Nies and Scholz defined quantum Martin-Löf randomness (q-MLR) for states (infinite qubitstrings). We define a notion of quantum Solovay randomness and show it to be equivalent to q-MLR using purely linear algebraic methods. Quantum Schnorr randomness is then introduced. A quantum analogue of the law of large numbers is shown to hold for quantum Schnorr random states. We introduce quantum-K, (
QK) a measure of the descriptive complexity of density matrices using classical prefix-free Turing machines and show that the initial segments of weak Solovay random and quantum Schnorr random states are incompressible in the sense of
QK. Several connections between Solovay randomness and
K carry over to those between weak Solovay randomness and
QK. We then define
QKC, using computable measure machines and connect it to quantum Schnorr randomness. We then explore a notion of `measuring' a state. We formalize how `measurement' of a state induces a probability measure on the space of infinite bitstrings. A state is `measurement random' (
mR) if the measure induced by it, under any computable basis, assigns probability one to the set of Martin-Löf randoms. I.e., measuring a
mR state produces a Martin-Löf random bitstring almost surely. While quantum-Martin-Löf random states are
mR, the converse fails: there is a
mR state,
ρ which is not quantum-Martin-Löf random. In fact, something stronger is true. While
ρ is computable and can be easily constructed, measuring it in any computable basis yields an arithmetically random sequence with probability one. So, classical randomness can be generated from a computable state which is not quantum random. We conclude by studying the asymptotic von Neumann entropy of computable states.
Von Neumann defined the informational entropy of a density matrix ρ by the expression S(ρ)=−tr(ρ ln ρ). [J. Von Neumann, Die Mathematischen Grundlagen der Quantummechanik (Springer-Verlag, Berlin, 1932)]. Here, starting from the definitions of Shannon entropy and of Renyi entropy of random variables, and using some (reasonable) rules of inference, two expressions are obtained for the quantum entropy of a given square (deterministic) matrix, irrespective of any probabilistic framework and/or any randomization technique. These definitions apply directly to linear operators on Hilbert spaces even when they are not of positive trace class, and they contain Von Neumann entropy as a special case. These new measures of uncertainty could provide new approaches to some problems related to structure complexity.
Quantum Martin-Löf randomness (q-MLR) for infinite qubit sequences was introduced by Nies and Scholz [J. Math. Phys. 60(9), 092201 (2019)]. We define a notion of quantum Solovay randomness, which is equivalent to q-MLR. The proof of this goes through a purely linear algebraic result about approximating density matrices by subspaces. We then show that random states form a convex set. Martin-Löf absolute continuity is shown to be a special case of q-MLR. Quantum Schnorr randomness is introduced. A quantum analog of the law of large numbers is shown to hold for quantum Schnorr random states.
, named after John von Neumann, is an extension of the classical concept of entropy to the field of quantum mechanics. From a numerical perspective, von Neumann entropy can be computed simply by computing all eigenvalues of a density matrix, an operation that could be prohibitively expensive for large-scale density matrices. We present and analyze three randomized algorithms to approximate von Neumann entropy of real density matrices: our algorithms leverage recent developments in the Randomized Numerical Linear Algebra (RandNLA) literature, such as randomized trace estimators, provable bounds for the power method, and the use of random projections to approximate the eigenvalues of a matrix. All three algorithms come with provable accuracy guarantees and our experimental evaluations support our theoretical findings showing considerable speedup with small loss in accuracy.
In quantum information geometry, the curvature of von-Neumann entropy and relative entropy induce a natural metric on the space of mixed quantum states. Here we use this information metric to construct a random matrix ensemble for states and investigate its key statistical properties such as the asymptotic eigenvalue density and mean entropy. We present an algorithm for generating these entropy-based random density matrices, thus providing a new recipe for random state generation that differs from the well established Hilbert-Schmidt and Bures-Hall ensemble approaches. We also prove a duality between the entropy-based state ensemble and a random Hamiltonian model constructed from the thermodynamic length over the set of Gibbs states. This Hamiltonian model is found to display Wigner level repulsion, implying that the dual state ensemble can be realised as a random Gibbs state with respect to a class of chaotic Hamiltonians. As an application we use our model to compute the survival probability of a randomly evolved thermofield double state, predicting a ramp and plateau over time that is characteristic of quantum chaos. For other applications, the entropy-based ensemble can be used as an uninformative prior for Bayesian quantum state or Hamiltonian tomography.
Consider a source of pure quantum states with density matrix ρ and von Neumann entropy S . It is known that the quantum information of the source may be faithfully compressed to 5 qubits/signal (asymptotically for long sequences) and that this limit is optimal if we demand high fidelity for long strings of signals. The usual method for achieving this compression, based on the typical subspace of the source, requires knowledge of the density matrix. We describe a new “universal” compression scheme which does not even require knowledge of ρ. More precisely, suppose that the source is known to have von Neumann entropy less than some given bound S 0 but is otherwise completely unspecified. Our method will faithfully compress the quantum information of any such source to S 0 qubits/signal. It is known that S is always less than or equal to the Shannon entropy H of the prior distribution of the source. According to Shannon’s theorem, H provides the limit for compression of classical signals, so one may ask the question: what is the physical origin of the extra compression available in the quantum case compared to the classical case (where the prior probabilities are the same)? One may suggest that this effect is a consequence of possible non-orthogonality of signal states. However we demonstrate that it is possible to make each pair of signal states of a source more parallel (keeping the prior probabilities the same) while increasing the von Neumann entropy. This casts doubt on the conventional wisdom that overlap of quantum states is a measure of their (non-)distinguishability. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.