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Consider a tripartite state \boxed{\rho_{ABC}} consisting of d-dimensional qudits. Let A, B, C contain NA, NB, and NC qubits, respectively. Suppose there exist a channel R acting on the Hilbert space and outputing to the Hilbert space of AB such that <Math> || R[\rho_{BC}] - \rho_{ABC} ||_1 = \epsilon<\Math>. One can s... | d^NC | Math/Physics | Easy | Advanced Reasoning | Quantum Information | Yifan Zhang | null |
Given a TFIM Hamiltonian \boxed{H_L = \sum_{i=1}^{L-1} X_i X_{i+1} + \sum_{i=1}^L Z_i} of size L, rewrite <Math>H_{L+1}</Math> as a function of <Math>H_L</Math> | H_{L+1} = H_L \otimes I + X_L X_{L+1} + Z_{L+1} | Math/Physics | Easy | Advanced Reasoning | Condensed Matter Physics | Roger | null |
Given a TFIM Hamiltonian <Math>H_L = \sum_{i=1}^{L-1} X_i X_{i+1} + \sum_{i=1}^L Z_i</Math> of size L, rewrite <Math>H_{2L+2}</Math> as a function of <Math>H_L</Math> such that the expression splits the total system into a bipartite system connected by 2 sites, one is <Math>H_L^{sys}</Math> represents the system, the o... | H_{L+1} = H_L \otimes I_{L+2} + I_{L+2}\otimes H_L + X_L X_{L+1} + X_{L+1} X_{L+2} + X_{L+2}X_{L+3} + Z_{L+1} + Z{L+2} | Math/Physics | Medium | Advanced Reasoning | Condensed Matter Physics | Roger | null |
We can expand the power of the adjoint of the sum of operators <Math>sum_{i=1}^n A_i</Math> with an operator B into a power form, in short <Math>ad^k_{sum_{i=1}^n A_i} B = (?)^k</Math> | (sum_i ad_{A_i})^k (B) | Math/Physics | Medium | Advanced Reasoning | Quantum Information | Roger | null |
Given a fermionic model <Math>H_L = \sum_{i=1}^{L-1} [a^{\dagger}_i a_{i+1} + a_i a^{\dagger}_{i+1} +n_i n{i+1}] </Math>, write it into a qubit format with Jordan Wigner transformation | H_L=\sum_{i=1}^{L-1}[\tfrac12\bigl(\sigma_i^{x}\sigma_{\,i+1}^{x}+\sigma_i^{y}\sigma_{\,i+1}^{y})+\tfrac14\bigl(\sigma_i^{z}\sigma_{\,i+1}^{z}+\sigma_i^{z}+\sigma_{\,i+1}^{z}+1)] | Math/Physics | Medium | Advanced Reasoning | Condensed Matter Physics | Yuxuan Zhang | null |
Consider a bipartite state <Math>\rho_{AB}</Math> consisting of d-dimensional qudits. Let A, B contain NA, NB qubits, respectively. Find a bound on <Math>||\rho_{AB} - \rho_A \otimes I_B ||_1<\Math> that is a second moment quantity of <Math>\rho_{AB} - \rho_A \otimes I_B<\Math>. I_B is the maximally mixed state on B. E... | d^{(NA+NB)/2} ||\rho_{AB} - \rho_A \otimes I_B ||_2 | Math/Physics | Medium | Advanced Reasoning | Quantum Information | Yifan Zhang | null |
Consider a bipartite state <Math>\rho_{AB}</Math> consisting of d-dimensional qudits. Let A, B contain NA, NB qubits, respectively. Find a bound on <Math>||\rho_{AB} - \rho_A \otimes \rho_B ||_1<\Math> that is a fourth moment quantity of <Math> \rho_{AB} <\Math>. I am expecting an answer that is a function of. <Math>\r... | d^{3(NA+NB)/4} 4 ||\rho_{AB} - \rho_A \otimes \rho_B ||_2^2 | Math/Physics | Hard | Advanced Reasoning | Quantum Information | Yifan Zhang | null |
Using combinatorics, write down the size of the total Fock space for m different modes. Consider the following cases: Spinless Fermions. Output the LaTeX for final answers only without specifying the variable. | 2^m | Math/Physics | Easy | Basic Knowledge | Condensed Matter Physics | Yuxuan Zhang | null |
Using combinatorics, write down the size of each Fock sub-spaces for m different modes and n different particles. Consider the following cases: Spinless Fermions. Output the LaTeX for final answers only without specifying the variable. | \binom{m}{\,n} | Math/Physics | Easy | Basic Knowledge | Condensed Matter Physics | Yuxuan Zhang | null |
Consider the following Hamiltonian: <Math>
\hat{H} = -\frac{1}{2} \sum_{n=1}^N \left\{
\left[J_0 + (-1)^{n-1} J_1 \right]
\left[
\frac{1+\gamma}{2} \, \sigma_n^x \sigma_{n+1}^x +
\frac{1-\gamma}{2} \, \sigma_n^y \sigma_{n+1}^y
\right]
+ h \, \sigma_n^z
\right\}
- \frac{\Omega}{2} \sum_{n=1}^N \left(
\sigma_{n-1}^x \sig... | Yes | Math/Physics | Hard | Advanced Reasoning | Quantum Information | Yuxuan Zhang | null |
You are given three gates. <Math> Z_1(\theta_1) = exp(i \theta_1 |1><1|_1) <\Math> is a phase gate on qubit 1. <Math> Z_2(\theta_2) = exp(i \theta_2 |1><1|_2) <\Math> is a phase gate on qubit 2. <Math> R = exp(i \theta_r |11><11|) <\Math> is a phase gate acting on qubit 1 and 2. Let <Math> U = Z_1 R Z_2 <\Math>. How to... | \theta_1=3\pi/2, \theta_2=3\pi/2, \theta_r=\pi | Math/Physics | Easy | Advanced Reasoning | Quantum Information | Yifan Zhang | null |
Rényi-$\alpha$ negativity is defined as:
<Math>
\mathcal{N}_\alpha(\rho) = -\log \left( \operatorname{Tr} \left[ \left( \rho^{T_A} \right)^\alpha \right] / \operatorname{Tr} \left[ \rho ^\alpha \right]\right)
<\Math>
Given two-qubit density matrix: <Math> \rho = \begin{bmatrix} 0.5782 & 0.2853 - 0.1241i & 0.0008 - 0.00... | 0.586 | Math/Physics | Medium | Advanced Reasoning | Quantum Information | Yuxuan Zhang | null |
the power of an adjoint of tensor product <Math>ad^k_{A\otimes B}</Math> can be expand into what form of tensor product of adjoint operator <Math> (?)^k </Math>, use <Math>\sigma</Math> to annotate the signs of commutator, e.g <Math>ad_{A,\sigma}</Math> where <Math>ad_{A,+}</Math> represents anti-commutator, <Math>ad_{... | \frac{1}{2^k} (\sum_{\sigma_i\in{+,-} ad_{A,\sigma_i} \otimes ad_{B,-\sigma_i})^k | Math/Physics | Hard | Advanced Reasoning | Quantum Information | Xiu-Zhe Luo | null |
One of the key conceptual hurdles that was overcome in the development of neutral atom-based optical clocks is the need to trap and confine the atoms (using an optical lattice). However, the accuracy and precision of such a
clock degrades significantly in the presence of differential AC Stark shift between the two leve... | \Delta E_g=I \frac{\left|d_{e g}\right|^2}{\hbar \epsilon_0 c} \frac{\omega_{e g}}{\omega^2-\omega_{e g}^2}<\Math> \n <Math>\Delta E_e=-I \frac{\left|d_{e g}\right|^2}{\hbar \epsilon_0 c} \frac{\omega_{e g}}{\omega^2-\omega_{e g}^2} | Math/Physics | Hard | Advanced Reasoning | Quantum Information | Xiang Zou | null |
What is the change in the transition frequency ωeg due to the presence of the trapping light? | \Delta \omega_{e g}=\frac{\Delta E_e-\Delta E_g}{\hbar}=-\frac{\left|d_{e g}\right|^2}{\hbar^2 \epsilon_0 c} \frac{2 I \omega_{e g}}{\omega^2-\omega_{e g}^2} | Math/Physics | Hard | Advanced Reasoning | Quantum Information | Xiang Zou | null |
Now consider a three-level atom, where the excited state |e⟩is additionally coupled to
a higher-energy level |f⟩with transition frequency ω_fe and dipole moment d_fe. Find the AC Stark shift for the state |e⟩taking into account both
states |f⟩and |g⟩. | \Delta E_e=\frac{I}{\hbar \epsilon_0 c}\left(\frac{\omega_{f e}\left|d_{f e}\right|^2}{\omega^2-\omega_{f e}^2}-\frac{\omega_{e g}\left|d_{e g}\right|^2}{\omega^2-\omega_{e g}^2}\right) | Math/Physics | Hard | Advanced Reasoning | Quantum Information | Xiang Zou | null |
At what laser frequency ω_m does the transition frequency ω_eg become
independent of the trap laser intensity I? | \omega_m=\sqrt{\frac{\left(2 \omega_{f e}\left|d_{e g}\right|^2-\omega_{e g}\left|d_{f e}\right|^2\right) \omega_{e g} \omega_{f e}}{2 \omega_{e g}\left|d_{e g}\right|^2-\omega_{f e}\left|d_{f e}\right|^2}} | Math/Physics | Hard | Advanced Reasoning | Quantum Information | Xiang Zou | null |
A material perfect clock is in uniform circular motion with angular speed ω and radius R in flat spacetime. Is there any physical restriction on \omega and R? | \omega*R <1 | Math/Physics | Medium | Advanced Reasoning | Astrophysics/Cosmology | Xiang Zou | null |
After one full circle, how much will the reading of the perfect clock advance? | \frac{2\pi}{\omega}\sqrt{1-\omega^2R^2} | Math/Physics | Hard | Advanced Reasoning | Astrophysics/Cosmology | Xiang Zou | null |
Some student said that the following invariant interval for some physical spacetime is problematic, ds^2 =du^2 +2dudv+dv^2 +dy^2 +dz^2. He is correct, why can you infer the determinant of the metric tensor represented by this invariant interval? | 0 | Math/Physics | Medium | Advanced Reasoning | Astrophysics/Cosmology | Xiang Zou | null |
How many different quantum states are there in a k-design? Assume the Hilbert space size is $d$, output your answer in an equation using combinatorics only | \binom{d + k - 1}{k} | Math/Physics | Easy | Basic Knowledge | Quantum Information | Yuxuan Zhang | null |
For a spherical 3-design with N states, sample each quantum state according to the probability distribution of each state. What is the probability that a sampled state has a probability weight higher than 1/N? Output a number or expression only. | 1/6 | Math/Physics | Hard | Advanced Reasoning | Quantum Information | Yuxuan Zhang | null |
T_n = \sum_i R_i 1(X_0>0), where R_i is the rank of X_i in the sorted order |X_1|\leq |X_2|.... Decompose T_n into a linear combination of U-statistic of order 1 and a U-statistic of order 2. (Hint: Express the result as aU_2+bU_2, where the coefficients a and b depend on n. U_2 = \frac{\sum_{i<j} 1(X_1+X_j>0))}{\binom... | T_n = \binom{n}{2}U_2 + nU_1 | Math/Physics | Medium | Advanced Reasoning | Probability/Statistics | Shunan Zheng | null |
During the supernovae explosion, it will eject a large amount of material into the surrounding environment (chracterized by number density n_H), with mass given by M_{ej}, and release amount of energy E_{SN}. Assume that during its evolution phase, the ionizing radiation has lower the ambient density of supernovae such... | t_{free expansion} ~ M_{\star}^{5/6} n_H^{-1/3}E_{SN}^{-1/2} | Math/Physics | Medium | Advanced Reasoning | Astrophysics/Cosmology | Saiyang Zhang | Assume the expansion halts when M_ej is equal to the mass swept away by the ejected mass: M_ej ~ M_swept = 4/3 pi r_{free expansion}^3 mu n_H m_p, where mu is the molecular weight of the universe. We derive the sweeping radius as r_{free expansion} ~(3M_{ej}/4 pi m_p n_H)^{1/3}. By the conservation of energy, the time ... |
If we are living in a universe with primordial black holes as the only dark matter candidate (i.e. \Omega_{DM} = \Omega_{PBH}), and all of them are centered around some mass M_{PBH}, with number density given by n_{PBH}. What is the dominant change (Poisson fluctuation) induced by PBHs to the linear power spectrum P(k)... | P(k) \propto 1/n_{PBH}.
| Math/Physics | Hard | Advanced Reasoning | Astrophysics/Cosmology | Saiyang Zhang | Assume PBHs are located at spatial coordinate \mathbf{x}, the overdense field at spatial point is given by: \delta_{iso}(\mathbf{x})\equiv \frac{\delta \rho_{PBH}}{\Bar{\rho}_{ DM}}=\frac{1}{n_{PBH}} \sum_i \delta_D\left(\mathbf{x}-\mathbf{x}_i\right)-1. Taking the two point correlation of the density fluctuation, we h... |
Work out the temperature profile (T(r)) of the black hole of mass M_{BH} accretion disk with radial denpendence (r), assume that black hole accretes at a rate \dot{M}, and assume that half of the graviational potential energy is converted into thermal heating energy, radiates away like black body. Express the answer in... | T(t) = (G M_{BH}\dot{M} / 8 \pi \sigma_{SB} r^3)^{1/4} | Math/Physics | Medium | Advanced Reasoning | Astrophysics/Cosmology | Saiyang Zhang | For a mass element M move from r+\delta r to r, the change in potential energy can be written as \delta E = GM_{BH}M \delta r / r^2. If half of the energy is converted to luminosity, we have \delta L = 2G M_{BH}\dot{M} / 2r^2 \delta r. If each segment of the accretion disk emits like a black body, the luminosity can be... |
In a standard LCDM universe, if a Black Hole seed is born in an ideal environment and always growing at an Eddington rate, with radiation efficiency \epsilon of 0.1, what will be the mass of the BH at z =10 if it is born with a mass of 1000 Solar mass (express in M_\odot ) at z=20? (Round to one significant figure) | 6 \times 10^5 M_\odot | Math/Physics | Medium | Advanced Reasoning | Astrophysics/Cosmology | Saiyang Zhang | The characteristic Salpter time(e-fold growing time) scale for BH growth is 5\times 10^7 yr, where M(t) = M_{ini} exp(t/t_{Salpter}). Therefore, the time difference between redshift 20 and redshift 10 is about 3\times 10^8 yrs. Therefore, the BH mass estimated at z = 10 is about 4 \times 10^5 solar mass. |
Consider a dark matter halo at redshift $z$, capable of cooling through molecular hydrogen (H$\_2\$). The halo is characterized by its virial temperature $T_{\rm vir}$, molecular hydrogen cooling function $\Lambda(T, n_{\rm H})$, and hydrogen number density $n_{\rm H}$. To determine the minimum mass of such a halo capa... | M_{t_H-\rm cool} \approx 1.54 \times 10^{5} M_\odot \left(\frac{1+z}{31}\right)^{-2.074}. | Math/Physics | Hard | Advanced Reasoning | Astrophysics/Cosmology | Saiyang Zhang | To find the minimum mass, we first equate the cooling time $\tau_{\rm cool}$ to the local Hubble time $t_H(z)$: \frac{3 k_B T_{\rm vir}(M,z)}{2 \Lambda(T_{\rm vir}, n_{\rm H}) f_{\rm H_2}} = t_H(z). Plugging in the expressions for $T_{\rm vir}$, $\Lambda$, and $n_{\rm H}$, we obtain: \tau_{\rm cool} \approx 3.46 \times... |
The lifetime of a lightbulb is an exp(\lambda) random variable. Replacement happens every fixed T time unit, starting at time 0. What is the expected time when the lightbulb is discovered to have failed. Let D be the time when the machine is discovered to be down. |
E[D] = \frac{T}{1-e^{-\lambda T}}
| Math/Physics | Medium | Advanced Reasoning | Probability/Statistics | Shunan Zheng | null |
The lifetime of a lightbulb follows exp(1/\theta), where the expected lifetime \theta is unknown. In one experiment, test N lightbulbs and record their exact lifetime y_1,...,y_N. In another experiment, test M lightbulbs. At some time t, record if the lightbulbs are still burning with indicators Z_1,...,Z_M. Z_i = 1 i... | Q(\theta | \theta_t) = -(N + M) \ln \theta - \frac{1}{\theta} ( N \bar{y} + Z(t + \theta_t) + (M - Z) ( \theta_t - \frac{t \exp(-t/\theta_t)}{1 - \exp(-t/\theta_t)} ) )
| Math/Physics | Hard | Advanced Reasoning | Probability/Statistics | Shunan Zheng | null |
Let ${N(t), t \ geq 0}$ be an RP with integer valued inter-renewal times with pmf
$P(X_n = 0) = 1-\alpha$, $P(X_n = 1) = \alpha$, for $n \geq 1$, where $0<\alpha<1$. Compute $P(N(t) = k)$ for k = 0,1,2,....
| P(N(t) = k) = \binom{t}{k} (1-\alpha)^{k-t}\alpha^{t+1}
| Math/Physics | Medium | Advanced Reasoning | Probability/Statistics | Shunan Zheng | null |
Assume the distribution of the population G_i is: $G_i(m = 1) \sim N(\mu^{(i)}, \sigma_i^2), \quad (i = 1, 2)$. According to the distance-based classification rule (assume $\mu^{(1)} > \mu^{(2)}$, $\sigma_1 < \sigma_2$):
\[
\begin{cases}
x \in G_1, & \text{if } \mu^* < x < \mu^{(1)} \\
x \in G_2, & \text{if } x \leq ... | P(2\mid 1) = \Phi(\frac{\mu^{(2)}-mu^{(1)}}{\sigma_1+\sigma_2}) +\Phi(\frac{\mu^{(2)}-mu^{(1)}}{\sigma_2-\sigma_1}) | Math/Physics | Medium | Advanced Reasoning | Probability/Statistics | Shunan Zheng | null |
how many elements are in the group described by the following relation: \begin{aligned} & {\left[(-1)^F\right]^2=1, \quad \mathrm{C}^2=1, \quad \mathrm{R}^2=1, \quad(-1)^F \mathrm{C}=\mathrm{C}(-1)^F} \\ & \mathrm{R}(-1)^F=(-1)^F \mathrm{R}, \quad \mathrm{RC}=\mathrm{C}(-1)^F \mathrm{R} \\ & \mathrm{T}^2=1, \quad \math... | 16 | Math/Physics | Medium | Advanced Reasoning | Condensed Matter Physics | Wucheng Zhang | null |
What is the group described by the following relation: \begin{aligned} & {\left[(-1)^F\right]^2=1, \quad \mathrm{C}^2=1, \quad \mathrm{R}^2=1, \quad(-1)^F \mathrm{C}=\mathrm{C}(-1)^F} \\ & \mathrm{R}(-1)^F=(-1)^F \mathrm{R}, \quad \mathrm{RC}=\mathrm{C}(-1)^F \mathrm{R} \\ & \mathrm{T}^2=1, \quad \mathrm{~T}(-1)^F=(-1)... | D_4 \times \mathbb{Z}_2 | Math/Physics | Medium | Advanced Reasoning | Condensed Matter Physics | Wucheng Zhang | null |
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