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## Hot answers tagged probability

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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4 votes
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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: $|\... • 24.8k 4 votes Accepted ### Existence of a two-outcome measurement$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$is not necessarily positive, ... • 6,718 3 votes ### Spoofing XQUATH with the Feynman method The paper does not specify the exact algorithm or class of distributions$\mathcal{D}$for which such algorithm fails to refute XQUATH, and some classes of distributions$\mathcal{D}$do not satisfy ... • 326 3 votes Accepted ### Probability of measuring one qubit from the state of two qubits If we have the state$|\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 ... • 13.8k 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 ... • 4,977 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 ... • 5,743 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$and adding one "free" variable every time you have a ... • 1,416 3 votes Accepted ### Why do probablity distribution with orthogonal suppor have maximal Kolmogorov distance? Note that if$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 ... • 5,449 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\... • 4,254 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' ... • 5,743 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 ... • 632 3 votes Accepted ### Question regarding the output probability of a quantum circuit Let$\mathcal{H}_A$denote the Hilbert space of qubits in partition$A$and similarly for$\mathcal{H}_\bar{A}$. Define the operator$P:=Q|0^n\rangle\langle 0^n|Q^\dagger$and write its Schmidt ... • 22.3k 3 votes Accepted ### What is the quantum analogue of$P_{XY} = P_{Y|X}P_X$The classical conditional distribution of$Y$given$X\$ is defined as the joint distribution divided by the marginal distribution. Note that we can reconstruct the joint distribution if we know all ...
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