Bagaimana pengukuran satu qubit memengaruhi yang lain?

Untuk mewakili keadaan komputer kuantum, semua qubit berkontribusi pada satu vektor keadaan (ini adalah salah satu perbedaan utama antara komputasi kuantum dan klasik seperti yang saya pahami). Pemahaman saya adalah bahwa itu mungkin untuk mengukur hanya satu qubit dari sistem multi qubit. Bagaimana mengukur bahwa satu qubit memengaruhi keseluruhan sistem (khususnya, bagaimana hal itu memengaruhi vektor keadaan)?

Ada banyak cara berbeda dalam memandang qubit, dan formalisme vektor negara hanyalah salah satunya. Dalam pengertian linear-aljabar umum pengukuran diukur ke atas dasar. Di sini saya akan memberikan wawasan dengan contoh dari sudut pandang Pauli yang dapat diamati, yaitu model sirkuit QC yang biasa.

Pertama, yang menarik adalah dasar vektor negara disediakan - setiap operator pengukuran dilengkapi dengan set status eigen, dan pengukuran apa pun yang Anda lihat (mis. $X,Y,Z, XX, XZ$ , dll.) tentukan dasar yang paling baik bagi Anda untuk menulis vektor keadaan. Cara termudah untuk menjawab pertanyaan Anda adalah jika Anda tahu dasar mana yang menarik bagi Anda, dan yang lebih penting, apakah itu bolak-balik dengan pengukuran yang baru saja Anda buat .

Jadi demi kesederhanaan, katakanlah Anda mulai dengan dua qubit berpasangan dalam status arbitrer yang ditulis dalam $Z$ dasar untuk kedua qubit:

$$| \psi \rangle = a | 0_{Z} \rangle \otimes | 0_{Z} \rangle +b | 0_{Z} \rangle \otimes | 1_{Z} \rangle + c | 1_{Z} \rangle \otimes | 0_{Z} \rangle + d | 1_{Z} \rangle \otimes | 1_{Z} \rangle$$

Yang paling sederhana mungkin pengukuran Anda bisa membuat akan , yaitu Z operator pada qubit pertama, diikuti oleh Z 2 , yang Z operator pada qubit kedua. Apa yang dilakukan pengukuran? Ini memproyeksikan negara ke salah satu status eigen. Anda dapat menganggap ini sebagai menghilangkan semua kemungkinan jawaban yang tidak konsisten dengan jawaban yang baru saja kita ukur. Sebagai contoh, katakanlah kita mengukur Z 1 dan memperoleh hasil 1 , maka negara yang dihasilkan kita akan memiliki akan:$Z_{1}$$Z$$Z_{2}$$Z$$Z_{1}$$1$

$$| \psi \rangle = \frac{1}{\sqrt{|c|^{2} +|d|^{2}}} \left(c | 1_{Z} \rangle \otimes | 0_{Z} \rangle + d | 1_{Z} \rangle \otimes | 1_{Z} \rangle \right)$$

Perhatikan bahwa koefisien di depan hanya untuk renormalisasi. Jadi probabilitas kami ukur adalah $Z_{2}=0$$\frac{1}{|c|^{2} +|d|^{2}} |c^{2}|$. Note this is different from the probability we had in the initial state, which was $|a|^{2}+|c|^{2}$.

Suppose the next measurement you make does not commute with the previous one, however. This is trickier because you have to implement a change of basis on the state vector in order to understand the probabilities. With Pauli measurements, though, it tends to be easy since the eigenbases relate in a nice way, that is:

$$| 0_{Z} \rangle = \frac{1}{\sqrt{2}} (|0_{X}\rangle + |1_{X} \rangle )$$

$$| 1_{Z} \rangle = \frac{1}{\sqrt{2}} (|0_{X}\rangle - |1_{X} \rangle )$$

Cara yang baik untuk memeriksa pemahaman Anda: Berapa probabilitas mengukur setelah Z 1 = 1 pengukuran di atas? Berapa probabilitas jika kita belum membuat Z 1 pengukuran? Kemudian pertanyaan yang lebih rumit adalah dengan melihat operator produk yang bertindak di kedua qubit sekaligus, misalnya, bagaimana pengukuran Z 1 Z 2 = + 1 mempengaruhi keadaan awal? Di sini Z 1 Z 2 mengukur produk dari dua operator.$X= +1$$Z_{1}=1$$Z_{1}$$Z_{1}Z_{2}=+1$$Z_{1}Z_{2}$

$n$$\lvert \psi \rangle \in \mathcal H_2^{\otimes n}$$\mathcal H_2 \cong \mathbb C^2$

$$\lvert \psi \rangle = \sum_{x \in \{0,1\}^n} u_x \lvert x \rangle$$ for some coefficients $u_x \in \mathbb C$ such that $\sum_x \lvert u_x \rvert^2 = 1$.

• $$\begin{aligned} \lvert \varphi_0 \rangle &= \!\!\!\!\!\sum_{x' \in \{0,1\}^{n-1}}\!\!\!\!\!\! u_{0x'} \,\lvert0\rangle \lvert x' \rangle, \\ \lvert \varphi_1 \rangle &= \!\!\!\!\!\sum_{x' \in \{0,1\}^{n-1}}\!\!\!\!\!\! u_{1x'} \,\lvert1\rangle \lvert x' \rangle,\end{aligned}$$ and let $\lvert \psi_0 \rangle = \lvert \varphi_0 \rangle \big/\! \sqrt{\langle \varphi_0 \vert \varphi_0 \rangle}\,$ and $\,\lvert \psi_1 \rangle = \lvert \varphi_1 \rangle \big/\! \sqrt{\langle \varphi_1 \vert \varphi_1 \rangle}\,$. It is not too difficult to show that, if you measure the first qubit and obtain the state $\lvert 0 \rangle$, the state of the entire system "collapses" to $\lvert \psi_0 \rangle$, and if you obtain $\lvert 1 \rangle$ what you obtain is $\lvert \psi_1 \rangle$.

  This is broadly analogous to the idea of conditional probability distributions: you might think of $\lvert \psi_0 \rangle$ as the state of the system conditioned on the first qubit being $\lvert 0 \rangle$, and $\lvert \psi_1 \rangle$ as the state of the system conditioned on the first qubit being $\lvert 1 \rangle$ (except of course that the story is a bit more complicated, on account of the fact that the first qubit is not "secretly" in either the state $0$ or $1$).

• The above is not strongly dependent on measuring the first qubit: we can define $\lvert \varphi_0 \rangle$ and $\lvert \varphi_1 \rangle$ in terms of fixing any particular bit in the bit string $x$ to either $0$ or $1$, summing over only those components which are consistent with either the choice $0$ or $1$, and proceeding as above.

• The above is also not strongly dependent on measuring in the standard basis, as Emily indicates. If we wish to consider measuring the first qubit in the basis $\lvert \alpha \rangle, \lvert \beta \rangle$, where $\lvert \alpha \rangle = \alpha_0 \lvert 0 \rangle + \alpha_1 \lvert 1 \rangle$ and $\lvert \beta \rangle = \beta_0 \lvert 0 \rangle + \beta_1 \lvert 1 \rangle$, we define

  $$\begin{aligned} \lvert \varphi_0 \rangle &= \Bigl(\lvert \alpha \rangle\!\langle \alpha \lvert \otimes I^{\otimes n-1}\Bigr)\lvert \psi\rangle = \!\!\!\!\!\sum_{x' \in \{0,1\}^{n-1}}\!\!\!\!\!\! \bigl(\alpha_0^\ast u_{0x'} + \alpha_1^\ast u_{1x'}\bigr) \,\lvert\alpha\rangle \lvert x' \rangle\,, \\ \lvert \varphi_1 \rangle &= \Bigl(\lvert \beta\rangle\!\langle \beta \lvert \otimes I^{\otimes n-1}\Bigr)\lvert \psi\rangle = \!\!\!\!\!\sum_{x' \in \{0,1\}^{n-1}}\!\!\!\!\!\! \bigl(\beta_0^\ast u_{0x'} + \beta_1^\ast u_{1x'}\bigr) \,\lvert\beta\rangle \lvert x' \rangle\,, \end{aligned}$$ and then proceeding as above.

Less formally-stated than the other answers, but for beginners I like the intuitive method outlined by Prof. Vazirani in this video.

Suppose you have a general two-qbit state:

$|\psi\rangle = \begin{bmatrix} \alpha_{00} \\ \alpha_{01} \\ \alpha_{10} \\ \alpha_{11} \end{bmatrix} = \alpha_{00}|00\rangle + \alpha_{01}|01\rangle + \alpha_{10}|10\rangle + \alpha_{11}|11\rangle$

Now suppose you measure the most-significant (leftmost) qbit in the computational basis (as in, collapse it to either $|0\rangle$ or $|1\rangle$). There are two questions we might ask:

1. What is the probability that the measured qbit collapses to $|0\rangle$? What about $|1\rangle$?
2. What is the state of the 2-qbit system after measurement?

For the first question, the intuitive answer is this: take the sum of squares of all amplitudes associated with the value for which you want to find the probability of collapse. So, if you want to know the probability of the measured qbit collapsing to $|0\rangle$, you'd look at the amplitudes associated with cases $|00\rangle$ and $|01\rangle$, because those are the cases where the measured qbit is $|0\rangle$. Thus:

$P[|0\rangle] = |\alpha_{00}|^2 + |\alpha_{01}|^2$

Similarly, for $|1\rangle$ you look at the amplitudes associated with cases $|10\rangle$ and $|11\rangle$, so:

$P[|1\rangle] = |\alpha_{10}|^2 + |\alpha_{11}|^2$

As for the state of the 2-qbit system after measurement, what you do is cross out all the components of the superposition which are inconsistent with the answer you got. So, if you measured $|0\rangle$, then the state after measurement is:

$\require{cancel} |\psi\rangle = \alpha_{00}|00\rangle + \alpha_{01}|01\rangle + \cancel{\alpha_{10}|10\rangle} + \cancel{\alpha_{11}|11\rangle} = \alpha_{00}|00\rangle + \alpha_{01}|01\rangle$

However, this state is not normalized - the sum of squares does not add up to 1, and so you have to normalize it:

$|\psi\rangle = \frac{\alpha_{00}|00\rangle + \alpha_{01}|01\rangle}{\sqrt{|\alpha_{00}|^2 + |\alpha_{01}|^2}}$

Similarly, if you measured $|1\rangle$ then you'd get:

$\require{cancel} |\psi\rangle = \cancel{\alpha_{00}|00\rangle} + \cancel{\alpha_{01}|01\rangle} + \alpha_{10}|10\rangle + \alpha_{11}|11\rangle = \alpha_{10}|10\rangle + \alpha_{11}|11\rangle$

Normalized:

$|\psi\rangle = \frac{\alpha_{10}|10\rangle + \alpha_{11}|11\rangle}{\sqrt{|\alpha_{10}|^2 + |\alpha_{11}|^2}}$

And that's how you calculate the action of measuring one qbit in a multi-qbit state, in the simplest case!