第９回目

細谷暁夫  ２０２０・１１・5

量子コンピューターのための量子力学

1.3

EPR

EPR

|ψ >=

√

1

2

[

|0 >

A

|1 >

B

−|1 >

A

|0 >

B

]

(19)

|0 >

A

|1 >

A

|1 >

B

|0 >

B

U

_A

_{⊗ 1}

_B

9

復習

|ψ >=

!

ij

C

_ij

_{|i >}

_A

_{⊗|j >}

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_A

_{⊗ H}

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(20)

|ψ >

U

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!

= U

_A

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!

ij

C

_ij

U

_A

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_A

_{⊗|j >}

_B

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_A

_{⊗ H}

_B

(21)

B

ρ

!

_B

:= T r

_A

[

_|ψ

!

>< ψ

!

_|]

T r

_A

[U

_A

_{|i >}

_A

< i

!

_|U

_A

†

]

T r

_A

[U

_A

_{|i >}

_A

< i

!

_|U

_A

†

] = T r

_A

[U

_A

†

U

_A

_{|i >}

_A

< i

!

_{|] = T r}

_A

[

_{|i >}

_A

< i

!

_{|] = δ}

_ii

!

(22)

ρ

!

_B

= ρ

_B

B

ρ

_B

U

_A

|ψ >

EPR

ρ

_B

=

1

₂

50 : 50

10

エンタングルメント

を使った

量子テレポテーション

については

最後にやります

ご質問にお答えします

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は何かアンケートを

取ります。

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S(P : Q) = S(P : Q, M ) _{≥ S(P}! : Q!, M!) ≥ S(P! : M!). (15)   一番左の等式は M が純粋状態なので自明。次の不等式は相互情報量の 量子操作 Λ による単調性から導ける。最後の不等式は相互情報量の単調性 S(A : B!, C) _{≥ S(A : B)} から来る。 ただし、ここに S(P! : Q!, M!) などは、量子操作後の密度演算子 ρP!Q!M! を用いて定義したものである。 ρP!Q!M! = Λ(ρP QM) = ! x,y p_{x|x"#x| ⊗} "E_yρ_x"E_{y ⊗ |y"#y|.} (16) これから ρP!M! = T r_Q[ρP!Q!M!] = ! x,y p_xp(y_{|x)|x"#x|⊗|y"#y| =} ! x,y p(x, y)_{|x"#x|⊗|y"#y|} (17) を得る。ただし、 p(y_{|x) = T r[Ey}ρ_x] (18) は、状態が x であるときに y という測定結果を得る「条件付き確率」である。 ベイズ則 pxp(y|x) = p(x, y) を用いると、結合確率 p(x, y) と関係がついて 上の式を得る。 従って、 S(P!, M!) = S(ρP!M!) = −#_x,y p(x, y) log p(x, y) =: Scl(X, Y ). 以上の準備のもとに定理を証明しよう。  不等式（15）の際右辺は S(P! : M!) = S_cl(X : Y )。一方、不等式（15） の際左辺は S(P : Q) = S(P ) + S(Q) − S(#_x p_{x|x"#x| ⊗ ρx}) = S({px}) + S(ρ) − [S({px}) + #_x pxS(ρx)] = S(ρ) − #x pxS(ρx) =: χ(ρ). (証明終)

3

量子テレポテーション

 アリスの状態 α_|0"A + β|1"A を、（α, β ∈ C の値を知ることなしに）別の 場所にいるボブの位置に再現することを目標とする。そのためにアリスとボ ブが、EPR 対

Alice

Bob

古典通信

ベルl測定をして結果をボブに知らせる

α|0 >

_A

+ β|1 >

_A

EPR対を共有

量子テレポーテーション

α

α, β

の値を Aliceも知らない

Alice

Bob

古典通信

AliceからBell測定の結果を聞いて

それに応じて手適切なユニタリー変換をする

α|0 >

_B

+ β|1 >

_B

ボブのところに以前アリスのところに

あった量子情報を再現する

図 2: 量子テレポテーション 1 √ 2[|0"A ⊗ |1"B − |1"A ⊗ |0"B] (19) を共有する。全体の状態は 1 √ 2(α|0"A + β|1"A)[|0"A ⊗ |1"B − |1"A ⊗ |0"B] (20) となる。簡単な式変形で、この状態は |ψ" := |ψ1"(α|1"B − β|0"B) +|ψ2"(α|1"B + β|0"B) +|ψ3"(−α|0"B + β|1"B) +|ψ4"(−α|0"B − β|1"B) と書き直せる。ここに、 |ψ1" = 1 √ 2[|0"A|0"A + |1"A|1"A], |ψ2" = 1 √ 2[|0"A|0"A − |1"A|1"A] |ψ3" = 1 √ 2[|1"A|0"A + |0"A|1"A], |ψ4" = 1 √ 2[|0"A|1"A − |1"A|0"A] は Bell 状態と呼ばれる互いに直交する状態である。 ここで、アリスが Bell 状態を測定するとしよう。例えば、_|ψ1" であったと しよう。すると、第３公理によりボブの状態は α_{|1"B − β|0"B} (21) に移行する。アリスはその結果をボブに古典通信（例えば電話で）で知らせ る。上の例だと _|ψ1" であったとボブに知らせる。ボブは、その情報に基づい て、ユニタリー変換 σ_zσ_x を行えば、ボブの状態は α_|0"B + β|1"_B (22) となり、アリスのところにあった初期状態と同じ形をしている。他の Bell 状 態の場合も同様である。  このプロトコルのポイントは、ベル測定の結果を古典通信で知らせると ころにある。この情報がなければ、ボブの状態を正しく α_|0"B + β|1"_B にす ることはできない。

注：古典通信を使うので

光速を超えることができない！

量子テレポテーションの実験

Experimental quantum teleportation

Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter, and Anton Zeilinger

Institut f¨

ur Experimentalphysik, Universit¨

at Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria

(Dated: December 11, 1997)

Quantum teleportation – the transmission and reconstruction over arbitrary distances of the state

of a quantum system – is demonstrated experimentally. During teleportation, an initial photon which

carries the polarization that is to be transferred and one of a pair of entangled photons are subjected

to a measurement such that the second photon of the entangled pair acquires the polarization of

the initial photon. This latter photon can be arbitrarily far away from the initial one. Quantum

teleportation will be a critical ingredient for quantum computation networks.

The dream of teleportation is to be able to travel by

simply reappearing at some distant location. An object

to be teleported can be fully characterized by its

prop-erties, which in classical physics can be determined by

measurement. To make a copy of that object at a

dis-tant location one does not need the original parts and

pieces – all that is needed is to send the scanned

in-formation so that it can be used for reconstructing the

object. But how precisely can this be a true copy of the

original? What if these parts and pieces are electrons,

atoms and molecules? What happens to their individual

quantum properties, which according to the Heisenberg’s

uncertainty principle cannot be measured with arbitrary

precision?

Bennett et al. [1] have suggested that it is possible

to transfer the quantum state of a particle onto another

particle – the process of quantum teleportation –

pro-vided one does not get any information about the state

in the course of this transformation. This requirement

can be fulfilled by using entanglement, the essential

fea-ture of quantum mechanics [2]. It describes correlations

between quantum systems much stronger than any

clas-sical correlation could be.

The possibility of transferring quantum information is

one of the cornerstones of the emerging field of

quan-tum communication and quanquan-tum computation [3].

Al-though there is fast progress in the theoretical

descrip-tion of quantum informadescrip-tion processing, the difficulties

in handling quantum systems have not allowed an equal

advance in the experimental realization of the new

pro-posals.

Besides the promising developments of

quan-tum cryptography [4] (the first provably secure way to

send secret messages), we have only recently succeeded

in demonstrating the possibility of quantum dense

cod-ing [5], a way to quantum mechanically enhance data

compression. The main reason for this slow

experimen-tal progress is that, although there exist methods to

pro-duce pairs of entangled photons [6], entanglement has

been demonstrated for atoms only very recently [7] and

it has not been possible thus far to produce entangled

states of more than two quanta.

Here we report the first experimental verification of

quantum teleportation. By producing pairs of entangled

photons by the process of parametric down-conversion

and using two-photon interferometry for analysing

en-tanglement, we could transfer a quantum property (in

our case the polarization state) from one photon to

an-other. The methods developed for this experiment will

be of great importance both for exploring the field of

quantum communication and for future experiments on

the foundations of quantum mechanics.

THE PROBLEM

To make the problem of transferring quantum

infor-mation clearer, suppose that Alice has some particle in a

certain quantum state

_{| i and she wants Bob, at a distant}

location, to have a particle in that state. There is

cer-tainly the possibility of sending Bob the particle directly.

But suppose that the communication channel between

Alice and Bob is not good enough to preserve the

neces-sary quantum coherence or suppose that this would take

too much time, which could easily be the case if

_{| i is}

the state of a more complicated or massive object. Then,

what strategy can Alice and Bob pursue?

As mentioned above, no measurement that Alice can

perform on

_{| i will be sufficient for Bob to reconstruct}

the state because the state of a quantum system cannot

be fully determined by measurements. Quantum systems

are so evasive because they can be in a superposition of

several states at the same time. A measurement on the

quantum system will force it into only one of these states

– this is often referred to as the projection postulate. We

can illustrate this important quantum feature by taking

a single photon, which can be horizontally or vertically

polarized, indicated by the states

_{| $i and | li. It can}

even be polarized in the general superposition of these

two states

| i = ↵| $i + | li

(1)

where ↵ and

are two complex numbers satisfying

|↵|

2

+

_{| |}

2

= 1. To place this example in a more

gen-eral setting we can replace the states

_{| $i and | li in}

Eq. (1) by

_{|0i and |1i, which refer to the states of any}

two-state quantum system. Superpositions of

_{|0i and |1i}

are called qubits to signify the new possibilities

intro-duced by quantum physics into information science [8].

arXiv:1901.11004v1 [quant-ph] 30 Jan 2019

Nature 390,

2

If a photon in state

_{| i passes through a polarizing}

beamsplitter – a device that reflects (transmits)

horizon-tally (vertically) polarized photons – it will be found in

the reflected (transmitted) beam with probability

_|↵|

2

(

_{| |}

2

). Then the general state

_{| i has been projected}

ei-ther onto

_{| $i or onto | li by the action of the}

measure-ment. We conclude that the rules of quantum mechanics,

in particular the projection postulate, make it impossible

for Alice to perform a measurement on

_{| i by which she}

would obtain all the information necessary to reconstruct

the state.

THE CONCEPT OF QUANTUM

TELEPORTATION

Although the projection postulate in quantum

me-chanics seems to bring Alice’s attempts to provide Bob

with the state

_{| i to a halt, it was realised by Bennett}

et al. [1] that precisely this projection postulate enables

teleportation of

_{| i from Alice to Bob. During}

teleporta-tion Alice will destroy the quantum state at hand while

Bob receives the quantum state, with neither Alice nor

Bob obtaining information about the state

_{| i. A key}

role in the teleportation scheme is played by an entangled

ancillary pair of particles which will be initially shared

by Alice and Bob.

Suppose particle 1 which Alice wants to teleport is in

the initial state

_{| i}

₁

= ↵

_{| $i}

₁

+

_{| li}

₁

(Fig. 1(a)), and

the entangled pair of particles 2 and 3 shared by Alice

and Bob is in the state:

| i

23

=

1

2

| $i

2

| li

3

| li

2

| $i

3

(2)

That entangled pair is a single quantum system in

an equal superposition of the states

_{| $i}

₂

_{| li}

₃

and

| li

2

| $i

3

.

The entangled state contains no

informa-tion on the individual particles; it only indicates that

the two particles will be in opposite states. The

impor-tant property of an entangled pair is that as soon as a

measurement on one of the particles projects it, say, onto

| $i the state of the other one is determined to be | li,

and vice versa. How could a measurement on one of the

particles instantaneously influence the state of the other

particle, which can be arbitrarily far away? Einstein,

among many other distinguished physicists, could

sim-ply not accept this “spooky action at a distance”. But

this property of entangled states has now been

demon-strated by numerous experiments (for reviews, see refs. 9

and 10).

The teleportation scheme works as follows. Alice has

the particle 1 in the initial state

_{| i}

₁

and particle 2.

Par-ticle 2 is entangled with parPar-ticle 3 in the hands of Bob.

The essential point is to perform a specific measurement

on particles 1 and 2 which projects them onto the

entan-FIG. 1. Scheme showing principles involved in quantum

tele-portation (a) and the experimental set-up (b).

(a) Alice

has a quantum system, particle 1, in an initial state which

she wants to teleport to Bob. Alice and Bob also share an

ancillary entangled pair of particles 2 and 3 emitted by an

Einstein-Podolsky-Rosen (EPR) source. Alice then performs

a joint Bell-state measurement (BSM) on the initial particle

and one of the ancillaries, projecting them also onto an

entan-gled state. After she has sent the result of her measurement as

classical information to Bob, he can perform a unitary

trans-formation (U) on the other ancillary particle resulting in it

being in the state of the original particle. (b) A pulse of

ul-traviolet radiation passing through a nonlinear crystal creates

the ancillary pair of photons 2 and 3. After retroflection

dur-ing its second passage through the crystal the ultraviolet pulse

creates another pair of photons, one of which will be prepared

in the initial state of photon 1 to be teleported, the other one

serving as a trigger indicating that a photon to be teleported

is under way. Alice then looks for coincidences after a beam

splitter BS where the initial photon and one of the ancillaries

are superposed. Bob, after receiving the classical information

that Alice obtained a coincidence count in detectors f1 and

f2 identifying the

_{| i}

12

Bell state, knows that his photon

3 is in the initial state of photon 1 which he then can check

using polarization analysis with the polarizing beam splitter

PBS and the detectors d1 and d2. The detector p provides

the information that photon 1 is under way.

理論的スキーム

実験のセットアップ

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_Nonenone

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マッハ・ツエンダー干渉計復習

干渉

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実験結果

4

FIG. 2. Photons emerging from type II down-conversion (see text). Photograph taken perpendicular to the propagation direction. Photons are produced in pairs. A photon on the top circle is horizontally polarized while its exactly opposite partner in the bottom circle is vertically polarized. At the intersection points their polarizations are undefined; all that is known is that they have to be di↵erent, which results in entanglement.

To make sure that photons 1 and 2 cannot be distin-guished by their arrival times, they were generated using a pulsed pump beam and sent through narrow-bandwidth filters producing a coherence time much longer than the pump pulse length [20]. In the experiment, the pump pulses had a duration of 200 fs at a repetition rate of 76 MHz. Observing the down-converted photons at a wavelength of 788 nm and a bandwidth of 4 nm results in a coherence time of 520 fs. It should be mentioned that, because photon 1 is also produced as part of an entangled pair, its partner can serve to indicate that it was emitted.

How can one experimentally prove that an unknown quantum state can be teleported? First, one has to show that teleportation works for a (complete) basis, a set of known states into which any other state can be de-composed. A basis for polarization states has just two components, and in principle we could choose as the ba-sis horizontal and vertical polarization as emitted by the source. Yet this would not demonstrate that teleporta-tion works for any general superpositeleporta-tion, because these two directions are preferred directions in our experiment. Therefore, in the first demonstration we choose as the ba-sis for teleportation the two states linearly polarized at 45 and +45 which are already superpositions of the horizontal and vertical polarizations. Second, one has to show that teleportation works for superpositions of these base states. Therefore we also demonstrate teleportation for circular polarization.

RESULTS

In the first experiment photon 1 is polarized at 45 . Teleportation should work as soon as photon 1 and 2

are detected in the _{| i}12 state, which occurs in 25%

of all possible cases. The _{| i}12 state is identified by

recording a coincidence between two detectors, f1 and f2, placed behind the beam splitter (Fig. 1(b)).

If we detect a f1f2 coincidence (between detectors f1 and f2), then photon 3 should also be polarized at 45 . The polarization of photon 3 is analysed by passing it through a polarizing beam splitter selecting +45 and 45 polarization. To demonstrate teleportation, only detector d2 at the +45 output of the polarizing beam splitter should click (that is, register a detection) once

de-tectors f1 and f2 click. Detector d1 at the 45 output

of the polarizing beam splitter should not detect a pho-ton. Therefore, recording a three-fold coincidence d2f1f2 (+45 analysis) together with the absence of a three-fold coincidence d1f1f2 ( 45 analysis) is a proof that the po-larization of photon 1 has been teleported to photon 3.

To meet the condition of temporal overlap, we change in small steps the arrival time of photon 2 by changing the delay between the first and second down-conversion by translating the retroflection mirror (Fig. 1(b)). In this way we scan into the region of temporal overlap at the beam splitter so that teleportation should occur.

Outside the region of teleportation, photon 1 and 2 each will go either to f1 or to f2 independent of one an-other. The probability of having a coincidence between f1 and f2 is therefore 50%, which is twice as high as inside the region of teleportation. Photon 3 should not have a well-defined polarization because it is part of an entan-gled pair. Therefore, d1 and d2 have both a 50% chance of receiving photon 3. This simple argument yields a

25% probability both for the 45 analysis (d1f1f2

coin-cidences) and for the +45 analysis (d2f1f2 coincoin-cidences) outside the region of teleportation. Figure 3 summa-rizes the predictions as a function of the delay. Success-ful teleportation of the +45 polarization state is then

characterized by a decrease to zero in the 45 analysis

(Fig. 3(a)), and by a constant value for the +45 analysis (Fig. 3(b)).

The theoretical prediction of Fig. 3 may easily be un-derstood by realizing that at zero delay there is a decrease to half in the coincidence rate for the two detectors of the Bell-state analyser, f1 and f2, compared with outside the region of teleportation. Therefore, if the polarization of photon 3 were completely uncorrelated to the others the three-fold coincidence should also show this dip to half. That the right state is teleported is indicated by the fact that the dip goes to zero in Fig. 3(a) and that it is filled to a flat curve in Fig. 3(b).

We note that equally as likely as the production of photons 1, 2 and 3 is the emission of two pairs of

down-5

FIG. 3. Theoretical prediction for the three-fold coincidence probability between the two Bell-state detectors (f1, f2) and one of the detectors analysing the teleported state. The sig-nature of teleportation of a photon polarization state at +45 is a dip to zero at zero delay in the three-fold coincidence rate with the detector analysing 45 (d1f1f2) (a) and a constant value for the detector analysis +45 (d2f1f2) (b). The shaded area indicates the region of teleportation.

converted photons by a single source. Although there is no photon coming from the first source (photon 1 is absent), there will still be a significant contribution to the three-fold coincidence rates. These coincidences have nothing to do with teleportation and can be identified by blocking the path of photon 1.

The probability for this process to yield spurious two-and three-fold coincidences can be estimated by taking into account the experimental parameters. The experi-mentally determined value for the percentage of spurious three-fold coincidences is 68%_{± 1%. In the experimental} graphs of Fig. 4 we have subtracted the experimentally determined spurious coincidences.

The experimental results for teleportation of photons polarized under +45 are shown in the left-hand column of Fig. 4; Fig. 4(a) and (b) should be compared with the theoretical predictions shown in Fig. 3. The strong decrease in the 45 analysis, and the constant signal for the +45 analysis, indicate that photon 3 is polarized along the direction of photon 1, confirming teleportation. The results for photon 1 polarized at 45 demonstrate that teleportation works for a complete basis for polar-ization states (right-hand column of Fig. 4). To rule out any classical explanation for the experimental results, we have produced further confirmation that our procedure works by additional experiments. In these experiments we teleported photons linearly polarized at 0 and at 90 , and also teleported circularly polarized photons. The

ex-FIG. 4. Experimental results. Measured three-fold coinci-dence rates d1f1f2 ( 45 ) and d2f1f2 (+45 ) in the case that the photon state to be teleported is polarized at +45 ((a) and (b)) or at 45 ((c) and (d)). The coincidence rates are plot-ted as function of the delay between the arrival of photon 1 and 2 at Alice’s beam splitter (see Fig. 1(b)). The three-fold coincidence rates are plotted after subtracting the spurious three-fold contribution (see text). These data, compared with Fig. fig:3, together with similar ones for other polarizations (Table I) confirm teleportation for an arbitrary state.

TABLE I. Visibility of teleportation in three-fold coincidences Polarization Visibility +45 0.63 _{± 0.02} 45 0.64 _{± 0.02} 0 0.66 _{± 0.02} 90 0.61 _{± 0.02} Circular 0.57 _{± 0.02}

perimental results are summarized in Table I, where we list the visibility of the dip in three-fold coincidences, which occurs for analysis orthogonal to the input polar-ization.

As mentioned above, the values for the visibilities are obtained after subtracting the o↵set caused by spurious three-fold coincidences. These can experimentally be ex-cluded by conditioning the three-fold coincidences on the detection of photon 4, which e↵ectively projects photon 1 into a single-particle state. We have performed this four-fold coincidence measurement for the case of teleporta-tion of the +45 and +90 polarizateleporta-tion states, that is, for two non-orthogonal states. The experimental results are shown in Fig. 5. Visibilities of 70%_{±3% are obtained} for the dips in the orthogonal polarization states. Here, these visibilities are directly the degree of polarization of the teleported photon in the right state. This proves

エンタングルした光子対

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Dだけが鳴る

理論予想

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