15 problems

Let $n$ be a positive integer, let $sigma$ be a density operator, and let $M^c$ denote the complement of $M$ in $\{1,2,\cdots,n\}$. For a subset $M$, write $\sigma^{\otimes M^c}$ f…

Let $S(\rho,k)=\frac{1}{2}\operatorname{Tr}[\rho^{1/2},k]^2$ be the Wigner–Yanase entropy of a state $\rho$ with respect to a conserved self-adjoint observable $k$. For an aggregat…

Let $n$ be a positive integer, let $[n]=\{1,2,\ldots,n\}$, and let $p$ be a probability distribution on the subsets of $[n]$. Let $M$ be a quantum state, let $v_M$ be a unit eigenv…

Let $n$ be a positive integer, let $[n]=\{1,2,\ldots,n\}$, and let $p$ be a probability distribution on the subsets of $[n]$. Let $M$ be a quantum state, let $v_M$ be a unit eigenv…

Let $\rho_1$ and $\rho_2$ be quantum states on a bipartite system $A B$ whose marginals on $B$ need not be identical. Let $d_A$ be the dimension of system $A$, let $S(A|B)_\rho$ de…

Approximate stabilizer-state conjecture. There exist a pure stabilizer state $\phi_{stab}$ and a constant $c$ independent of $n$ such that

Convolutional strong subadditivity. One has

Triangle inequality for the quantum Ruzsa divergence. Given a balanced convolution $\boxtimes_{s,t}$, the quantum Ruzsa divergence satisfies

Exact catalytic entropy conjecture. Such a density matrix $csigma$ and unitary $U$ exist if and only if $H(\rho)<H(\rho')$ and…

Quantum cross-entropy conjecture. The quantum cross entropy $S(\rho,\sigma)$ is the average number of qubits needed to encode data from a quantum source represented by $\rho$, when…

Entropy comparison conjecture. The entropy of the quantum-addition mixture is at most the entropy of the ordinary weighted mixture:

Let $\rho$ and $\sigma$ be quantum states, and for each positive integer $n$ define … Writing $S(\sigma||\rho)$ for the quantum relative entropy, the informatic entropy production…

For a tripartite pure state $\rho_{ABC}$, define the conditional sandwiched Rényi entropy by … where $\sigma_B$ ranges over states on $B$, $I$ is the identity operator, and…

Temporal entropy-gain conjecture. For $T,V\to\infty$, the temporal frame-averaging map gains the same entropy specified by the thermodynamic constraint.

Thermodynamic entropy-gain conjecture. The spatial frame-averaging map satisfies the thermodynamic entropy constraint above.