# Questions tagged [probability]

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

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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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### Find probability of a single qubit's measurement results from a 5 qubit state

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 ...
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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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### 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 ...
I am trying to understand the integration on page 4 of this paper. Consider a Haar random circuit $C$ and a fixed basis $z$. Each output probability of a Haar random circuit (given by $|\langle z | C |... 1answer 82 views ### What's the difference between$p(i|m)$and$p(m|i)$in measurement? Suppose we perform a measurement described by measurement operators$M_m$. If the initial state is$|{\psi_i}\rangle$, then the probability of getting result$mis \begin{align} p(m|i)=\| M_m|\... 1answer 188 views ### 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? I have trouble understanding two equations in the Nielsen & Chuang textbook. Suppose we perform a measurement described by the operator M_m. If the initial state is |\psi_i\rangle, then the ... 2answers 71 views ### Given averages of powers of position and momentum in quantum mechanics what information can be secured about the wave-function? Question If I tell you all the averages of powers of position and momentum in quantum mechanics can you tell me the value of the wave-function? What can you tell me about the wavefunction? Is there ... 0answers 46 views ### Is this inequality related to time-energy uncertainty true or testable? Background It is known: In all physical systems in which energy is bounded below, there is no self-adjoint observable that tracks the time parameter t. However I don't think this forbids any ... 2answers 218 views ### How to interpret complex probability of superposition state? I have one qubit and I apply two gates to it: H and T, which yields the following superposition: \frac{1}{\sqrt{2}} |0\rangle + \frac{1+i}{2}|1\rangle $$Now I want to calculate probability of 0 ... 1answer 73 views ### Additivity of Renyi entropy The Renyi entropy of order \beta, for a discrete probability distribution p is given by H_{\beta}(p) = \frac{1}{1 - \beta} ~\log \left( \sum_{i \in S} p(i)^{\beta} \right), \end{... 0answers 29 views ### RAC for XOR functions I need the optimal encoding protocol for 3 \rightarrow 1 Classical RAC such that the receiver is able to retrieve any one of the initial bits, as well as the XOR combinations of those bits. ( If a, ... 1answer 61 views ### Quantum Circuit to inverse the probability distribution I'm using Qiskit and after running the circuit, as we all know, we get a count dictionary such as ... 0answers 52 views ### In QAOA why do we need m \log(m) repititions to get at least F_{p}(\beta , \gamma) - 1 with probability of 1 - 1/m? In the original QAOA paper from Farhi https://arxiv.org/pdf/1411.4028.pdf, it is stated in chapter 2 last paragraph (page 6) that: when measuring F_{p}(\beta , \gamma) we get an outcome of at least ... 1answer 119 views ### Question about Haar random quantum states Let |\psi\rangle be a n qubit Haar-random quantum state. I am trying to show that in the limit of large n, for each z_{i} \in \{0, 1\}^{n},$$ |\langle 0|\psi\rangle|^{2}, |\langle 1|\psi\... 1answer 77 views ### Nielsen & Chuang Exercise 6.13: Standard deviation of classical counting algorithm\newcommand{\expectation}[1]{\mathop{\mathbb{E}} \left[ #1 \right] } \newcommand{\Var}{\mathrm{Var}}$From Nielsen & Chuang 10th edition page 261: Consider a classical algorithm for the counting ... 1answer 59 views ### Hamiltonian simulation: how can I incorporate the constant before each term? I got another follow-up question about Hamiltonian simulation from the previous post: if I perform the controlled time-evolution of the Hamiltonian: $$H_{3} = \alpha\ X_1\otimes Y_2 + \beta \ Z_1\... 0answers 72 views ### What is the relation between density matrices and phase-space probability distributions? According to its tag description, a density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical ... 1answer 62 views ### Relating quantum max-relative entropy to classical maximum entropy The quantum max-relative entropy between two states is defined as$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$where$\rho\leq \sigma$should be read as$\sigma - \...
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 \...