27 votes
Accepted

How does bra-ket notation work?

As already explained by others, a ket $\left|\psi\right>$ is just a vector. A bra $\left<\psi\right|$ is the Hermitian conjugate of the vector. You can multiply a vector with a number in the ...
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23 votes

How does bra-ket notation work?

You could think of $|0\rangle$ and $|1\rangle$ as two orthonormal basis states (represented by "ket"s) of a quantum bit which resides in a two dimensional complex vector space. The lines and brackets ...
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19 votes

How does bra-ket notation work?

What do all of these brackets and vertical lines mean? The notation $\left \lvert v \right \rangle$ means exactly the same thing as $\vec{v}$ or $\textbf{v}$, i.e. it denotes a vector whose name is "...
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  • 1,153
16 votes

How does bra-ket notation work?

The ket notation $|\psi\rangle$ means a vector in whatever vector space we're working in, such as the space of all complex linear combinations of the eight 3-bit strings $000$, $001$, $010$, etc., as ...
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14 votes

How does bra-ket notation work?

This leads me to conclude that there is some difference/reason why bra-ket is especially handy for denoting quantum algorithms. There's already an accepted answer and an answer that explains 'ket', '...
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  • 581
12 votes
Accepted

What is the meaning of the state $|1\rangle-|1\rangle$?

The second approach is to take the state after the CNOT $|11\rangle-|10\rangle$ and just as I did with the CNOT I would apply $\operatorname{H}$ gate only on the first qubit which is $|1\rangle-|1\...
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12 votes
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What do double wires mean in quantum circuits, and how do they relate to if statements?

The double lines are one common convention for classical bits in quantum circuit diagrams. In this case, they represent the bits arising from the measurements of the qubits ...
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12 votes
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Elementary question: is $\langle a|b \rangle$ the same as $\langle a||b\rangle$?

Your understanding is correct. $\langle a | b \rangle$ is shorthand for $\langle a||b\rangle$. Here is a good resource for linear algebra and Dirac notation for quantum computing.
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  • 2,576
11 votes
Accepted

How to compute the average value $\langle X_1 Z_2\rangle$ for a two-qubit system?

I suggest two different ways of trying to solve this, which will give you experience of different bits of the formulation of Quantum Information Theory. I'll give examples that are closely related to ...
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10 votes
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What does "bipartite" mean?

What does a bipartite system mean? I'll summarize the main definitions below (adapted from Quantiki). Bipartite system and states: If Alice's subsystem is described by the Hilbert space $\mathcal{H}...
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9 votes
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What does it mean to "measure an operator"?

Any operator Hermitian $A$ can be described using its eigenvalue decomposition $$ A=\sum_\lambda\lambda P_\lambda, $$ where $\{\lambda\}$ are the distinct eigenvalues and $P_{\lambda}$ are the ...
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9 votes

Is there any difference between "value of a qubit" and its "state"?

I'm not aware of any widespread technical distinction between the "value" and "state" of qubits. I'd expect any paper or textbook or presentation using such a distinction to define ...
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  • 21.8k
8 votes

What is the meaning of the state $|1\rangle-|1\rangle$?

Sanchayan Dutta's answer correctly points out that $|1\rangle-|1\rangle$ doesn't actually arise in your example problem, but it doesn't answer the question in the title: what is $|1\rangle-|1\rangle$? ...
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  • 639
8 votes

Quantum states are unit vectors... with respect to which norm?

Some terminology seems a little bit jumbled here. Quantum states are represented (within a finite dimensional Hilbert space) by complex vectors of length 1, where length is measured by the Euclidean ...
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8 votes
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What does $\lvert \psi \rangle$ mean?

The notation $\lvert \alpha \rangle$, $\lvert \psi \rangle$, etc. just indicates that the thing in question is a vector. Furthermore, it is extremely common that the following things are also intended ...
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8 votes
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What is the relation between POVMs and observables (as Hermitian operators)?

One way of looking at the relationship between POVMs and observables arises from identifying their counterparts in the theory of probability of which quantum mechanics can be thought of as an ...
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7 votes
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What is the meaning of writing a state in its Bloch representation?

Yes. The Bloch sphere formalism is used for geometrically representing pure and mixed states of two-dimensional quantum systems a.k.a qubits. Any pure state $|\Psi\rangle$ of a qubit can be written in ...
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7 votes

What would be the meaning of an $i$ in a qubit state $i\alpha|0\rangle+\beta|1\rangle$?

You have applied a $$ U = \begin{pmatrix} i &0\\ 0&1 \end{pmatrix} $$ gate. You have not affected the probabilities of measuring $0$ or $1$ in the computational basis but you have affected ...
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  • 3,503
7 votes
Accepted

What is the name for quantum gates that can be reversed?

symmetric? Again, can be used in other contexts, but it is the right word for this context because what you're talking about is something that's invariant under swap. Perhaps the first time you use ...
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7 votes
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What does the notation $\sigma_j^z$ mean for Pauli matrices?

What $\sigma^z_i$ means is that you've got a Pauli-$Z$ applied to qubit $i$, and nothing else on the other qubits (i.e. the identity). So, you could expand it as $$ I^{\otimes(i-1)}\otimes\sigma^z\...
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6 votes

What do we mean by the notation $\lvert \mathbf{x}, 0\rangle$?

Yes, $|\mathbf{x},0\rangle$ is a shorthand for $|\mathbf x\rangle\otimes |0\rangle$. Note that $|\mathbf x\rangle$ itself, with $\mathbf x = x_1x_2\dots x_N$ a bit string, is just a shorthand for $$ |...
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6 votes

Quantum states are unit vectors... with respect to which norm?

More mathematically, because $\mathbb{R}^n$ with an $L^p$ norm is a Hilbert space only for $p=2$.
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6 votes
Accepted

Quantum states are unit vectors... with respect to which norm?

Born's rule states that $|\psi(x)|^2 = P(x)$ which is the probability of finding the quantum system in the state $|x\rangle$ after a measurement. We need the sum (or integral!) over all $x$ to be 1: \...
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  • 11.9k
6 votes
Accepted

How to read Dirac notation (without algebra)?

The dirac notation itself, the $|$ and $\rangle$ parts is simply a notation to remind you that you're dealing with quantum states. What you write inside this `ket' is completely arbitrary. In the ...
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6 votes
Accepted

What does quantum error correction code notation stand for?

An $[\![n,k,d]\!]$ code is a quantum error correction code which encodes $ k$ qubits in an $ n$-qubit state, in such a way that any operation which maps some encoded state to another encoded state ...
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6 votes
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Circuit construction and Dirac notation of the following operation

As you say, any register on which you do nothing, use the identity, $I$. This is also going to be the case on $x_1,x_2,\ldots$ and $y_1,y_2,\ldots$. When you want to control something, use the ...
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6 votes
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Notation question: How would one read $|0\rangle (|0\rangle+|0\rangle)$?

As a starting point, the state $\Psi = \frac{\vert 0 \rangle + \vert 1 \rangle}{\sqrt{2}}$ is a superposition of states $\vert 0 \rangle$ and $\vert 1 \rangle$, both with amplitude $\frac{1}{\sqrt{2}}$...
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6 votes
Accepted

What are $|+\rangle$ and $|-\rangle$?

The set $\{ \left|+\right>, \left|-\right> \}$ is known as the polar basis. It easy to see that they are the result of applying the Hadamard transform $H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 &...
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  • 136
6 votes
Accepted

What does the notation $U(B,\beta) = \prod_{j =1}^n e^{-i \beta \sigma_j^x} $ mean in the context of QAOA?

I think there are two ways that you could denote the same thing. The first is what is done here: $$ \prod_{j =1}^n e^{-i \beta \sigma_j^x} $$ The second is $$ \bigotimes_{j-1}^ne^{-i \beta \sigma^x}, $...
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6 votes
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How to translate the Hadamard gate matrix into Dirac notation?

First remember that each matrix element can be written as outer products in Dirac notation: $$|0\rangle\langle 0| = \begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix},|1\rangle\langle 1| = \begin{...
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  • 11.9k

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