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2.5

# Measurement

We just described a single qubit and its important characteristics, its ability to support superposition and phase. But it’s important to note that we can’t directly see either of those characteristics. Instead, we must measure the qubit, which has some behaviors that are relatively intuitive and some that aren’t.

As we noted, the waves we have been describing are called probability amplitudes in quantum mechanics. These amplitudes determine the probability of finding a value when we measure the state.

## Measuring a Qubit

When we measure a qubit, we always find one of two states (usually either zero or one, but we’ll relax that a bit later). For a 50/50 state like our “ket plus” state

$$\sqrt{1/2}|0\rangle + \sqrt{1/2}|1\rangle$$,

there is a fifty percent probability of finding zero, and a fifty percent probability of finding one. We can calculate this by taking the absolute value of the square of the amplitude. For both $$|0\rangle$$ and $$|1\rangle$$, that is $$|\sqrt{1/2}|^2 = 1/2 = 50\%$$.

For the state

$$1/2|0\rangle + \sqrt{3/4}|1\rangle$$

there is a $$|1/2|^2 = 1/4 = 25\%$$ probability of finding zero and a 75% probability of finding one. We don’t get 0.75 when we measure it; instead, our quantum probability amplitudes determine what the probability is of finding one of the states.

## Collapse of the Wave Function

Not only does measuring the qubit give us a value based on the probability amplitudes, we also say that it collapses the wave function. What does this mean?

It means that after we find, for example, a zero, any amplitude for one has disappeared. We are left with 100% zero. There is no way to work backwards from the measurement result to determine anything about the original probability amplitudes, other than obviously the amplitude for the value that we measured was non-zero.

If you want to know more, you’ll just have to rerun your experiment a bunch of times, starting from the preparation of your qubit, to collect some statistics. For example, if you run your experiment 100 times and you find zero 49 times and one 51 times, you can infer that your experiment is creating a state that is about 50% zero and 50% one.

## Three Ways of Measuring One Qubit

The Bloch sphere is useful for thinking about measurements. (It’s one of the main reasons we introduced the concept.) The qubit’s vector can point to an arbitrary position on the sphere. If the vector is in the “northern” hemisphere, we are more likely to find a $$|0\rangle$$ state when we measure (recalling that $$|0\rangle$$ is at the north pole), whereas if the vector is in the “southern” hemisphere, we are more likely to find a $$|1\rangle$$ (since the $$|1\rangle$$ state is at the south pole). The state is then projected onto the state we measured, so that we only have $$|0\rangle$$ or $$|1\rangle$$ left. (Projected means that we move from our arbitrary vector onto one of those states.)

The $$|0\rangle$$ / $$|1\rangle$$ axis of the Bloch sphere is known as the $$Z$$ axis. In our discussion so far, we have assumed that measuring a qubit means looking at it in a way that projects it into one of those two states. Measuring a qubit is actually more general than that: we can pick any line through the center of Bloch sphere and measure the qubit, which will project the qubit onto one of the two places where the line meets the sphere.

To keep things relatively simple, we will assume that our measurements are along the $$X$$, $$Y$$, or $$Z$$ axis of the Bloch sphere. When we measure on the $$X$$ axis, we will project our qubit to one end of the $$X$$ axis, which we called our $$|+\rangle$$ and $$|-\rangle$$ states.

Measuring along the $$Y$$ axis is less common, so unlike the $$Z$$ and $$X$$ axes, the two ends don’t have such simple nicknames. The states at the two ends are
$$\sqrt{1/2}|0\rangle + (\pi/2)\sqrt{1/2}|1\rangle$$ or $$\sqrt{1/2}|0\rangle + (3\pi/2)\sqrt{1/2}|1\rangle.$$

## Relationship to Algorithms

Just a quick peek ahead: in fact, the wave function collapse can affect not just the particular qubit we are looking at, but the state of all of the qubits in our system, under some circumstances. We’ll see a little more of that when we talk about entanglement shortly, and a lot more when we talk about algorithms.

We will see that the entire goal of an algorithm is to use interference, which we have already talked about, to create states where measuring the outcome has a high probability of being the solution to our problem.

# 測定

これまでに、重ね合わせ状態や位相といった量子ビットの特性について学びました。 また、それらの量子ビットの情報はそのままダイレクトに見えるものではない、ということも重要な点の１つです。 量子ビットの状態を知るには、その量子ビットを「測定」する必要があります。

## 量子ビットの測定

$\sqrt{1/2}|0\rangle + \sqrt{1/2}|1\rangle$

で、50%の確率で0を観測し、50%の確率で1を観測します。振幅の絶対値の自乗で確率を求めることができるため、この場合、$$\vert0\rangle$$や $$\vert1\rangle$$が観測される確率は両方 $$\vert\sqrt{1/2}\vert^2 = 1/2 =50\%$$で計算することができます。

また、

$1/2|0\rangle + \sqrt{3/4}|1\rangle$

このような状態の場合、

0を観測する確率は$$\vert1/2\vert^2 = 1/4 = 25\%$$であり、同様の計算手法で1を観測する確率が75％であることが分かります。 あくまで、量子確率振幅は、それぞれの状態が観測される確率を決定するだけであり、測定を行うことで0.75という値が直接得られるわけではありません。

## 波動関数の収束

ある量子ビットの状態を詳細に知りたい場合、その量子ビットを準備して、それを測定する実験を複数回行う必要があります。例えば、もし100回の実験を行い、49回0を観測し、51回1を観測した場合、その量子ビットは50%で1、50%で0の重ね合わせ状態にあったと推測できます。

## 1つの量子ビットを測定する3つの方法

こういった単一量子ビットの測定について考える時、ブロッホ球は便利な表記法です。 量子ビットのベクトルは球面上の任意の点を指すことができます。 ベクトルが北半球にある場合、測定で$$\vert0\rangle$$を観測する確率が高くなります( $$\vert0\rangle$$が北極にあることを思い出してください)。 同様に、ベクトルが南半球にある場合は$$\vert1\rangle$$を観測する確率が高くなります。($$\vert1\rangle$$が南極にあることを思い出してください)。

ブロッホ球上での$$\vert0\rangle$$と$$\vert1\rangle$$を結ぶ軸は$$Z$$軸と呼ばれ、量子ビットを測定することは$$Z$$軸の両端の状態のうち、どちらか１つの状態へと射影することを仮定してきました。 実際は量子ビットの測定は常に$$Z$$軸を基底とする必要性はなく、ブロッホ球の原点を通る任意の軸にそって射影を行うことができます。

$$Y$$軸に沿って測定することはあまり一般的ではないため、 $$Z$$や$$X$$ 軸のように両端に名前はありません。 両端の状態はそれぞれ$$\sqrt{1/2}\vert0\rangle + (\pi/2)\sqrt{1/2}\vert1\rangle$$と $$\sqrt{1/2}\vert0\rangle + (3\pi/2)\sqrt{1/2}\vert1\rangle$$です。