quantum state by means of classical communication and quantum entanglement [4]. ... sition and momentum of the compound quantum system. Q+A. Then Alice ...
Continuous variable teleportation as a generalized thermalizing quantum channel Masashi Ban,1 Masahide Sasaki,2, 3 and Masahiro Takeoka2, 3
arXiv:quant-ph/0202172v1 28 Feb 2002
1
Advanced Research Laboratory, Hitachi, Ltd., 1-280 Higashi-Koigakubo, Kokubunji, Tokyo 185-8601, Japan 2 Communication Research Laboratory, 4-2-1 Nukui-Kitamachi, Koganei, Tokyo 184-8795, Japan 3 CREST, Japan Science and Technology (Dated: February 1, 2008) A quantum channel is derived for continuous variable teleportation which is performed by means of an arbitrary entangled state and the standard protocol. When a Gaussian entangled state such as a two-mode squeezed-vacuum state is used, the continuous variable teleportation is equivalent to the thermalizing quantum channel. Continuous variable dense coding is also considered. Both the continuous variable teleportation and the continuous variable dense coding are characterized by the same function determined by the entangled state and the quantum measurement. PACS numbers: 03.67.Hk, 03.65.Ud, 89.70.+c
Quantum information processing attracts much attention in quantum physics and information science [1, 2, 3], which gives deeper insight into the principles of quantum mechanics and provides novel communication systems such as quantum teleportation and quantum dense coding. Quantum teleportation transmits an unknown quantum state by means of classical communication and quantum entanglement [4]. Quantum dense coding sends classical information via a quantum channel and quantum entanglement [5]. Bowen and Bose have recently shown that a finite dimensional quantum teleportation with the standard protocol is equivalent to a generalized depolarizing quantum channel [6]. In this letter, we generalize this result to infinite dimensional quantum teleportation and we show that continuous variable quantum teleportation with the standard protocol is equivalent to a generalized thermalizing channel. We explain the standard protocol of continuous variable quantum teleportation [7, 8]. Alice and Bob first share an entangled quantum system in a quantum state ˆ AB which is arbitrary though a two-mode squeezedW vacuum state is usually used, where A and B stand for the quantum system at Alice’s and Bob’s sides. When Alice is given a quantum state ρˆQ to be teleported to Bob, she performs the simultaneous measurement of position and momentum of the compound quantum system Q+A. Then Alice sends the measurement outcome (x, p) to Bob via a classical communication channel. After receiving the measurement outcome, Bob applies the unitary transformation described by the displacement operˆ ˆ ator D(x, p) = ei(pˆx−xp) to the system B. Bob finally obtains the quantum state ρˆB which is, in an ideal case, identical to the quantum state ρˆQ to be teleported. The main result of this letter is that the completely positive map Lˆ which describes the continuous variable teleportation by means of an arbitrary entangled state and the standard protocol is given by
ˆρ = Lˆ
Z
∞
−∞
dx
Z
∞ −∞
ˆ ˆ † (x, p), dp P(x, p)D(x, p)ˆ ρD
(1)
with ˆ AB |ΨAB (x, p)i, P(x, p) = hΨAB (x, p)|W
(2)
ˆ B (x, p)]|ΦAB i, |ΨAB (x, p)i = [ˆ1A ⊗ D
(3)
where |ΦAB i is a completely entangled state of continuous variable Z ∞ 1 |ΦAB i = √ dx |xA i ⊗ |xB i, (4) 2π −∞ which satisfies the relations, 1 ˆB 1 , 2π 1 ˆA TrB |ΦAB ihΦAB | = 1 . 2π TrA |ΦAB ihΦAB | =
(5) (6)
In this state, the systems A and B have the same values of position and the opposite values of momentum, (ˆ xA − xˆB )|ΦAB i = (ˆ pA + pˆB )|ΦAB i = 0.
(7)
When P(x, p) = P (x)P (p) and P (x) is a Gaussian function of x, the completely positive map Lˆ becomes the thermalizing quantum channel. Hence we refer to the completely positive map Lˆ as a generalized thermalizing quantum channel. In a discrete quantum teleportation of a N -dimensional Hilbert space, this corresponds to a generalized depolarizing quantum channel obtained by Bowen and Bose [6]. When a pure state ρˆ = |ψihψ| is teleported, the fidelity Fψ is calculated by the formula Z ∞ Z ∞ ˆ Fψ = dx dp P(x, p)|hψ|D(x, p)|ψi|2 . (8) −∞
−∞
The property of the simultaneous measurement of position and momentum is essential for deriving the comˆ The simultaneous measurement pletely positive map L. which yields the measurement outcome (x, p) is described by a projection operator, ˆ AB (x, p) = |ΦAB (x, p)ihΦAB (x, p)|, X
(9)
2 where |ΦAB (x, p)i is given by Z ∞ 1 dy |xA + y A i ⊗ |y B i eipy |ΦAB (x, p)i = √ 2π −∞ i h 1 ˆ A (x, p) ⊗ ˆ = D 1B |ΦAB i e− 2 ipx h i ˆ B (−x, p) |ΦAB i e− 12 ipx = ˆ 1A ⊗ D 1
= |ΨAB (−x, p)i e− 2 ipx ,
For quantum optical systems, the simultaneous measurement of position and momentum is implemented by a heterodyne detection [9].
(10)
Let us now derive the main result given by Eqs (1)–(4). Suppose that Alice and Bob share an arbitrary entangled ˆ AB and Alice has a quantum state ρˆQ which is state W to be teleported to Bob. The total quantum state ρˆQAB of Alice and Bob is given by
ˆ where D(x, p) is the displacement operator, ˆ D(x, p) = exp[i(pˆ x − xˆ p)] = exp(αˆ a† − αˆ a),
(11)
√ √ with a ˆ = (ˆ x + iˆ p)/ 2 and α = (x + ip)/ 2. The vector |ΦAB (x, p)i is the simultaneous eigenstate of positiondifference operator x ˆA −ˆ xB and momentum-sum operator A B pˆ + pˆ , which satisfies the relations, hΦAB (x, p)|ΦAB (x′ , p′ )i = δ(x − x′ )δ(p − p′ ), Z
∞
−∞
dx
Z
∞
−∞
(12)
dp |ΦAB (x, p)ihΦAB (x, p)| = ˆ 1A ⊗ ˆ1B . (13)
ˆ AB . ρˆQAB = ρˆQ ⊗ W
(14)
Alice performs the simultaneous measurement of position and momentum of the compound system Q+A, described ˆ QA (x, p), and informs Bob by the projection operator X of the measurement outcome (x, p). Then Bob applies ˆ B (x, p) to the system B and he the unitary operator D finally obtains the quantum state,
n h io ˆ QA (x, p) ⊗ ˆ1B ρˆQ ⊗ W ˆ B (x, p) TrQA X ˆ AB ˆ B † (x, p) D D i h , ρˆB (x, p) = ˆ AB ˆ QA (x, p) ⊗ ˆ1B ρˆQ ⊗ W TrQAB X
(15)
with probability, P (x, p) = TrQAB
i h ˆ AB . ˆ QA (x, p) ⊗ ˆ1B ρˆQ ⊗ W X
(16)
In deriving Eq. (15), we have used the state-reduction formula [10, 11, 12]. Hence the teleported quantum state ρˆB out of Bob is given, in average, by ρˆB out =
Z
∞
dx
−∞
Z
∞ −∞
io h n ˆ AB ˆ B † (x, p), ˆ QA (x, p) ⊗ ˆ1B ρˆQ ⊗ W ˆ B (x, p) TrQA X D dp D
(17)
which determines the completely positive map Lˆ of the continuous variable teleportation by means of an arbitrary entangled state and the standard protocol. ˆ AB in terms of To obtain the completely positive map Lˆ from Eq. (17), we expand the entangled quantum state W AB |Φ (x, p)i’s as ˆ AB = W
Z
∞
−∞
dx
Z
∞
dy −∞
Z
∞
−∞
du
Z
∞
dv FW (x, u; y, v)|ΦAB (x, u)ihΦAB (y, v)|,
(18)
−∞
ˆ AB |ΦAB (y, v)i. Note that the state vector |ΦAB (x, p)i satisfies the where we set FW (x, u; y, v) = hΦAB (x, u)|W relation, Z ∞ 1 hΦAB (x, u)|ΦBC (y, v)i = dz |z C − y C ihz A + xA | e−iuz+iv(z−y) . (19) 2π −∞
3 Substituting these equations into Eq. (17) and using |x + yi = e−iypˆ|xi, after some calculation, we obtain Z ∞ Z ∞ B ˆ B (−x, p)ˆ ˆ B † (−x, p). dp FW (x, p; x, p)D ρB D dx ρˆout =
(20)
−∞
−∞
where ρˆB is the quantum state of the system B, which is identical to that described by ρˆQ . Using Eq. (10), we can arrive at the result given by Eqs. (1)–(4). Therefore, we have found that the continuous variable teleportation by means of an arbitrary entangled state and the standard protocol is equivalent to the generalized thermalizing quantum channel. The continuous variable teleportation is usually performed by means of a two-mode squeezed-vacuum state |ΨAB SV i = † ˆ† ˆ r(ˆ AB 2 e a b −ˆab) |0A , 0B i [7, 8]. In this case, calculating P(x, p) = |hΨAB |Φ (x, p)i| , we obtain SV 2 Z ∞ Z ∞ x + p2 ˆ ˆ † (x, p), ˆρ = 1 D(x, p)ˆ ρD (21) Lˆ dx dp exp − 2π¯ nr −∞ 2¯ nr −∞
where n ¯ r = e−2r and r is the squeezing parameter of the two-mode squeezed-vacuum state |ΨAB SV i. The completely positive map Lˆ given by Eq. (21) is equivalent to the transfer-operator representation of the continuous variable teleportation [13]. When we denote the GlauberSudarshan P -function of the quantum state ρˆ as P (α), Eq. (21) can be rewritten as Z 2 d α ˆ ˆ † (α), ˆ Lˆ ρ= (22) P (α)D(α)ˆ ρth D π where ρˆth is the thermal state ρˆth
n ∞ n ¯r 1 X = |nihn|. 1+n ¯ r n=0 1 + n ¯r
(23)
This result explicitly shows that the continuous variable teleportation by means of the two-mode squeezedvacuum state is nothing but the thermalizing quantum
ˆ AB (x, p) = W
Z
∞
dy
−∞
Z
∞
−∞
dz
Z
∞
−∞
du
Z
channel. When the two-mode squeezed-vacuum state is shared through a noisy quantum channel, the parameter n ¯ r = e−2r appeared in Eq. (21) is replaced with n ¯ r = 1 − 1 − e−2r T [14, 15, 16], where T stands for the transmission coefficient of the noisy quantum channel. We next consider the continuous variable quantum dense coding by means of an arbitrary entangled state and the standard protocol in which Alice applies the uniˆ tary operator D(x, p) to encode some classical information and Bob performs the simultaneous measurement of position and momentum to extract the information [17, 18, 19, 20]. Suppose that Alice and Bob share an arˆ AB to transmit clasbitrary entangled quantum state W sical information via the continuous variable quantum dense coding. When Alice applies the unitary operator ˆ D(x, p) to her system and sends it to Bob, his quantum ˆ AB (x, p) is given by state W
∞
dv FW (y, u; z, v)|ΦAB (x + y, p + u)ihΦAB (x + z, p + v)|.
(24)
−∞
Thus the conditional probability (the channel matrix of the quantum dense coding) P (x′ , p′ |x, p) that Bob obtains the measurement outcome (x′ , p′ ) when Alice encoded (x, p) is calculated to be ˆ AB |ΦAB (x′ − x, p′ − p)i P (x′ , p′ |x, p) = hΦAB (x′ − x, p′ − p)|W ′ ′ ′ = FW (x − x, p − p; x − x, p′ − p) = P(x − x′ , p′ − p), ˆ AB is the two-mode squeezed-vacuum state, we obtain where P(x, p) is given by Eq. (2). When W (x′ − x)2 + (p′ − p)2 1 ′ ′ . exp − P (x , p |x, p) = 2π¯ nr 2¯ nr
In the strong squeezing limit (r ≫ 1), the channel ma-
(25)
(26)
trix P (x′ , p′ |x, p) is approximated as δ(x′ − x)δ(p′ − p)
4 and thus the continuous variable quantum dense coding can transmit the amount of classical information twice as that without quantum entanglement [18]. When the two-mode squeezed-vacuum state is sent through a noisy quantum channel, the parameter n ¯ r = e−2r appeared in Eq. (26) is replaced with n ¯ r = 1 − 1 − e−2r T [19, 20]. In summary, we have shown that the continuous variable quantum teleportation by means of an arbitrary entangled quantum state and the standard protocol is equivalent to the generalized thermalizing quantum channel. This is a continuous version of the result obtained by Bowen and Bose [6], in which they have shown that the discrete (the finite dimensional) quantum teleportation
with the standard protocol is equivalent to the generalized depolarizing quantum channel. In particular, when a two-mode squeezed-vacuum state is used as an entanglement resource, the continuous variable quantum teleportation becomes the thermalizing channel in which the number of thermal photons is given by n ¯ r = e−2r . The continuous variable quantum dense coding by means of an arbitrary entangled state the standard protocol has been also considered. As long as the standard protocol is applied, not only the continuous variable quantum teleportation but also the continuous variable quantum dense coding is completely determined by the function P(x, p) given by Eq. (2).
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