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Register a new userA quantum version of Wielandts inequality
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In this paper, Wielandts inequality for classical channels is extended to quantum channels. That is, an upper bound to the number of times a channel must be applied, so that it maps any density operator to one with full rank, is found. Using this bound, dichotomy theorems for the zero--error capacity of quantum channels and for the Matrix Product State (MPS) dimension of ground states of frustration-free Hamiltonians are derived. The obtained inequalities also imply new bounds on the required interaction-range of Hamiltonians with unique MPS ground state.
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For an even qudit dimension $dgeq 2,$ we introduce a class of two-qudit states exhibiting perfect correlations/anticorrelations and prove via the generalized Gell-Mann representation that, for each two-qudit state from this class, the maximal violation of the original Bell inequality is bounded from above by the value $3/2$ - the upper bound attained on some two-qubit states. We show that the two-qudit Greenberger-Horne-Zeilinger (GHZ) state with an arbitrary even $dgeq 2$ exhibits perfect correlations/anticorrelations and belongs to the introduced two-qudit state class. These new results are important steps towards proving in general the $frac{3}{2}$ upper bound on quantum violation of the original Bell inequality. The latter would imply that similarly as the Tsirelson upper bound $2sqrt{2}$ specifies the quantum analog of the CHSH inequality for all bipartite quantum states, the upper bound $frac{3}{2}$ specifies the quantum analog of the original Bell inequality for all bipartite quantum states with perfect correlations/ anticorrelations. Possible consequences for the experimental tests on violation of the original Bell inequality are briefly discussed.
Given a uniform, frustration-free family of local Lindbladians defined on a quantum lattice spin system in any spatial dimension, we prove a strong exponential convergence in relative entropy of the system to equilibrium under a condition of spatial mixing of the stationary Gibbs states and the rapid decay of the relative entropy on finite-size blocks. Our result leads to the first examples of the positivity of the modified logarithmic Sobolev inequality for quantum lattice spin systems independently of the system size. Moreover, we show that our notion of spatial mixing is a consequence of the recent quantum generalization of Dobrushin and Shlosmans complete analyticity of the free-energy at equilibrium. The latter typically holds above a critical temperature Tc. Our results have wide-ranging applications in quantum information. As an illustration, we discuss four of them: first, using techniques of quantum optimal transport, we show that a quantum annealer subject to a finite range classical noise will output an energy close to that of the fixed point after constant annealing time. Second, we prove Gaussian concentration inequalities for Lipschitz observables and show that the eigenstate thermalization hypothesis holds for certain high-temperture Gibbs states. Third, we prove a finite blocklength refinement of the quantum Stein lemma for the task of asymmetric discrimination of two Gibbs states of commuting Hamiltonians satisfying our conditions. Fourth, in the same setting, our results imply the existence of a local quantum circuit of logarithmic depth to prepare Gibbs states of a class of commuting Hamiltonians.
We formulate and prove the main properties of the generalized Gell-Mann representation for traceless qudit observables with eigenvalues in $[-1,1]$ and analyze via this representation violation of the CHSH inequality by a general two-qudit state. For the maximal value of the CHSH expectation in a two-qudit state with an arbitrary qudit dimension $dgeq2$, this allows us to find two new bounds, lower and upper, expressed via the spectral properties of the correlation matrix for a two-qudit state. We have not yet been able to specify if the new upper bound improves the Tsirelson upper bound for each two-qudit state. However, this is the case for all two-qubit states, where the new lower bound and the new upper bound coincide and reduce to the precise two-qubit CHSH result of Horodeckis, and also, for the Greenberger-Horne-Zeilinger (GHZ) state with an odd $dgeq2,$ where the new upper bound is less than the upper bound of Tsirelson. Moreover, we explicitly find the correlation matrix for the two-qudit GHZ state and prove that, for this state, the new upper bound is attained for each dimension $dgeq2$ and this specifies the following new result: for the two-qudit GHZ state, the maximum of the CHSH expectation over traceless qudit observables with eigenvalues in $[-1,1]$ is equal to $2sqrt{2}$ if $dgeq2$ is even and to $frac{2(d-1)}{d}sqrt{2}$ if $d>2$ is odd.
We introduce a concept of Kadison-Schwarz divisible dynamical maps. It turns out that it is a natural generalization of the well known CP-divisibility which characterizes quantum Markovian evolution. It is proved that Kadison-Schwarz divisible maps are fully characterized in terms of time-local dissipative generators. Simple qubit evolution illustrates the concept.
We prove a double-inequality for the product of uncertainties for position and momentum of bound states for 1D quantum mechanical systems in the semiclassical limit.
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