¯ y ⎟ ⎜ 2 2⎟ ⎟ f1 | κk |f1  f2 | κk |f2  f3 | κk |f3  f4 | κk |f4  ⎜ ⎜x¯3 y3 ⎟ ⎝ ⎠ x¯4 y4



x1 y¯1



⎜ ⎟  ⎜x y¯ ⎟  ⎜ 2 2⎟ = f1 | κk |f1  f2 | κk |f2  f3 | κk |f3  f4 | κk |f4  ⎜ ⎟ = 0. (17) ⎜x3 y¯3 ⎟ ⎝ ⎠ x4 y¯4 The perfect teleportation succeeds if and only if Eqs. (16) and (17) are satisfied for any k. It is easily checked that when the following equations ⎞⎛ ⎞ ⎛ |x1 |2 − |y1 |2 ω1 ω2 ω3 ω4 ⎟⎜ ⎟ ⎜ ⎜ω ω ω ω ⎟ ⎜|x |2 − |y |2 ⎟ 1 4 3⎟ ⎜ 2 2 ⎟ ⎜ 2 (18) ⎟⎜ ⎟ = 0, ⎜ ⎜ω3 ω4 ω1 ω2 ⎟ ⎜|x3 |2 − |y3 |2 ⎟ ⎠⎝ ⎠ ⎝ |x4 |2 − |y4 |2 ω4 ω3 ω2 ω1 ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ω1 ω2 ω3 ω4 x¯1 y1 ω1 ω2 ω3 ω4 x1 y¯1 ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎜ω ω ω ω ⎟ ⎜x¯ y ⎟ ⎜ω ω ω ω ⎟ ⎜x y¯ ⎟ 1 4 3⎟ ⎜ 2 2⎟ 1 4 3⎟ ⎜ 2 2⎟ ⎜ 2 ⎜ 2 (19) ⎟⎜ ⎟=⎜ ⎟⎜ ⎟=0 ⎜ ⎜ω3 ω4 ω1 ω2 ⎟ ⎜x¯3 y3 ⎟ ⎜ω3 ω4 ω1 ω2 ⎟ ⎜x3 y¯3 ⎟ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ω4 ω3 ω2 ω1 x¯4 y4 ω4 ω3 ω2 ω1 x4 y¯4 are satisfied, the perfect teleportation is realized. One can see that if the entangled state is maximal, namely ω1 = ω2 = ω3 = ω4 = 14 , the above equations are satisfied. On the other hand, the above conditions are also satisfied for the following quantities: 1. eigenvalues {ωi }4i=1 , 1 ω1 + ω2 = ω3 + ω4 = , (20) 2  2. the basis vector |x = 4i=1 xi |fi , 1 1 1 1 (21) (x1 , x2 , x3 , x4 ) = √ , √ , √ , √ , 4 4 4 4  3. the basis vector |y = 4i=1 yi |fi , 1 1 1 1 (22) (y1 , y2 , y3 , y4 ) = √ , √ , − √ , − √ . 4 4 4 4

QUANTUM TELEPORTATION FOR NONMAXIMAL ENTANGLED STATES

65

Note that the entangled state satisfying condition (20) is a nonmaximal entangled state (the detail of this argument is discussed in the next subsection). Therefore this example shows that perfect teleportation succeeds even if a nonmaximal entangled state is used. Next, we construct unitary keys {Uk } in the above case. From Eqs. (21) and (22), we represent the basis vectors of input |x and |y as 1 |x = √ (|f1  + |f2  + |f3  + |f4 ) , 4 1 |y = √ (|f1  + |f2  − |f3  − |f4 ) , 4

(23) (24)

and introduce the state vectors defined by 1 |w = √ (|f1  − |f2 ) , 2

1 |z = √ (|f3  − |f4 ) . 2

The set {|x , |y , |w , |z} is an ONB of C4 . Next we consider a unitary operator ⎞ ⎛ √ 2 0 1 1 ⎟ ⎜ √ ⎟ 1 1 − 2 0 1 ⎜ ⎟ ⎜ T =√ ⎜ (25) √ ⎟, ⎟ 4⎜ 1 −1 0 2 ⎝ √ ⎠ 1 −1 0 − 2 such that T |f1  = |x ,

T |f2  = |y ,

T |f3  = |w ,

T |f4  = |z .

On the other hand, the basis vectors of the output are defined by   x = √f gk |x , k x| κk |x

  y = √f gk |y k y| κk |y

in Eq. (6). Due to the Schatten decomposition of f ∗ f , the operator f is represented as 4  √   eiθi ωi f˜i fi | , f = (26) i=1

 4  with an ONB f˜i . Put the two more vectors: i=1           x = eiθ1 √ω1 f˜1 + eiθ2 √ω2 f˜2 + eiθ3 √ω3 f˜3 + eiθ4 √ω4 f˜4 , 1           y = eiθ1 √ω1 f˜1 + eiθ2 √ω2 f˜2 − eiθ3 √ω3 f˜3 − eiθ4 √ω4 f˜4 . 1

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Y. TANAKA, M. ASANO, and M. OHYA

        w = eiθ1 2ω2 f˜1 − eiθ2 2ω1 f˜2 , 1         z = eiθ3 2ω4 f˜3 − eiθ4 2ω3 f˜4 , 1     orthogonal to x1 and y1 , one can give the unitary operator ⎞ ⎛ √ √ √ eiθ1 2ω2 0 eiθ1 ω1 eiθ1 ω1 ⎟ ⎜ √ √ √ ⎟ ⎜eiθ2 ω eiθ2 ω2 −eiθ2 2ω1 0 2 ⎟ ⎜ V =⎜ √ ⎟ √ √ iθ3 ⎜eiθ3 ω3 −eiθ3 ω3 0 e 2ω4 ⎟ ⎠ ⎝ √ iθ4 √ iθ4 √ iθ4 e ω4 −e ω4 0 −e 2ω3  4          to x1 , y1 , w1 , z1 . By using the for basis transformation from f˜i i=1 operators T and V , one can construct a unitary key U1 : ⎞ ⎛ R1 O ⎠ E, U1 = T V ∗ = ⎝ O R2 where

⎞ √ √ √ √ ω1 + ω2 ω2 − ω1 R1 = ⎝√ √ √ ⎠, √ ω1 − ω2 ω2 + ω1 ⎞ ⎛ √ √ √ √ ω3 + ω4 ω4 − ω3 R2 = ⎝√ √ √ √ ⎠, ω3 − ω4 ω4 + ω3

E = diag e−iθ1 , e−iθ2 , e−iθ3 , e−iθ4 . ⎛

In similar way, one can construct other keys Uk : ⎧ ⎪ TV∗ (k = 1, 5) , ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ⎪ T V X2 (k = 2, 6) , ⎪ ⎪ ⎪ ⎪ ⎪ T V ∗ X3 (k = 3, 7) , ⎪ ⎪ ⎪ ⎪ ⎨ T V ∗X (k = 4, 8) , 4 , Uk = ∗ ⎪ 2T X = 9, 13) , V (k ⎪ 2 ⎪ ⎪ ⎪ ⎪ ∗ ⎪ ⎪ 2T X2 V X2 (k = 10, 14) , ⎪ ⎪ ⎪ ⎪ ⎪ 2T X2 V ∗ X3 (k = 11, 15) , ⎪ ⎪ ⎪ ⎩ 2T X V ∗ X (k = 12, 16) , 2

4

where X1 , X2 , X3 and X4 are the operators in Eq. (13).

QUANTUM TELEPORTATION FOR NONMAXIMAL ENTANGLED STATES

3.3.

67

A nonmaximal entangled state in a perfect two-level teleportation

The entangled state vector defined by the operator f of Eq. (26) is represented as |ξ  =

4 

|j  ⊗ f |j 

j =1

=

4  j =1

=

4  k=1

=

4 

|j  ⊗

4 

 √   eiθk ωk f˜k fk | |j 

k=1 4    √   fk |j  |j  ⊗ f˜k eiθk ωk k=1

√     eiθk ωk f¯k ⊗ f˜k ,

k=1

   4 where f¯k = 4j =1 fk |j  |j  k=1 are an ONB of C4 and {ωk }4k=1 are eigenvalues of f ∗ f. As mentioned in Eq. (20), {ωk }4k=1 should satisfy ω1 + ω2 = ω3 + ω4 = 1/2 in our perfect teleportation model. In a sense, √ such a nonmaximal √ entangled state has the same property as an EPR pair like 1/ 2 |0 ⊗ |0 + 1/ 2 |1 ⊗ |1. However, note that this |ξ  and the EPR pair are not equivalent in the following sense: There are no local unitary operations U1 and U2 such that 1 1 (U1 ⊗ U2 ) |ξ  = √ |0 ⊗ |0 + √ |1 ⊗ |1 , 2 2 where |0 and |1 are vectors in C4 . One can show it by rewriting |ξ  as   1 1   |φ + |ξ  = φ 2 2 with two normalized vectors           |φ = eiθ1 2ω1 f¯1 ⊗ f˜1 + eiθ2 2ω2 f¯2 ⊗ f˜2 , (27)             φ = eiθ3 2ω3 f¯3 ⊗ f˜3 + eiθ4 2ω4 f¯4 ⊗ f˜4 . (28)    Generally,   |φ and φ are not factorizable, and it is impossible to transform |φ  and φ into |0 ⊗ |0 and |1 ⊗ |1 by local operations. 4.

A generalization of the two-level teleportation In the previous section, we explained the concept of two-level teleportation and showed an example of its realization. In this section, we describe the general process of two-level teleportation.

68 4.1.

Y. TANAKA, M. ASANO, and M. OHYA

A generalized Bell measurement

In the two-level teleportation, the set of measurable states {| k }16 k=1 by the 16 operators {gk }k=1 of Eq. (12) has an extended form of the Bell measurement. In this subsection, we discuss the general form of an extended Bell measurement. The example of the operators {gk }16 k=1 in Eq. (12) is constructed by the phase operator Xi and the basis-permutation operator Yi . Here we change the index k = 1, 2, 3, · · · , 16 to (α, β) = (1, 1), (2, 1), (3, 1), · · · , (4, 4), and rewrite {gk }16 k=1 as gα,β ≡ Xα Yβ . Further we express the operators Xα and Yβ in Eqs. (13) and (14) as 4   1  Xα = √ exp iθj(α) |j  j | , 4 j =1

Yβ =

4    ηβ (i) i| , i=1

 where the phases

θj(α)

4 j,α=1

are given by

  θ1(1) , θ2(1) , θ3(1) , θ4(1) = (0, 0, 0, 0),   θ1(2) , θ2(2) , θ3(2) , θ4(2) = (0, π, π, 0),   θ1(3) , θ2(3) , θ3(3) , θ4(3) = (0, π, 0, π),   θ1(4) , θ2(4) , θ3(4) , θ4(4) = (0, π, π, 0), and {ηα }4α=1 mean permutations on the set L4 = {1, 2, 3, 4}: ⎛ ⎞ ⎛ 1 2 3 4 1 2 3 ⎠, η2 = ⎝ η1 = ⎝ 1 2 3 4 2 1 4 ⎛ ⎛ ⎞ 1 2 3 1 2 3 4 ⎠, η4 = ⎝ η3 = ⎝ 4 3 2 3 4 1 2

⎞ 4 3 4

⎠, ⎞ ⎠.

1

In this representation, we change the expression of the basis {|fi }4i=1 to {|i}4i=1 , for further discussions. When the permutation ηβ (j ) is described as a binary operation on L4 , i.e. β  j ≡ ηβ (j ), the operation  is given by the following table,

QUANTUM TELEPORTATION FOR NONMAXIMAL ENTANGLED STATES



1 2 3 4

1

1 2 3 4

2

2 1 4 3

3

3 4 1 2

4

4 3 2 1

69

Such table is called the Latin square, in which each of 4 different numbers appeares exactly once in each row and exactly once in each column. This algebra (L4 , ) corresponds to the Klein four-group, which is well known in group theory. N  From the above discussion, we define the generalized Bell basis  α,β α,β=1 in CN ⊗ CN . In this generalization, the operators gα,β are defined as gα,β

N 1  2π α = X α Yβ = √ exp i j |β ◦ j  j | , N N j =1

where the operators Xα and Yβ have the forms of N 1  2π α Xα = √ exp i j |j  j | , N N j =1 Yβ =

N 

|β ◦ i i| .

i=1

In the above equation, the symbol ◦ means a binary operation on LN = {1, 2, · · · , N }  N that is given by an N × N Latin square. Then the Bell bases  α,β α,β=1 are written as N    ∗  α,β = |ei  ⊗ |ei  gα,β i=1

N N   2π α 1  β ◦ j |ei  |ei  exp −i =√ j |j  ⊗ N N j =1 i=1 N   2π α 1  exp −i j |j  ⊗ f¯β◦j , =√ N N j =1

(29)

  N where f¯β◦j denotes i=1 β ◦ j |ei  |ei  . Note that our definition contains the

70

Y. TANAKA, M. ASANO, and M. OHYA

generalization by Bennett et al. [2], N−1    2π α ψα,β = √1 exp i j |j  ⊗ |(j + β) modN  , N N j =0

(30)

(α, β = 0, 1, · · · , N − 1) 4.2.

Condition for a perfect two-level teleportation with N = 2n

In order to discuss the perfect two-level teleportation on 2n-dimensional systems, we rewrite the success conditions of Eqs. (8) and (9) by means of indices α, β : 2n 

i| κα,β |j  x¯i xj − y¯i yj = 0,

i,j =1 2n 

i| κα,β |j  x¯i yj =

i,j =1

2n 

i| κα,β |j  y¯i xj .

i,j =1

In the generalized Bell states, the operators gα,β are characterized by a binary operation ◦ , which is given by a 2n × 2n Latin square. The operator κα,β in the above equation is described by ∗ f ∗ f gα,β = κα,β = gα,β

2n 1  ωα◦j |j  j | . 2n j =1

When we give the basis of input state vector | = a |x + b |y by n 1  |x = √ |i , n i=1

2n 1  |y = √ |i , n i=n+1

one can rewrite the above success conditions in the following simple form, n 

ωα◦j =

j =1

2n 

1 ωα◦j = . 2 j =n+1

(31)

This result indicates that the relation between the eigenvalues of f ∗ f and the binary operation ◦ is very important for perfect teleportation. For example, let the eigenvalues ωi be such that n  i=1

ωi =

2n 

1 ωi = , 2 i=n+1

(32)

then an appropriate binary operation ◦ is given by a 2n × 2n Latin square with the following structure,

QUANTUM TELEPORTATION FOR NONMAXIMAL ENTANGLED STATES



1

1 .. .

···

n

n+1

···

K1,1

K1,2

K2,1

K2,2

71

2n

n n+1 .. . 2n Here, K1,1 and K2,2 are n×n Latin sub-squares, whose components consist of the numbers {1, 2, · · · , n}. The K1,2 and K2,1 consist of the numbers {n + 1, n + 2, · · · , 2n}. For example, in the case of N = 6 (n = 3), an appropriate binary operation ◦ has the following table of Latin square, ◦

1

2

3

4

5

6

1

1

2

3

4

5

6

2

2

3

1

5

6

4

3

3

1

2

6

4

5

4

4

5

6

1

2

3

5

5

6

4

2

3

1

6

6

4

5

3

1

2

Expansion to m-level teleportation In this section, we extend the two-level teleportation to the m-level teleportation. We consider mn-dimensional Hilbert spaces Cmn , and the input state vector of an m-level state is m  | = ai |xi 

5.

where ai ∈ C with and such that

mn

i=1

i=1

|ai | = 1, and state vectors |xi  are mutually orthogonal 2

in  1 |j  . |xi  = √ n j =(i−1)n+1

The output state is given by m   − 1  − 1 ai f gα,β κα,β 2 |xi  . f gα,β κα,β 2 | = i=1

72

Y. TANAKA, M. ASANO, and M. OHYA

As discussed in Section 3.1, we obtain the success conditions of perfect m-level teleportation if the basis |xi  satisfies x1 | κα,β |x1  = x2 | κα,β |x2  = · · · = xm | κα,β |xm  , if i = i ,

xi | κα,β |xi  = 0

∗ for the operator κα,β = gα,β f ∗ f gα,β . Note that the operator κα,β can be written as ∗ κα,β = gα,β f ∗ f gα,β

=

mn 1  ωα◦j |j  j | , mn j =1

where ◦ is a binary operation on Lmn = {1, 2, · · · , mn} given by an (mn) × (mn) Latin square. One can describe the above success condition as n 

2n 

ωα◦j =

j =1

mn 

ωα◦j = · · · =

j =n+1

ωα◦j =

j =(m−1)n+1

1 . m

(33)

When the eigenvalues ωi of f ∗ f satisfy the relation m 

ωk =

k=1

2m  k=m+1

mn 

ωk = · · · =

ωk ,

k=m(n−1)+1

there is an (mn) × (mn) Latin square for the appropriate operation ◦ giving the following table, ◦ 1 .. .

1

···

n

n + 1 · · · 2n

K1,1

K1,2

K2,1

K2,2

(m − 1)n + 1 · · · mn ···

K1,m

n n+1 .. .

K2,m

2n .. . (m − 1)n + 1 .. . mn

Km,1

..

Km,2

.

···

.. . Km,m

QUANTUM TELEPORTATION FOR NONMAXIMAL ENTANGLED STATES

73

Here Ki,j are n × n Latin squares each of which consists of the numbers {(i ∗ j − 1) n + 1, (i ∗ j − 1) n + 2, · · · , (i ∗ j ) n} , where ∗ is a binary operation on Lm = {1, 2, · · · , m}, given by another m × m Latin square. For example, in the case of 6-dimensional spaces, i.e. N = 6(m = 3, n = 2), such tables for the operations ◦ and ∗ are given by ◦

1 2

3 4

5 6

1

1 2

3 4

5 6

2

2 1

4 3

6 5

3

3 4

5 6

12

4

4 3

6 5

21

5

5 6

1 2

3 4

6

6 5

2 1

4 3



1 2 3

1

1 2 3

2

2 3 1

3

3 1 2

and

6.

.

Conclusion We discussed a teleportation model with nonmaximal entangled state. This model, called the two-level teleportation, has been constructed in terms of the Kossakowski and Ohya teleportation scheme. In our model, measurable states are represented by means of algebra theory; a binary operation specified with a Latin square determines a set of measurable states, see Eq. (29). Such forms contain the wellknown Bell states and their extension for N -dimensional spaces. In this sense, our representation gave a generalized form of multipartite maximal entangled states. By this generalization, we reconsidered the two-level teleportation, and we extended it to an m-level teleportation. Through this study, we made the clear condition of nonmaximal entangled state for a perfect teleportation, see Eqs. (31) and (33). Nonmaximal entangled states satisfying such conditions we call the m-level entangled state. In usual quantum information theory, the ideal resources are maximal entangled states. However, as shown in the present study, nonmaximal entangled states have potentiality to be useful resources. Furthermore, we believe that the m-level entangled

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Y. TANAKA, M. ASANO, and M. OHYA

states are a kind of realistic entangled states in quantum physical systems. So, this study is related very much to [14] with respect to two-level entangled state. REFERENCES [1] M. Asano, M. Ohya and Y. Tanaka: Complete m-level quantum teleportation based on Kossakowski-Ohya scheme, Quantum Bio-Informatics II: From Quantum Information to Bio-Informatics, 19–29, 2009. [2] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W. K. Wootters: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels, Phys. Rev. Lett. 70(13) (1993), 1895–1899. [3] D. Boschi, S. Branca, F. De Martini, L. Hardy and S. Popescu: Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein–Podolsky–Rosen channels, Phys. Rev. Lett. 80(6) (1998), 1121–1125. [4] D. Bouwmeester, J.W. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger: Experimental quantum teleportation, Nature 390(6660) (1997), 575–579. [5] S.L. Braunstein and H.J. Kimble: Teleportation of continuous quantum variables, Phys. Rev. Lett. 80(4) (1998), 869–872. [6] M.S. Kim, W. Son, V. Buˇzek and P.L. Knight: Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement, Phys. Rev. A 65(3) (2002), 32323. [7] A. Kossakowski and M. Ohya: New scheme of quantum teleportation, Infinite Dimensional Analysis Quantum Probability and Related Topics 10(3) (2007), 411. [8] A. Luis and L.L. Sanchez-Soto: Phase-difference operator, Phys. Rev. A 48(6) (1993), 4702–4708. [9] J.S. Neergaard-Nielsen, B.M. Nielsen, C. Hettich, K. Molmer and E.S. Polzik: Generation of a superposition of odd photon number states for quantum information networks, Phys. Rev. Lett. 97(8) (2006), 83604. [10] J.W. Noh, A. Foug`eres and L. Mandel: Measurements of the probability distribution of the operationally defined quantum phase difference, Phys. Rev. Lett. 71(16) (1993), 2579–2582. [11] M. Ohya and D. Petz: Quantum entropy and its use, Springer, Berlin, 2004. [12] A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri and P. Grangier: Generation of optical ’Schr¨odinger cats’ from photon number states, Nature 448(7155) (2007), 784–786. [13] D. T. Pegg and S. M. Barnett: Phase properties of the quantized single-mode electromagnetic field, Phys. Rev. A 39(4) (1989), 1665–1675. [14] Y. Tanaka, M. Asano and M. Ohya: Physical realization of quantum teleportation for a nonmaximal entangled state, Phys. Rev. A 82(2) (2010), 022308. [15] C. Wang, F.G. Deng, Y.S. Li, X.S. Liu and G.L. Long: Quantum secure direct communication with high-dimension quantum superdense coding, Phys. Rev. A 71(4) (2005), 44305.