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7 votes
Accepted

What does $M_m |\psi_i\rangle$ mean in the equation $p(m|i)=\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle$?

Measurement postulate The statement you are asking about is a postulate of quantum mechanics, so it cannot be mathematically derived from other facts in the theory. Instead, it is justified by its ...
Adam Zalcman's user avatar
  • 23.2k
7 votes
Accepted

Confusion about the output distribution of Haar random quantum states

The two facts are connected in that they both arise as a result of rotational invariance of the Haar measure. We will derive them in the case of large $n$ since this is when the Porter-Thomas ...
Adam Zalcman's user avatar
  • 23.2k
7 votes
Accepted

Can all mixed states be written as a convex combination $\rho=\sum_j p_j |\psi_j\rangle\langle \psi_j|$?

Given $\rho$ and a fixed ensemble $\{ |\psi_i \rangle \}$ it might not be possible to write $\rho$ as $\sum_i p_i |\psi_i \rangle \langle \psi_i |$. For example, let $| + \rangle = \frac{1}{\sqrt{2}} ...
biryani's user avatar
  • 1,016
6 votes
Accepted

Quantum Amplitude Estimation vs Quantum Phase Estimation

Ok let's break this down. Firstly the success probability for QPE and QAE are defined slightly differently. With QPE there are two error bounds $|\hat{x} - x| < \epsilon$ to consider $\epsilon < ...
Sam Palmer's user avatar
6 votes
Accepted

Computing expectation value of $|\langle z|C|0^n\rangle|^2$ over Haar random circuit

The issue that easily leads to confusion is the dual role played by output bitstring probability. It enters the computation of the average in two ways. On one hand, it determines how often one sees ...
Adam Zalcman's user avatar
  • 23.2k
5 votes
Accepted

Nielsen & Chuang Exercise 6.13: Standard deviation of classical counting algorithm

Since the classical algorithm samples "uniformly and independently $𝑘$ times from the search space", equation $(5)$ should be, $P(X_i=1, X_j=1)= P(X_i=1)P(X_j=1)=\frac{M^2}{N^2}$ instead. ...
Nan's user avatar
  • 321
5 votes
Accepted

Compute the large $n$ distribution of $|\langle z_i|\psi\rangle|^2$ over Haar random quantum states

In the following, I'll show the evaluation of the probability densities of the transition probabilities: $|\langle \psi | z\rangle^2$ and their pairwise independence. I didn't work out the full mutual ...
David Bar Moshe's user avatar
5 votes
Accepted

How to interpret complex probability of superposition state?

You forgot to take the absolute value. The Born rule for computing measurement outcome probability from the state vector amplitudes says that the probability is the square of the magnitude of the ...
Adam Zalcman's user avatar
  • 23.2k
5 votes
Accepted

Prove the additivity of the Renyi entropy: $H_{\beta}(p \times r) = H_{\beta}(p) + H_{\beta}(r)$

This only holds if the two distributions are independent. In this case $$ \begin{aligned} H_{\beta}(p \times q) &= \frac{1}{1-\beta} \log\left( \sum_{i,j}(p(i) q(j))^{\beta} \right) \\ &= \...
Rammus's user avatar
  • 5,988
5 votes

Negative Probability — Reality vs Description

"I understand that quantum physics supports the concept that the probability of a qubit collapsing into (say) 1, can be negative or positive [...] If that is wrong, please say so; I understand ...
user1271772 No more free time's user avatar
5 votes
Accepted

Generating random, but non-uniform state

Rejection sampling is a good fit and works without any changes, simply by plugging the desired distribution $p(\psi)$ into the standard algorithm. Let$^1$ $M:=\max_{\psi\in\mathbb{CP}^1} p(\psi)$. To ...
Adam Zalcman's user avatar
  • 23.2k
4 votes

How to interpret complex probability of superposition state?

Just to note that the complex number also specify a quantum phase of a qubit. In this particular case $$ \frac{1+i}{2} = \frac{1}{\sqrt{2}}\mathrm{e}^{i\frac{\pi}{4}}, $$ so the relative phase is $\...
Martin Vesely's user avatar
4 votes
Accepted

What's the difference between $p(i|m)$ and $p(m|i)$ in measurement?

These are classical conditional probabilities used extensively in Bayesian probability. Suppose Alice can prepare any of a number of pure states $|\psi_i\rangle$. She chooses $i$ randomly from ...
Adam Zalcman's user avatar
  • 23.2k
4 votes
Accepted

Properties of frames in quasiprobability representation

The authors are certainly thinking about finite frames. In this case, your statement is correct, since the number of elements in every spanning set is at least the vector space dimension. As glS ...
Markus Heinrich's user avatar
4 votes

How to express a probability distribution $P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$ in terms of a trace of a density matrix?

First, recall that $\mathrm{tr} A = \sum_i \langle i|A|i \rangle$. Each equation is then a sum where all terms are products of $P(z)$ and three other quantities. Further, the sum in the first equation ...
Adam Zalcman's user avatar
  • 23.2k
4 votes

Applying Hadamard gate to $\sqrt{3/4}|0\rangle + \sqrt{1/4}|1\rangle$

It seems like you are trying to apply a Hadamadard gate to the state $\frac{\sqrt{3}}{2}|0\rangle + \frac{1}{2}|1\rangle$. This would go as follows: $$ \begin{align} H\left[\frac{\sqrt{3}}{2}|0\rangle ...
epelaez's user avatar
  • 2,915
4 votes

How do I get correct measurement probabilities in ZX calculus?

OK, I made two mistakes. Both were corrected by a closer reading of the paper "Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions". First, as was pointed out by ...
jjgoings's user avatar
  • 221
4 votes
Accepted

Confusion on the probability of measuring first qubit of a separable mixed state

They're the same because $$\sum_z |\langle z|\tilde U|\tilde x \rangle|^2=1$$ for any unitary $\tilde U$ and input $|\tilde x\rangle$. This follows directly from the normalization of probabilities: $|\...
glS's user avatar
  • 26k
4 votes
Accepted

Existence of a two-outcome measurement $M$ such that the induced distributions differs between different density matrices

Yes, you can use $\rho - \sigma$ to construct a measurement where the measurement statistics differ for $\rho$ compared to $\sigma$. The issue is that $\rho - \sigma$ is not necessarily positive, ...
forky40's user avatar
  • 7,213
4 votes
Accepted

Bound on success Probability for Regev's factoring algorithm

This is just Markov's inequality that states that for a positive random variable $X$ and $a > 0$ then $$ P( X \geq a ) \leq \frac{E(X)}{a}. $$ see the wikipedia: https://en.wikipedia.org/wiki/...
Frederik Ravn Klausen's user avatar
4 votes
Accepted

40th Question IBM Sample test

Qubits are initially in the ∣0⟩ state. The Rz gate, which is a rotation around the z-axis, only changes the phase of the qubit states and not the probabilities of measuring 0 or 1 in the computational ...
Bram's user avatar
  • 652
3 votes
Accepted

Probability of measuring one qubit from the state of two qubits

If we have the state $|\psi \rangle = a_{00}|00\rangle +a_{01}|01\rangle +a_{10}|10\rangle +a_{11}|11\rangle $ then the probability of the second qubit being in the state $|1\rangle$ is the ...
KAJ226's user avatar
  • 13.9k
3 votes
Accepted

Quantum supremacy: shallow depth Haar random circuits and unitary designs

First of all, that does not imply anything for shorter (constant/logarithmic) depths. Moreover, the 2-design property does not imply that the outcome distribution is the same as for Haar-random ...
Markus Heinrich's user avatar
3 votes
Accepted

How to find the POVM that optimally distinguishes between two given states?

The optimal probability of guessing correctly is $$ \frac12 + \frac12 \Big\|\frac23 \rho_0 - \frac13 \rho_1 \Big\|_1 $$ where $\| X \|_1 = \mathrm{Tr}[\sqrt{X^* X}]$ is the Schatten 1-norm. This ...
Rammus's user avatar
  • 5,988
3 votes
Accepted

Estimating output amplitudes of quantum circuits as GapP functions

Given a description of $ U_x $ you can efficiently find a decription of $ \textbf{B} $ and $ \phi $ by iterating over the gates of $U_x$ and adding one "free" variable every time you have a ...
tsgeorgios's user avatar
  • 1,426
3 votes
Accepted

Why do probablity distribution with orthogonal suppor have maximal Kolmogorov distance?

Note that if $p_{i}q_{i} = 0\,\,\forall i$, then for all $i$ either $p_{i} = 0$, $q_{i} = 0$, or both are $0$. Divide $\{i\} = \{1,\ldots,N\}$ into those $i$ for which these three different things ...
JSdJ's user avatar
  • 5,559
3 votes

What is the probability $\Pr(\|U-I\|_{\rm op}<\varepsilon)$ of a Haar-random unitary being close to the identity?

$U-I$ is a normal matrix so $||U-I||_{op}$ is its eigenvalue with the largest magnitude. The eigenvalue equation for this matrix is $$(U-I)|\psi\rangle=\lambda|\psi\rangle,$$ so $$|\lambda|^2=(\cos\...
Quantum Mechanic's user avatar
3 votes
Accepted

Relating quantum max-relative entropy to classical maximum entropy

As far as I'm aware there isn't much of a meaningful connection. The corresponding entropy for $D_{\max}$ is the min-entropy (written $H_{\min}$ or $H_{\infty}$). It measures a sort of `worst case' ...
Rammus's user avatar
  • 5,988
3 votes

Spoofing XQUATH with the Feynman method

The paper does not specify the exact algorithm or class of distributions $\mathcal{D}$ for which such algorithm fails to refute XQUATH, and some classes of distributions $\mathcal{D}$ do not satisfy ...
fiktor's user avatar
  • 326
3 votes
Accepted

Why is sampling from probability distributions generated by specific quantum circuits classically intractable?

The problem that led to an important role of IQP circuits is Boson Sampling. Boson sampling algorithm has to sample from a distribution based on the permanent of some complex matrix. Computing complex ...
3yakuya's user avatar
  • 632

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