# Questions tagged [probability]

For questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities.

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### How to prove that the mutual information is subadditive?

Let $\mathbf x=(x_1,...,x_n)$ and $\mathbf y=(y_1,...,y_n)$ be two vectors of random variables. To make things concrete, assume that Alice sends each component $x_j$ through a noisy channel to Bob, ...
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### Probability resulting from uncertainty when the measuring device exactly clicks? [migrated]

Background Let's say I have $2$ set of eigenkets of observables of a system $|x_i \rangle$ and $|p_j \rangle$ (which do not commute). Let's say I have a non-ideal detector in the sense the ...
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### Spoofing XQUATH with the Feynman method

Consider the XQUATH conjecture for random quantum circuits, as mentioned here. (XQUATH, or Linear Cross-Entropy Quantum Threshold Assumption). There is no polynomial-time classical algorithm that ...
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### How is the probability of success for Simon's algorithm determined?

In step 3 of Simon's algorithm, we are told to "Repeat until there are enough such $y$’s that we can classically solve for $s$." It then goes on: The above are from this course notes. I am ...
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### Marginal output probability of first bit for constant-depth circuits

Consider a constant depth $1\text{D}$ quantum circuit, which is applied to the input state $|0^{n}\rangle$, and whose output is measured in the standard basis. You can assume that the gates of the ...
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### Properties of frames in quasiprobability representation

Let $\mathbb{C}^{d}$ be a complex Euclidean space. Let $\mathsf{H}(\mathbb{C}^{d})$ be the set of all Hermitian operators, mapping vectors from $\mathbb{C}^{d}$ to $\mathbb{C}^{d}$. I had some ...
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### How to find the POVM that optimally distinguishes between two given states?

A quantum state preparation machine emits a state $\rho_0$ with probability $2/3$ and emits the state $\rho_1$ with probability $1/3$. We aim to make the best guess which one is it using a set of two ...
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### Measurement probability of a state from three hilbert spaces

I'm curious how to find the probability measurement of a state when one qubit is measured. For example this state: $|\gamma\rangle = \frac{1}{\sqrt{2}}(| 010 \rangle + | 101 \rangle )$ Assuming these ...
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### Quantum supremacy: shallow depth Haar random circuits and unitary designs

I had a confusion about shallow depth Haar random quantum circuits. In this paper, in Section B (related works), it is mentioned that Haar random quantum circuits form approximate $2$-designs only ...
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### 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?

I have been given and expression for a probability distribution $$P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$$ and I have been asked to show that the above expression can be ...
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### Why do probablity distribution with orthogonal suppor have maximal Kolmogorov distance?

Can anyone explain why the $l_1$ distance has the property that probability distributions $P,Q$ with orthogonal support (meaning that the product $p_iq_i$ vanishes for each value of $i$) are at a ...
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### Quantum Amplitude Estimation vs Quantum Phase Estimation

Quick question concerning the probability of success after a phase estimation algorithm vs an amplitude estimation algorithm. Given the calculation on the wikipedia page, the probability of measuring ...
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I got another follow-up question about Hamiltonian simulation from the previous post: if I perform the controlled time-evolution of the Hamiltonian: $$H_{3} = \alpha\ X_1\otimes Y_2 + \beta \ Z_1\... 0answers 76 views ### What is the relation between density matrices and phase-space probability distributions? According to its tag description, a density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical ... 1answer 65 views ### Relating quantum max-relative entropy to classical maximum entropy The quantum max-relative entropy between two states is defined as$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$where \rho\leq \sigma should be read as \sigma - \... 1answer 107 views ### Dirichlet distribution: posteriors and priors of distribution Let |\psi\rangle \in \mathbb{C}^{2n} be a random quantum state such that  |\langle z| \psi \rangle|^{2}  is distributed according to a \text{Dirichlet}(1, 1, \ldots, 1) distribution, for z \in \... 1answer 223 views ### What is the probability \Pr(\|U-I\|_{\rm op}<\varepsilon) of a Haar-random unitary being close to the identity? If one generates an n\times n Haar random unitary U, then clearly \Pr(U=I)=0. However, for every \epsilon>0, the probability$$\Pr(\|U-I\|_{\rm op}<\varepsilon) should be positive. How ...
I have a tensor product of a 5 qubit state $|h\rangle$. From this I want to calculate the probability of the 2nd qubit being in state $|1\rangle$. Can someone show me how I can do this? I know I can ...