利用者:Ykhope/sandbox

爆弾検査問題(Elitzur–Vaidman bomb tester)

爆弾検査問題の図解。A - 光子エミッタ。B - 試験対象の爆弾。C、D - 光子検出器。左下と右上の角の鏡はビームスプリッタ。 Bomb-testing problem diagram. A – photon emitter, B – bomb to be tested, C, D – photon detectors. Mirrors in the lower left and upper right corners are semi-transparent.

爆弾検査問題は、無相互作用測定を使って爆弾を爆発させることなく爆弾の機能を検証する量子力学の思考実験である。1993年にAvshalom ElitzurとLev Vaidmanによって考案された。発表後、実験によって理論通り機能することが確認された。

The Elitzur–Vaidman bomb-tester is a quantum mechanics thought experiment that uses interaction-free measurements to verify that a bomb is functional without having to detonate it.

It was conceived in 1993 by Avshalom Elitzur and Lev Vaidman.

Since their publication, real-world experiments have confirmed that their theoretical method works as predicted.[1]

爆弾検査は、光子や電子などの素粒子の2つの特性である非局所性と粒子と波動の二重性を利用している。 爆弾が爆発する可能性が50％あるが、粒子を重ね合わせにすることで、爆弾が爆発を引き起こすことなく機能することを実験で確認できる。

The bomb tester takes advantage of two characteristics of elementary particles, such as photons or electrons: nonlocality and wave-particle duality.[2] By placing the particle in a quantum superposition, the experiment can verify that the bomb works without ever triggering its detonation, although there is a 50% chance that the bomb will explode in the effort.

キャットステート(Cat state)

キャットステートは、量子コンピュータでシュレーディンガーの猫の後に名付けられた。異なる二つの状態である重ね合わせである。 個々の重ね合わせ状態は、古典的か量子的であるが巨視的には重要な基準である。 キャットステートは、一つか多くの形態か量子である可能性があり、特に単一の粒子の場合はもつれさせる必要がない。 グリーンバーガー＝ホーン＝ツァイリンガー状態と比べ、定義上は複数の異なる粒子、形態とそのもつれからなる。 In quantum computing, the cat state, named after Schrödinger's cat, is a quantum superposition of two macroscopically distinct states. The individual states being superposed could be classical or quantum, but their macroscopicity is an important criterion. A cat state could be of one or more modes or particles, and does not necessarily need entanglement, especially for the single-particle case. This is in contrast to the Greenberger–Horne–Zeilinger state, which by definition consists of multiple distinct particles or modes and their entanglement. In other quantum mechanics contexts, according to The New York Times for example, physicists view the cat state as composed of two diametrically opposed conditions at the same time, such as the possibilities that a cat be alive and dead at the same time. This is sometimes connected to the many worlds hypothesis by proponents of the many worlds interpretation of quantum mechanics. More prosaically, a cat state might be the possibilities that six atoms be spin up and spin down, as published by a team led by David Wineland at NIST, December 1, 2005. Large cat states have also been experimentally created using photons by a team led by Jian-Wei Pan at University of Science and Technology of China, for instance, four-photon entanglement, five-photon entanglement, six-photon entanglement, eight-photon entanglement, and five-photon ten-qubit cat state. This spin up/down formulation was proposed by David Bohm, who conceived of spin as an observable in a version of thought experiments formulated in the 1935 EPR paradox.

背景(Background)

爆弾検査問題は、無相互作用測定である。 この発案は、相互作用することなく物体の情報を取得することは目新しいことではない。 例えば、二つの箱があり、一つには何かが入っている、もう一つには何も入っていない。 もし一つの箱を開けてみて何もなければ、他の箱には何かが入っていることを開けることなく知ることが出来る。

The bomb test is an interaction-free measurement.

The idea of getting information about an object without interacting with it is not a new one. For example, there are two boxes, one of which contains something, the other of which contains nothing.

If you open one box and see nothing, you know that the other contains something, without ever opening it.[2]

この実験のルーツは、二重スリット実験とその他ともっと複合的なコンセプトであるシュレーディンガーの猫、ホィーラーの遅延選択実験から触発されている。 素粒子の振る舞いは、我々のマクロの世界での実験では、異なりすぎる。 波動か粒子のように振る舞うことが出来る(粒子と波動の二重性を参照)。 波動の状態を重ね合わせと呼ぶ。 この状態で、粒子の特性として例えば、場所を特定出来ない。としても重ね合わせ状態では、多くかすべての可能性が等しく現実的である。 よって、もし複数の場所に存在可能であれば、全ての場所の存在する。 粒子の波が注意深く崩壊する直前に場所が一度だけ特定出来る。（波動関数の崩壊によって？） 粒子の実際の状態だけでなく、崩壊前に存在する他の状態や場所の情報を収集出来る。 粒子の状態や場所が不確実であっても、可能である。

This experiment has its roots in the double-slit experiment and other, more complex concepts it inspired, including Schrodinger's cat, and Wheeler's delayed choice experiment.[3]

The behavior of elementary particles is very different from what we experience in our macroscopic world. They can behave like a wave or like a particle (see wave–particle duality). When they are in a wave state, they are in what is called a "superposition". In this state, some properties of the particle, for example, its location, are not definite. While in a superposition, any and all possibilities are equally real. So, if it can exist in more than one location, it does exist in them all[clarification needed]. The particle's wave can later be "collapsed" by observing it, at which time its location once again becomes definite. Information can then be gleaned not only about the actual state of the particle, but also other states, or locations in which it existed before the collapse.

This is possible even though the particle was never factually in those states or locations.

動作原理(How it works)

→「マッハ・ツェンダー干渉計」も参照

Figure 1: An illustration of the experiment using a Mach–Zehnder interferometer

Figure 2: Legend for Fig. 1

感光性爆弾の不発弾を収集することを考察する。 たった一つの光子を爆弾のトリガが吸収し、検出することで爆弾は爆発する。 不発弾のトリガには、センサーがない。それで光子は吸収できない。 よって、不発弾は光子を検出せずに爆発しない。 不発弾ではない爆弾を含む全ての爆弾の中から、爆発させることなく不発弾を特定することは可能か？

Consider a collection of light-sensitive bombs, of which some are duds.

When their triggers detect any light, even a single photon, the light is absorbed and the bomb explodes. The triggers on the dud bombs have no sensor, so the photon can't be absorbed.[4] Thus, the dud bomb will not detect the photon and will not detonate.

Is it possible to determine which bombs are functional and which are duds without detonating all of the live ones?

部位(Components)

• 感光性爆弾：不発弾であるかどうかは不明。
• 光子エミッタ：実験目的の一つの光子を導く。
• 光子：放射された後、箱の下へ行く。
• 箱の中身：
  • 初期半透鏡：光子が箱に入るとこのビームスプリッターに遭遇する。光子は鏡を透過して箱の中の下のパスへ行くか反射して90度の角度で箱の上のパスへ行く。
  • 検査する爆弾：箱の中の下のパスに配置する。活性弾で光子が到達すると爆発して自身と光子が破壊される。不発弾であれば、光子が下のパスを通過する。
  • 通常鏡のペア：これらは両方のパスに位置する。二つ目のビームスプリッタへ向むかうように二つのパスに配置する。
  • 初期半透鏡と同一の二つ目のビームスプリッタ：下のパスと上のパスを交差し、箱の出口に初期半透鏡の対向に位置する。（その後、通常鏡によって向けなおされる）
• 光子検出器のペア：箱の外側に位置し、二つ目のビームスプリッタと直線上にある。光子はどちらでも検出できるが、同時に検出されない。

• A light-sensitive bomb: it's not known whether it's live or a dud.
• A photon emitter: it produces a single photon for the purposes of the experiment.
• A photon: after being emitted, it travels through the box below.
• A "box" which contains:
  • An initial half-silvered mirror: the photon enters the box when it encounters this "beam splitter".The photon will either pass through the mirror and travel along the "lower path" inside the box, or be reflected at a 90-degree angle and travel along the box's "upper path".
  • The bomb in question: it's placed inside the box beforehand on the "lower path".If it's live and comes into contact with a photon, it will detonate and destroy itself and the photon.If, however, the bomb is a dud, the photon passes it by and continues on its way along the lower path.
  • A pair of ordinary mirrors: they're located on each path. They are positioned to redirect the photon so that the two paths intersect one another at the same position as the second beam splitter.
  • A second beam splitter identical to the initial one: it's positioned opposite the initial one, at the intersection between the lower path and upper path (after they have been redirected by the ordinary mirrors), at the exit of the box.
• A pair of photon detectors: they're located outside the box, aligned with the second beam-splitter. The photon can be detected at either or neither, but never both.

パート１：重ね合わせ(Part 1: The Superposition)

爆弾検査器内の重ね合わせは、曲がった半透鏡によって作られ、光子は透過するか90度の角度で反射する。(図3を参照)どちらの可能性も等しい。どちらのケースでも光子が重ね合わせ状態になる。単一の粒子は両方とも半透鏡を透過する。その瞬間、単一の粒子は、二つの異なる場所に存在する。 上のパスか下のパスに沿って、粒子が通常鏡に入り、他のルートへ向けなおされる。それから二つの目の半透鏡と交差する。他の側では、検出器のペアをどちらかの検出器が光子を検出し、両方から検知できないように配置する。両方とも検出しない可能性もある。この結果を基準として活性爆弾では、50%の確率で爆発し、25%の確率で爆発させずに機能があることを確認できて、25%の確率で結果が出ない。

A superposition in the bomb tester is created with an angled half-silvered mirror, which allows a photon to either pass through it, or be reflected off it at a 90-degree angle (see figure 3). There is equal probability it will do either. The photon enters a superposition, in which it does both. The single particle both passes through, and is reflected off the half-silvered mirror. From that moment on, the single photon exists in two different locations. Along both the upper and lower path, the particle will encounter an ordinary mirror, positioned to redirect the two routes toward one another. They then intersect at a second half-silvered mirror. On the other side, a pair of detectors are placed such that the photon can be detected by either detector, but never by both. It is also possible that it will not be detected by either. Based on this outcome, with a live bomb, there is a 50% chance it will explode, a 25% chance it will be identified as good without exploding and a 25% chance there will be no result.

パート２：爆弾(Part 2: The Bomb)

→「Cat state」を参照

Figure 4: If the bomb is live, it will absorb the photon and detonate. If its a dud, the photon is unaffected and continues along the lower path.

感光性爆弾は下のパスに沿って配置される。もし活性爆弾なら光子が到達すると爆発し破壊される。不発弾であれば、光子の影響を受けない。(図4を参照)この実験の原理を理解するために爆弾は一種の観察者であり、光子が遭遇すると一種の観察であることを知ることが重要である。ゆえに光子の重ね合わせが崩壊すると、光子が上と下のパスに沿って行く。光子が到達するのは活性爆弾か検出器であるとはいえ、どちらか片方しかない。しかし、シュレーディンガーの猫のように

A light-sensitive bomb is placed along the lower path. If the bomb is good, when a photon arrives, it will explode and both will be destroyed. If it's a dud, the photon will pass by unaffected (see figure 4). To understand how this experiment works, it is important to know that the bomb is a kind of observer and that this encounter is a kind of observation. It can therefore collapse the photon's superposition, in which the photon is travelling along both the upper and lower paths. When it reaches the live bomb, or the detectors, however, it can only have been on one or the other. But, like the radioactive material in the box with Schrödinger's famous cat, upon its encounter with the half-silvered mirror at the beginning of the experiment, the photon, paradoxically does and does not interact with the bomb. According to the authors, the bomb both explodes and doesn't explode.[5] This is only in the case of a live bomb, however. In any event, once observed by the detectors, it will have only traveled one of the paths.

Part 3: The second half-silvered mirror

When two waves collide, the process by which they affect each other is called interference. They can either strengthen each other by "constructive interference", or weaken each other by "destructive interference".[6] This is true whether the wave is in water, or a single photon in a superposition. So even though there is only one photon in the experiment, because of its encounter with the half-silvered mirror, it acts like two. When "it" or "they" are reflected off the ordinary mirrors, it will interfere with itself as if it were two different photons. But that's only true if the bomb is a dud. A live bomb will absorb the photon when it explodes and there will be no opportunity for the photon to interfere with itself. When it reaches the second half-silvered mirror, if the photon in the experiment is behaving like a particle (in other words, if it is not in a superposition), then it has a fifty-fifty chance it will pass through or be reflected and be detected by one or the other detector. But that's only possible if the bomb is live. If the bomb "observed" the photon, it detonated and destroyed the photon on the lower path, therefore only the photon that takes the upper path will be detected, either at Detector C or Detector D.

Part 4: Detector C and Detector D

Detector D is the key to confirming that the bomb is live. The two detectors and the second half-silvered mirror are precisely aligned with one another. Detector C is positioned to detect the particle if the bomb is a dud and the particle traveled both paths in its superposition and then constructively interfered with itself. Detector D is positioned to detect the photon only in the event of destructive interference—an impossibility (see figure 6). In other words, if the photon is in a superposition at the time it arrives at the second half-silvered mirror, it will always arrive at detector C and never at detector D. If the bomb is live, there is a 50/50 chance that the photon took upper path. If it "factually" did so, then it "counter-factually" took the lower path (see figure 7). That counter-factual event destroyed that photon and left only the photon on the upper to arrive at the second half-silvered mirror. At which point it will, again, have a 50/50 chance of passing through it or being reflected off it, and, subsequently, it will be detected at either of the two detectors with the same probability. This is what makes it possible for the experiment to verify the bomb is live without actually blowing it up.[7]

Results

With a dud, the photon will always arrive at Detector C. With a live bomb, there can be three possible outcomes:

   No photon was detected (50% chance).
   The photon was detected at C (25% chance).
   The photon was detected at D (25% chance).

These correspond with the following conditions of the bomb being tested:

1. No photon was detected: The bomb exploded and destroyed the photon before it could be detected. This is because the photon in fact took the lower path and triggered the bomb, destroying itself in the process. There is a 50% chance that this will be the outcome if the bomb is live. 2. The photon was detected at C: This will always be the outcome if a bomb is a dud, however, there is a 25% chance that this will be the outcome if the bomb is live. If the bomb is a dud, this is because the photon remained in its superposition until it reached the second half-silvered mirror and constructively interfered with itself. If the bomb is live, this is because the photon in fact took the upper path and reflected off the second half-silvered mirror. 3. The photon was detected at D: The bomb is live but unexploded. That's because the photon in fact took the upper path and passed through the second half-silvered mirror, something possible only because there was no photon from the lower path with which it could interfere. This is the only way that a photon can ever be detected at D. If this is the outcome, the experiment has successfully verified that the bomb is live despite the fact that the photon never "factually" encountered the bomb itself. There is a 25% chance that this will be the outcome if the bomb is live.[7]

(Note: The diagram and explanation in Figure 7 unfortunately reverses the positions of detectors C & D with respect to the diagram at the top of the page. The explanation in this section refers to the initial diagram at the top of this page.)

If the result is 2, the experiment is repeated. If the photon continues to be observed at C and the bomb doesn't explode, it can eventually be concluded that the bomb is a dud.[8]

With this process 25% of live bombs can be identified without being detonated, 50% will be detonated and 25% remain uncertain.[8] By repeating the process with the uncertain ones, the ratio of identified non-detonated live bombs approaches 33% of the initial population of bombs. See the "Experiments" section below for a modified experiment that can identify the live bombs with a yield rate approaching 100%.

Many-worlds interpretation

The authors point out that the ability to obtain information about the bomb's functionality without ever "touching" it, appears to be a paradox. That, they claim, is based on the assumption that there is only a single "real" result.[3] But according to the many-worlds interpretation, each possible state of a particle's superposition is real. Therefore, the particle does actually interact with the bomb and it does explode, just not in our "world".[5]

Experiments

In 1994, Anton Zeilinger, Paul Kwiat, Harald Weinfurter, and Thomas Herzog actually performed an equivalent of the above experiment, proving interaction-free measurements are indeed possible.[9]

In 1996, Kwiat et al. devised a method, using a sequence of polarising devices, that efficiently increases the yield rate to a level arbitrarily close to one. The key idea is to split a fraction of the photon beam into a large number of beams of very small amplitude and reflect all of them off the mirror, recombining them with the original beam afterwards.[9][10] It can also be argued that this revised construction is simply equivalent to a resonant cavity and the result looks much less shocking in this language, see Watanabe and Inoue (2000).

In 2016, Carsten Robens, Wolfgang Alt, Clive Emary, Dieter Meschede, and Andrea Alberti[11] demonstrated that the Elitzur–Vaidman bomb testing experiment can be recast in a rigorous test of the macro-realistic worldview based on the violation of the Leggett–Garg inequality using ideal negative measurements. In their experiment they perform the “bomb test” with a single atom trapped in a polarization-synthesized optical lattice. This optical lattice enables interaction-free measurements by entangling the spin and position of atoms.