Describe the state of the qubit at points A, B and C. How does this demonstrate that we need the “ket” (or the vector) representation of qubits, rather than just describing them in terms of probabilities (e.g. “this is a qubit with a 50% probability of being a 1”).
121251
I'm a computer science student and soon I will have a math exam. I'm really struggling with this preparation question. Also, includes the following:     How does this demonstrate that we need the “ket” (or  the  vector) representation of qubits, rather than just describing them in terms of probabilities (eg. “this is a qubit with 50% probability of being 1”)?   Could you help me please?
122916
I've been searching for a quantum algorithm to compute the the inner product between two $n$-qubit quantum states, namely $\langle\phi|\psi\rangle$, which is in general a complex number. One can get $|\langle\phi|\psi\rangle|^2$ through thte SWAP test for multiple qubits, but this is not really the inner product as the information of the real and imaginary parts are lost. I came across this , which claims that the task can be done by simply using CZ gates. I'm not sure if I am mistaken, but it seems to me that such a circuit works for computational basis states, but not for general quantum states. Could somebody help me confirm if this is actually the case? If the above algorithm doesn't work for general quantum states, what algorithm would you suggest?
96624
Can qubit be inside Bloch Sphere, i.e. its length is less than 1? If yes, how we represent that state since we have only parameter for angles and not the length (norm) of the vector?
90699
Let's say we have a wavefunction $\psi$ and a measurement operator $\hat A$.  I understand how eigenvalues and eigenvectors of $\hat A$ describe the possible outcomes of the measurement.  I also understand that the average measurement can be computed as $\langle \psi|\hat A|\psi\rangle$.  It's still not clear to me what the direct meaning of $\hat A |\psi\rangle$ is. It is a wavefunction; how does its corresponding quantum state relate (in physical terms) to the original state $\psi$ and the measurement $\hat A$?
76094
I am currently trying to establish a clear picture of pure/mixed/entangled/separable/superposed states. In the following I will always assume a basis of $|1\rangle$ and $|0\rangle$ for my quantum systems. This is what I have so far:   superposed: A superposition of two states which a system $A$ can occupy, so $\frac{1}{\sqrt{2}}(|0\rangle_A+|1\rangle_A)$ seperable: $|1\rangle_A|0\rangle_B$ A state is called separable, if its an element of the (tensor)product basis of system $A$ and $B$ (for all possible choices of bases) entangled: $\frac{1}{\sqrt{2}}(|0\rangle_A|1\rangle_B+|1\rangle_A|0\rangle_B)$ is not a state within the product basis (again for all possible bases). mixed state: Is a statistical mixture, so for instance $|1\rangle$ with probability $1/2$ and $|0\rangle$ with probability $1/2$ pure state: Not a mixed state, no statistical mixture   I hope that the above examples and classifications are correct. If not it would be great if you could correct me. Or add further cases, if this list is incomplete.   On wikipedia I read about      Another way to say this is that while the von Neumann entropy of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero.   which is perfectly fine. However I also read on wikipedia a criterion for :     Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.   So does this imply that if I look at the subsystems of an entangled state, that they are in a mixed state? Sounds strange... What would be the statistical mixture in that case?  Moreover I also wanted to ask, whether you had further illustrative examples for the different states I tried to describe above. Or any dangerous cases, where one might think a state of one kind to be the other?
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In this diagram     what would be the beginning state for Q?  Since the first NOR for S and R rely on previous results, there must be something for the first iteration?  NOTE: I'm in a first year digital logic class, so the question is for theoretical use (table making, various homework problems it applies to, etc.), not actual implementation.  Just for things like "if R is __ and S is __ what is Q?"  Simple things like that.
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In the  for quantum mechanics, the first postulate states that a quantum state is a vector in a separable Hilbert space. It means it is assumed the Hilbert space has a basis with at most infinite countable elements (cardinality). In other words, it states that the energy of an arbitrary system is discrete. Is it always true? If not, can you make an specific example in nature that its eigen-energies are uncountable?
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When constructing the matrices for the two CNOT based on the target and control qubit, I can use reasoning:  "If $q_0$==$|0\rangle$, everything simply passes through", resulting in an Identity matrix style $\begin{bmatrix}1&amp;0\\0&amp;1\end{bmatrix}$ in the top left. "If $q_0==|1\rangle$, we need to let $q_0$ pass and swap $q_1$, resulting in a Pauli X $\begin{bmatrix}0&amp;1\\1&amp;0\end{bmatrix}$ in the bottom right.  $CNOT \equiv \begin{bmatrix}1&amp;0&amp;0&amp;0\\0&amp;1&amp;0&amp;0\\0&amp;0&amp;0&amp;1\\0&amp;0&amp;1&amp;0\end{bmatrix}$  "If $q_1$==$|0\rangle$, everything simply passes through", results in leaving $|00\rangle$ and $|10\rangle$ unaffected. "If $q_1==|1\rangle$, we need to let $q_1$ pass and swap $q_0$, mapping $|01\rangle$ to $|11\rangle$ and  $|11\rangle$ to $|01\rangle$  $CNOT \equiv \begin{bmatrix}1&amp;0&amp;0&amp;0\\0&amp;0&amp;0&amp;1\\0&amp;0&amp;1&amp;0\\0&amp;1&amp;0&amp;0\end{bmatrix}$  This all seems to check out, but here comes my question: I would like to know if there is a more mathematical way to express this, just as there would be when combining for instance two hadamard gates: $H \otimes H \equiv \frac{1}{\sqrt{2}}\begin{bmatrix}1&amp;1\\1&amp;-1\\ \end{bmatrix} \otimes \frac{1}{\sqrt{2}}\begin{bmatrix}1&amp;1\\1&amp;-1\\ \end{bmatrix} = \frac{1}{2}\begin{bmatrix}1&amp;1&amp;1&amp;1\\1&amp;-1&amp;1&amp;-1\\1&amp;1&amp;-1&amp;-1\\1&amp;-1&amp;-1&amp;1 \end{bmatrix}$  And a bonus question:  How I can, using notation like "CNOT", show which qubit is the control bit and which qubit the target bit?
71269
I understand how normal logic gates operate and are implemented using digital circuits and electronics but I wad wondering how qubits are physically implemented.   In other words is a physical qubit a mechanical structure or an electronic circuit or optical circuit or something else.  There's often alot of information online about qubits and how they are different from normal bits but I have never seen any information about how they are possibly physically implemented.
12801
In the book &quot;Quantum Mechanics and Path Integrals&quot; Feynman &amp; Hibbs state that  the probability $P(b,a)$ to go from point $x_a$ at time $t_a$ to the point $x_b$ at the time $t_b$ is $P(b,a) = \|K(b,a)\|^2$ of an amplitude $K(b,a)$ to go from $a$ to $b$. This amplitude is the sum of contributions $\phi[x(t)]$ from each path. $$ K(b,a) = \sum_{\text{paths from $a$ to $b$}} \phi[x(t)]$$ The contributions of a path has a phase proportional to the action $S$: $$ \phi[x(t)] = \text{const}\ e^{(i/\hbar)S[x(t)]}$$  Why must the contribution of a path be $\sim e^{(i/\hbar)S[x(t)]}$? Can this be somehow derived or explained? Why can't the contribution of a path be something else e.g. $\sim \frac{S}{\hbar}$, $\sim \cos(S/\hbar)$, $\log(S/\hbar)$ or $e^{- (S[x(t)]/\hbar)^2}$ ? Edit: I have to admit that in the first version of this question, I didn't exclude the possibility to derive the contribution of a path directly from Schrödinger's equation. So answers along this line are valid although not so interesting. I think when Feynman developed his formalism his goal was to find a way to quantize systems, which cannot be treated by Schrödinger's equation, because they cannot be described in terms of a Hamiltonian (e.g. the Wheeler-Feynman absorber theory). So I think a good answer would explain Feynman's Ansatz without referring to Schrödinger's equation, because I think Schrödinger's equation can only handle a specific subset of all the systems that can be treated by Feynman's more general principle.
54680
I have a question about an argument used in Schwabl's "Advanced Quantum Mechanics" concerning the properties of the Klein-Gordan-Equation (see page 120):    Since the eigenenergies of free solutions are $E= \pm \sqrt{p^2c^2+m^2c^4}$the energy states aren't bounded from below. But I don't understand why that  then K-G-equation provide a scalar theory that does not contain spin and then could only describe particles with zero spin.   Intuitively, I guess because that spin can't regard by a scalar equation but I find this "argument" too squishy and would like to know a more plausible argument.
106830
I am working on the following exercise:      Let $\mathcal{C} = (\mathcal{X} , P, \mathcal{Y})$ be a channel with $\mathcal{X} = \mathcal{Y} = \{0,1\}$ and transition matrix       $$ P = \begin{bmatrix}  7/8 &amp;1/8 \\ 1/8 &amp;7/8  \end{bmatrix}.$$      The input probabilities are $P[X = 0] = 0.3, P[X = 1] = 0.7$. Each input bit is encoded by   transmitting it through the channel $2n + 1$ times, and the decoding function is the majority   bit (the bit $0$ or $1$ that appears more times in the sequence of $2n + 1$ output bits).   Let $f(n)$ be the probability that the decoded bit is different from the encoded bit.      Express $f(n)$ as a sum of the form $f(n) = \sum_{k=\ldots}^\ldots (\ldots)$.   My attempt: From $P$ we read that the probabilites for the channel makes a wrong output, i.e.: $p(0 \mid 1)$ and $p(1 \mid 0)$, are both $1/8$. So the error probability is given by the probability that at least  $n+1$ many mistakes occur. We can calculate this via  $$\sum_{k=n+1}^{2n+1} 0.7 \cdot p(0 \mid 1)^k + 0.3 \cdot p(1 \mid 0)^k = \sum_{k=n+1}^{2n+1} (1/8)^k$$  Is this correct?
2405
I have always taken these kinds of things for granted. Well of course $a(b+c)=ab+ac$! But why? The thought randomly popped in my head, and I realized that I could not prove it.  Perhaps we should take some time to prove these "obvious" things. I feel like if I cannot prove it, then I do not fully understand it. I would really like a proof of this most basic law, because I have been pondering over it for quite a while. Thanks!  Edit: This proof is for real numbers.
52203
Is it true that the states of a quantum system are represented by vectors in a Hilbert space?   When does the term "ray" come into play in this context?
6953
Could someone please provide an explanation for the origin of ? (Ideally someone who I have been speaking with on the h-bar)  Any advanced maths beyond basic calculus will most probably leave me at a loss, though I do not mind a challenge! Please assume little prior knowledge, as over the past few days I have discovered that much of my understanding surrounding the process as virtual particle pairs is completely wrong.
36397
Quantum algorithms frequently use bra-ket notation in their description. What do all of these brackets and vertical lines mean? For example: $|ψ⟩=α|0⟩+β|1⟩$  While this is arguably a question about mathematics, this type of notation appears to be used frequently when dealing with quantum computation specifically. I'm not sure I have ever seen it used in any other contexts.     Edit  By the last part, I mean that it is possible to denote vectors and inner products using standard notation for linear algebra, and some other fields that use these objects and operators do so without the use of bra-ket notation.   This leads me to conclude that there is some difference/reason why bra-ket is especially handy for denoting quantum algorithms. It is not an assertion of fact, I meant it as an observation. "I'm not sure I have seen it used elsewhere" is not the same statement as "It is not used in any other contexts".
109412
I've read , I've read , I've read , but I still just don't get it. How does a Quantum Computer work? It sounds very intriguing, but I just can't wrap my mind around it. I would greatly appreciate it if someone could explain it to me as they would explain the basics of it to their 7 year old nephew (who apparently is probably about equal to me in their ability to understand this concept!)   Typically analogies work best when explaining concepts to the unenlightened, but this is quantum theory, so I'm not sure any analogy could exist that can explain such a bizarre concept ;)