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To represent a quantum computer's state, all the qubits contribute to one state vector (this is one of the major differences between quantum and classical computing as I understand it). My understanding is that it's possible to measure only one qubit out of a system of multiple qubits. How does measuring that one qubit affect the whole system (specifically, how does it affect the state vector)?

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There are a lot of different ways of looking at qubits, and the state vector formalism is just one of them. In a general linear-algebraic sense a measurement is projection onto a basis. Here I will provide insight with an example from the Pauli observable point of view, that is the usual circuit model of QC.

Firstly, it's of interest which basis the state vector is being provided in-- every measurement operator comes with a set of eigenstates, and whatever measurements you look at (eg. $X,Y,Z, XX, XZ$, etc.) determine the basis that might be best for you to write the state vector in. The easiest way to answer your question is if you know which basis is of interest to you, and more importantly, whether it commutes with the measurement you just made.

So for simplicity's sake, let's say you start with two coupled qubits in an arbitrary state written in the $Z$-basis for both qubits:

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

The simplest possible measurements you could make would be $Z_{1}$, that is the $Z$ operator on the first qubit, followed by $Z_{2}$, the $Z$ operator on the second qubit. What does measurement do? It projects the state into one of the eigenstates. You can think of this as eliminating all possible answers that are inconsistent with the one we just measured. For instance, say we measure $Z_{1}$ and obtain the outcome $1$, then the resulting state we would have would be:

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

Note that the coefficient out front is just for renormalization. So our probability of measuring $Z_{2}=0$ is $\frac{1}{|c|^{2} +|d|^{2}} |c^{2}|$. 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:

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

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

A good way to check your understanding: What is the probability of measuring $X= +1$ after the $Z_{1}=1$ measurement above? What is the probability if we have not made the $Z_{1}$ measurement? Then a more complicated question is to look at product operators that act on both qubits at once, for instance, how does a measurement of $Z_{1}Z_{2}=+1$ affect the initial state? Here $Z_{1}Z_{2}$ measures the product of the two operators.

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    $\begingroup$ Nice and simple answer. I think it is important to note, that what you describe is only true if you a) perform projective measurements and b) you know the outcome of the measurement. Just keep in mind that in general you will need mixed states to describe the post-measurement state. $\endgroup$ – M. Stern Mar 20 '18 at 0:35
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Suppose that, prior to measurement, your $n$-qubit system is in some state $\lvert \psi \rangle \in \mathcal H_2^{\otimes n}$, where $\mathcal H_2 \cong \mathbb C^2$ is the Hilbert space of a single qubit. Write $$ \lvert \psi \rangle = \sum_{x \in \{0,1\}^n} u_x \lvert x \rangle $$ for some coefficients $u_x \in \mathbb C$ such that $\sum_x \lvert u_x \rvert^2 = 1$.

  • If you are measuring the first qubit in the standard basis, define $$\begin{aligned} \lvert \varphi_0 \rangle &= \!\!\!\!\!\sum_{x' \in \{0,1\}^{n-1}}\!\!\!\!\!\! u_{0x'} \,\lvert0\rangle \lvert x' \rangle, \\ \lvert \varphi_1 \rangle &= \!\!\!\!\!\sum_{x' \in \{0,1\}^{n-1}}\!\!\!\!\!\! u_{1x'} \,\lvert1\rangle \lvert x' \rangle,\end{aligned}$$ 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.

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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!

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