We study a multilinear optimization problem for tensors with bounded Frobenius norm, where each of the $k$ local factors of dimension $d$ is chosen from a convex set. We give a deterministic algorithm running in time $d^{O(k)}$ for constant additive approximations. As applications, we obtain the first polynomial-time algorithms at constant accuracy for weak membership in the nuclear-norm unit ball of high-order tensors and for multipartite quantum separability in Frobenius norm. Our approach views tensor optimization as a cooperative multiprover game and combines this perspective with a recursive spectral compression procedure. We then show that related ideas admit a quantum implementation when the input state is given through copies, rather than an explicit classical description. In these settings we obtain an algorithm to solve Frobenius-norm separability testing using $\text{poly}(k)$ copies and $\text{poly}(k\log d)$ time.We study a multilinear optimization problem for tensors with bounded Frobenius norm, where each of the $k$ local factors of dimension $d$ is chosen from a convex set. We give a deterministic algorithm running in time $d^{O(k)}$ for constant additive approximations. As applications, we obtain the first polynomial-time algorithms at constant accuracy for weak membership in the nuclear-norm unit ball of high-order tensors and for multipartite quantum separability in Frobenius norm. Our approach views tensor optimization as a cooperative multiprover game and combines this perspective with a recursive spectral compression procedure. We then show that related ideas admit a quantum implementation when the input state is given through copies, rather than an explicit classical description. In these settings we obtain an algorithm to solve Frobenius-norm separability testing using $\text{poly}(k)$ copies and $\text{poly}(k\log d)$ time.

quant-pharXiv:2610.03637Oct 5, 20260 citations

On diagonal quantum channels

Amir R. Arab

arXiv:2011.00237Reports on Mathematical PhysicsAug 1, 20212 citations

ON CAPACITIES OF QUANTUM CHANNELS

Masanori Ohya, Dėnes Petz, Noboru Watanabe

WORLD SCIENTIFIC eBooksFeb 1, 200836 citations

Quantum coherence measures for generalized Gaussian wave packets under a Lorentz boost

Arnab Mukherjee, Soham Sen, Sunandan Gangopadhyay

In this paper, we consider a single-particle, spin-momentum entangled state and measure the effect of relativistic boost on quantum coherence. The effect of the relativistic boost on single-particle generalized Gaussian wave packets is studied. The coherence of the wave function as measured by the boosted observer is studied as a function of the momentum and the boost parameter. Using various formulations of coherence, it is shown that in general, the coherence decays with the increase in momentum of the state, as well as the boost applied to it. A more prominent loss of coherence due to relativistic boost is observed for a single-particle electron than that of a neutron. The analysis is carried out with a generalized Gaussian wave packet of the form $\mathcal{N}{p}^{n}exp(\ensuremath{-}\frac{{p}^{2}}{{\ensuremath{\sigma}}^{2}})$, with $n$ being the ``generalization parameter'' and $\mathcal{N}$ denoting the appropriate normalization constant. We also obtain a range for parameter $n$ appearing in the wave packet. The upper bound is found to have a dependence on the mass of the particle and the width of the Gaussian wave packet. We have obtained the Frobenius-norm measure of coherence, ${l}_{1}$ - and ${l}_{2}$ -norm measure of coherence, and relative entropy of coherence for a $(1+1)$ - and $(3+1)$ -dimensional analysis. Corresponding to each of the cases, we observe that the ${l}_{1}$ -norm measure of coherence is equal to the Frobenius-norm measure of coherence. We have analyzed the scenario for which such a beautiful coincidence can occur. Finally, we have plotted different measures of coherence for the electron as well as the neutron for different values of the width of the wave function $\ensuremath{\sigma}$, boost parameter $\ensuremath{\beta}$, and generalization parameter $n$ .In this paper, we consider a single-particle, spin-momentum entangled state and measure the effect of relativistic boost on quantum coherence. The effect of the relativistic boost on single-particle generalized Gaussian wave packets is studied. The coherence of the wave function as measured by the boosted observer is studied as a function of the momentum and the boost parameter. Using various formulations of coherence, it is shown that in general, the coherence decays with the increase in momentum of the state, as well as the boost applied to it. A more prominent loss of coherence due to relativistic boost is observed for a single-particle electron than that of a neutron. The analysis is carried out with a generalized Gaussian wave packet of the form $\mathcal{N}{p}^{n}exp(\ensuremath{-}\frac{{p}^{2}}{{\ensuremath{\sigma}}^{2}})$, with $n$ being the ``generalization parameter'' and $\mathcal{N}$ denoting the appropriate normalization constant. We also obtain a range for parameter $n$ appearing in the wave packet. The upper bound is found to have a dependence on the mass of the particle and the width of the Gaussian wave packet. We have obtained the Frobenius-norm measure of coherence, ${l}_{1}$- and ${l}_{2}$-norm measure of coherence, and relative entropy of coherence for a $(1+1)$- and $(3+1)$-dimensional analysis. Corresponding to each of the cases, we observe that the ${l}_{1}$-norm measure of coherence is equal to the Frobenius-norm measure of coherence. We have analyzed the scenario for which such a beautiful coincidence can occur. Finally, we have plotted different measures of coherence for the electron as well as the neutron for different values of the width of the wave function $\ensuremath{\sigma}$, boost parameter $\ensuremath{\beta}$, and generalization parameter $n$.

arXiv:2407.06599Physical Review ANov 12, 20241 citation

Factorizations of quantum channels

Jeremy Levick

arXiv:1808.05048Linear Algebra and its ApplicationsMay 7, 20182 citations

Noncommutative Marcus-Ree inequality and Erdős channels

Repana Devendra, Chaman Kumar Sahu

Unital quantum channels are the noncommutative analogues of doubly stochastic matrices. In 1959, Marcus and Ree showed that the squared Frobenius norm of a doubly stochastic matrix is at most its maximal diagonal sum. We prove a quantum version of this inequality for mixed unitary channels on $\mathbb{M}_d$ and give an example showing that it fails for general unital channels. As in the classical case, we call the channels attaining equality as Erdős channels. We show that the trace-preserving conditional expectation onto any unital $C^*$ -subalgebra of $\mathbb{M}_d$ is an Erdős channel. It is also locally isolated among channels of the same Choi rank. Next, we introduce quantum RCDS channels. These are the noncommutative analogue of doubly stochastic matrices whose supported permutations all have the same diagonal sum. We prove a structure theorem for them and use it to characterize the quantum RCDS Erdős channels. We give an explicit form, up to unitary equivalence, for the Erdős channels of Choi rank $2$ , and further show that there are uncountably many unitary equivalence classes of Erdős channels unlike the classical case. We also characterize the Erdős channels among mixed Weyl-unitary channels of any Choi rank. For qubits, we prove that there are exactly four unitary equivalence classes of Erdős channels and they are represented by the identity channel and uniform averages of Pauli channels. Finally, restricting to the diagonal channels, our results recover the Marcus--Ree inequality, the structure of RCDS doubly stochastic matrices, and the characterization of Erdős.Unital quantum channels are the noncommutative analogues of doubly stochastic matrices. In 1959, Marcus and Ree showed that the squared Frobenius norm of a doubly stochastic matrix is at most its maximal diagonal sum. We prove a quantum version of this inequality for mixed unitary channels on $\mathbb{M}_d$ and give an example showing that it fails for general unital channels. As in the classical case, we call the channels attaining equality as Erdős channels. We show that the trace-preserving conditional expectation onto any unital $C^*$-subalgebra of $\mathbb{M}_d$ is an Erdős channel. It is also locally isolated among channels of the same Choi rank. Next, we introduce quantum RCDS channels. These are the noncommutative analogue of doubly stochastic matrices whose supported permutations all have the same diagonal sum. We prove a structure theorem for them and use it to characterize the quantum RCDS Erdős channels. We give an explicit form, up to unitary equivalence, for the Erdős channels of Choi rank $2$, and further show that there are uncountably many unitary equivalence classes of Erdős channels unlike the classical case. We also characterize the Erdős channels among mixed Weyl-unitary channels of any Choi rank. For qubits, we prove that there are exactly four unitary equivalence classes of Erdős channels and they are represented by the identity channel and uniform averages of Pauli channels. Finally, restricting to the diagonal channels, our results recover the Marcus--Ree inequality, the structure of RCDS doubly stochastic matrices, and the characterization of Erdős.

arXiv:2610.02294Oct 1, 20260 citations

We study a multilinear optimization problem for tensors with bounded Frobenius norm, where each of the

k

local factors of dimension

d

is chosen from a convex set. We give a deterministic algorithm running in time

d^{O(k)}

for constant additive approximations. As applications, we obtain the first polynomial-time algorithms at constant accuracy for weak membership in the nuclear-norm unit ball of high-order tensors and for multipartite quantum separability in Frobenius norm. Our approach views tensor optimization as a cooperative multiprover game and combines this perspective with a recursive spectral compression procedure. We then show that related ideas admit a quantum implementation when the input state is given through copies, rather than an explicit classical description. In these settings we obtain an algorithm to solve Frobenius-norm separability testing using

\text{poly}(k)

copies and

\text{poly}(k\log d)

time.

n

being the ``generalization parameter'' and

\mathcal{N}

denoting the appropriate normalization constant. We also obtain a range for parameter

n

appearing in the wave packet. The upper bound is found to have a dependence on the mass of the particle and the width of the Gaussian wave packet. We have obtained the Frobenius-norm measure of coherence,

{l}_{1}

- and

{l}_{2}

-norm measure of coherence, and relative entropy of coherence for a

(1+1)

- and

(3+1)

-dimensional analysis. Corresponding to each of the cases, we observe that the

{l}_{1}

-norm measure of coherence is equal to the Frobenius-norm measure of coherence. We have analyzed the scenario for which such a beautiful coincidence can occur. Finally, we have plotted different measures of coherence for the electron as well as the neutron for different values of the width of the wave function $\ensuremath{\sigma}$, boost parameter $\ensuremath{\beta}$, and generalization parameter

n

\mathbb{M}_d

and give an example showing that it fails for general unital channels. As in the classical case, we call the channels attaining equality as Erdős channels. We show that the trace-preserving conditional expectation onto any unital

C^*

-subalgebra of

\mathbb{M}_d

is an Erdős channel. It is also locally isolated among channels of the same Choi rank. Next, we introduce quantum RCDS channels. These are the noncommutative analogue of doubly stochastic matrices whose supported permutations all have the same diagonal sum. We prove a structure theorem for them and use it to characterize the quantum RCDS Erdős channels. We give an explicit form, up to unitary equivalence, for the Erdős channels of Choi rank

2

, and further show that there are uncountably many unitary equivalence classes of Erdős channels unlike the classical case. We also characterize the Erdős channels among mixed Weyl-unitary channels of any Choi rank. For qubits, we prove that there are exactly four unitary equivalence classes of Erdős channels and they are represented by the identity channel and uniform averages of Pauli channels. Finally, restricting to the diagonal channels, our results recover the Marcus--Ree inequality, the structure of RCDS doubly stochastic matrices, and the characterization of Erdős.