22
votes
Accepted
How can classical bits be copied if qubits cannot be copied?
TL;DR: The ban on copying is not nearly as universal as it might seem. No-cloning theorem actually allows copying as long as it is limited to orthogonal states. Classical information is the type of ...
- 20.8k
16
votes
Accepted
Why doesn't the Gottesman-Knill theorem render quantum computing almost useless?
To my mind, this theorem is not very well stated in this form, if taken out of context. Where it says "phase gates", this may be misleading. It means specifically just $S=\sqrt{Z}$ and not what I ...
- 55.7k
9
votes
Why are diagonal Hamiltonians considered classical?
Classical Hamiltonians
By the spectral theorem, for every Hamiltonian there exists a basis in which it is diagonal. Thus, it is not correct to say that diagonal Hamiltonians are classical since this ...
- 20.8k
6
votes
Why are commuting density operators said to be "classical states"?
Usually, one defines classical states by first defining some "classical" orthonormal basis. Then a classical state is any state which is diagonal in this basis. Every classical state is then ...
- 5,355
5
votes
Accepted
What are some examples of uncomputability with quantum computers?
There was some initial characterization of quantum computers as going beyond Turing. But this is not the way we see things today.
Around 1982, Richard Feynman stated of his proposed quantum computer/...
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5
votes
Why are diagonal Hamiltonians considered classical?
While Adam's very detailed answer is probably emaculate, it's a bit long so for people that want a shorter answer, I'll give a much more compact alternative. This is not at all to challenge or try to ...
- 13.5k
5
votes
Accepted
Are inseparable states with positive partial transpose nonlocal?
This question was solved in 2014 by Vértesi and Brunner: they found a quantum state with positive partial transposition that violated a Bell inequality. The conjecture that all states with positive ...
- 2,372
5
votes
Why doesn't the Gottesman-Knill theorem render quantum computing almost useless?
Another way to think about this: To simulate what goes on in a quantum computer we have to do a lot of matrix math using $(2^N \times 2^N)$ matrices$^1$, and the action of (most) of the clifford gates ...
- 6,133
5
votes
Does a classical computer really require $2^n$ complex numbers to represent the state of $n$ qubit quantum computer?
If you treat the gate sequence as fixed then by the same logic you can treat the actual gates as fixed. No parameters is better than polynomial number of them :)
But the problem is not with this. ...
- 6,701
4
votes
Accepted
How does quantum contextuality relate to mutually commuting observables?
Is this just bad phrasing (or a typo) on Wikipedia's side, or am I missing something?
It does sound like bad phrasing. The idea here is that our set of observables is not necessarily mutually (...
- 17.2k
4
votes
Accepted
Understanding Hardy's proof of "nonlocality without inequalities"
Answering your precise question: mixing the four scenarios is not particular to Hardy's argument, it is done in all nonlocality proofs. The fundamental assumption is that the distribution of the ...
- 2,372
3
votes
Accepted
Quantum fourier transform with classical vibrations
The problem is that the classical analogue has to have physical elements and operations for each amplitude, instead of for each qubit.
Here is a quantum circuit for the 16-amplitude Fourier transform:
...
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3
votes
Accepted
Why are commuting density operators said to be "classical states"?
Nonclassicality in general
I should start by pointing out that there is no univocal notion of "(non)classicality". To name a few examples, in the context of quantum optics one might call a ...
glS♦
- 23.4k
3
votes
Why are diagonal Hamiltonians considered classical?
To pose a very simple answer to compete with all these complex (but also excellent) answers: the Ising model is a classical Hamiltonian because it is diagonal as it's written and therefore all of its ...
- 139
2
votes
Classical and quantum limits to classical copying?
You seem to be mixing two very different concepts here. Quantum cloning is talking about the absolute limits of what is theoretically possible in a perfect world. In this absolute theoretical limit, ...
- 55.7k
2
votes
If classical physics emerges in some limit of quantum mechanics, shouldn't there be intermediate classical-quantum computers?
You're presumably thinking of a spectrum with classical mechanics at one end and quantum mechanics at another, with some hazy "classical-quantum" in between. That's not a great way to think about it. ...
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2
votes
Accepted
Does the CHSH inequality fully characterise the local polytope?
Not quite. Consider the following no-signalling distribution $PR_1$ which I will write in the form
$$
\begin{pmatrix}
p(00|00) & p(01|00) & p(00|01) & p(01|01) \\
p(10|00) & p(11|00) ...
- 5,355
2
votes
Does a classical computer really require $2^n$ complex numbers to represent the state of $n$ qubit quantum computer?
I believe the issue you are missing is entanglement, which is an essential resource in quantum computing algorithms. Since we generate entanglement between these qubits, we can no longer think of ...
2
votes
Does a classical computer really require $2^n$ complex numbers to represent the state of $n$ qubit quantum computer?
The issue is that you are confusing the notions of Komogorov complexity and computational complexity. Kolmogorov complexity (roughly) means the smallest amount of data that you need to provide in ...
- 2,641
2
votes
Does a classical computer really require $2^n$ complex numbers to represent the state of $n$ qubit quantum computer?
As Danylo Y have answered, the key is you don't need to read out the entire quantum state at the end of the quantum algorithm to get your answer. There is another algorithm, called HHL algorithm, ...
- 13.7k
2
votes
Accepted
Why is the quantum discord of $\rho$ zero iff $\rho=\sum_j p_j \pi_j\otimes \rho_j$ for mutually orthogonal projections $\pi_j$?
$$I(A:B)=S(A)+S(B)-S(AB)$$
$$J(A_{\{\Pi_{i}\}}:B)=S(A_{\{\Pi_{i}\}})+S(B)-S(A_{\{\Pi_{i}\}}B)$$
$$I(A:B)-J(A_{\{\Pi_{i}\}}:B)=S(A)-S(AB)-S(A_{\{\Pi_{i}\}})+S(A_{\{\Pi_{i}\}}B)$$
Since $$\rho = \sum_j ...
- 1,115
2
votes
Circuit from finite group of gates and classical simulations
Let me give an answer which is not really intended as an answer (in that it doesn't address what I suspect the question is aimed at. For example, it does not cover the case of Clifford gates), and ...
- 55.7k
1
vote
How is the additivity of accessible information, $\frac{1}{n} I_{\rm acc}(\rho^{\otimes n})=I_{\rm acc}(\rho)$, proved?
I'm assuming you are referring to this paper:
Uncertainty, Monogamy, and Locking of Quantum Correlations.
In proposition 6, it's not clear to me if they are referring to the same product state that ...
- 1,775
1
vote
Does a classical computer really require $2^n$ complex numbers to represent the state of $n$ qubit quantum computer?
Consider an $n$ qubit system. For each qubit you need to store 2 complex numbers, making $2n$ numbers in total. But if these qubits are entangled, then you can no longer store those numbers separetely....
- 461
1
vote
Does the CHSH inequality fully characterise the local polytope?
Yes. As you've effectively said, all cases satisfying (2) are in a polytope and therefore convex. All the vertices of that polytope are deterministic strategies, and so every point inside the polytope ...
- 55.7k
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