We construct quantum mechanical observables and unitary operators which, if implemented in physical systems as measurements and dynamical evolutions, would contradict the Church-Turing thesis which lies at the foundation of computer science. We conclude that either the Church-Turing thesis needs revision, or that only restricted classes of observables may be realized, in principle, as measurements, and that only restricted classes of unitary operators may be realized, in principle, as dynamics.
It is possible in principle to construct quantum mechanical observables and unitary operators which, if implemented in physical systems as measurements and dynamical evolution, would contradict the Church-Turing thesis, which lies at the foundation of computer science. Elsewhere we have argued that the quantum measurement problem implies a finite, computational model of the measurement and evolution of quantum states. If correct, this approach helps to identify the key feature that can reconcile quantum mechanics with the Church-Turing thesis: finitude of the degree of fine-graining of Hilbert…
Quantum steering describes the ability of one observer to nonlocally affect the other observer's state through local measurements, which represents a new form of quantum nonlocal correlation and has potential applications in quantum information and quantum communication. In this paper, we propose a computable steering criterion that is applicable to bipartite quantum systems of arbitrary dimensions. The criterion can be used to verify a wide range of steerable states directly from a given density matrix without constructing measurement settings. Compared with the existing steering criteria, it…
Analytic quantifiers of the symmetric quantum discord for two-qubit X type states and block-diagonal states and the symmetric measurement induced nonlocality for any two qubit states are established on the basis of the quantum skew information.
It is shown that in two-state quantum theory, a generic quantum state can be described by a non-computable real number. In terms of this, the criterion for measurement outcome is simply and deterministically defined. This demonstration is based on a construction of the Riemann sphere whose points represent, not complex numbers, but divergent sequences with bivalent elements. Complex structure arises from self-similar properties of a set of operators which generate these sequences. In general, a rotation of (the coordinates of) the sphere maps a computable real to a non-computable real. This is…
We apply algorithmic information theory to quantum mechanics in order to shed light on an algorithmic structure which inheres in quantum mechanics. There are two equivalent ways to define the (classical) Kolmogorov complexity K(s) of a given classical finite binary string s. In the standard way, K(s) is defined as the length of the shortest input string for the universal self-delimiting Turing machine to output s. In the other way, we first introduce the so-called universal probability m, and then define K(s) as -log_2 m(s) without using the concept of program-size. We generalize the universal…
Quantum discord expresses a fundamental non-classicality of correlations more general than quantum entanglement. We combine the no-local-broadcasting theorem, semidefinite-programming characterizations of quantum fidelity and quantum separability, and a recent breakthrough result of Fawzi and Renner about quantum Markov chains to provide a hierarchy of computationally efficient lower bounds to quantum discord. Such a hierarchy converges to the surprisal of measurement recoverability introduced by Seshadreesan and Wilde, and provides a faithful lower bound to quantum discord already at the lowe…
Nies and Scholz introduced the notion of a state to describe an infinite sequence of qubits and defined quantum-Martin-Lof randomness for states, analogously to the well known concept of Martin-Löf randomness for elements of Cantor space (the space of infinite sequences of bits). We formalize how 'measurement' of a state in a basis induces a probability measure on Cantor space. 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. Equivalently, a state is mR if and only if measuring it in any com…
Our aim is to experimentally study the possibility of distinguishing between quantum sources of randomness--recently proved to be theoretically incomputable--and some well-known computable sources of pseudo-randomness. Incomputability is a necessary, but not sufficient "symptom" of "true randomness". We base our experimental approach on algorithmic information theory which provides characterizations of algorithmic random sequences in terms of the degrees of incompressibility of their finite prefixes. Algorithmic random sequences are incomputable, but the converse implication is false. We have …
We study a system of two bosonic fields subject to two-mode squeezing in the presence of dissipation. We find the Lie algebra governing the dynamics of the problem and use the Wei-Norman method to determine the solutions. Using this scheme we arrive at a closed form expression for an infinitely dimensional density operator which we use to calculate the degree of entanglement (quantified by Horodeckis' negativity) between the modes. We compare our result to the known continuous variable entanglement measures. We analyse the conditions for entanglement generation and the influence of thermal env…
Nonlocality, which is the key feature of quantum theory, has been linked with the uncertainty principle by fine-grained uncertainty relations, by considering combinations of outcomes for different measurements. However, this approach assumes that information about the system to be fine-grained is local, and does not present an explicitly computable bound. Here, we generalize above approach to general quasi-fine-grained uncertainty relations (QFGURs) which applies in the presence of quantum memory and provides conspicuously computable bounds to quantitatively link the uncertainty to entanglemen…
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…
We study dynamics of genuine entanglement for quantum states of three and four qubits under non-Markovian dephasing. Using a computable entanglement monotone for multipartite systems, we find that GHZ state is quite resilient state whereas the W state is the most fragile. We compare dynamics of chosen quantum states with dynamics of random pure states and weighted graph states.
Variational Quantum Eigensolver (VQE) is a promising algorithm for near-term quantum machines. It can be used to estimate the ground state energy of a molecule by performing separate measurements of $O(N^4)$ terms. Several recent papers observed that this scaling may be reducible to $O(N^3)$ by partitioning the terms into linear-sized commuting families that can be measured simultaneously. We confirm these empirical observations by studying the MIN-COMMUTING-PARTITION problem at the level of the fermionic Hamiltonian and its encoding into qubits. Moreover, we provide a fast, pre-computable pro…
The central idea of this work is the concept of prespace, a hypothetical structure that is postulated to underlie the fabric of space or space-time. I consider how such a structure could relate to space and space-time, and the implications of the existence of this structure for quantum theory. I compare space and space-time to other spaces used in physics, such as configuration space, phase space, and Hilbert space. I support the 'property view' of space, opposing both the traditional views of space and space-time, substantivalism and relationism. I argue that configuration space has a special…
We introduce an operational and statistically meaningful measure, the quantum tomographic transfer function, that possesses important physical invariance properties for judging whether a given informationally complete quantum measurement performs better tomographically in quantum-state estimation relative to other informationally complete measurements. This function is independent of the unknown true state of the quantum source, and is directly related to the average optimal tomographic accuracy of an unbiased state estimator for the measurement in the limit of many sampling events. For the ex…
Geometric Algebra and Calculus are mathematical languages encoding fundamental geometric relations that theories of physics seem to respect. We propose criteria given which statistics of expressions in geometric algebra are computable in quantum theory, in such a way that preserves their algebraic properties. They are that one must be able to arbitrarily transform the basis of the Clifford algebra, via multiplication by elements of the algebra that act trivially on the state space; all such elements must be neighbored by operators corresponding to factors in the original expression and not the…
We improve upon a recently introduced efficient quantum state reconstruction procedure targeted to states well-approximated by the multi-scale entanglement renormalization ansatz (MERA), e.g., ground states of critical models. We show how to numerically select a subset of experimentally accessible measurements which maximizes information extraction about renormalized particles, thus dramatically reducing the required number of physical measurements. We numerically estimate the number of measurements required to characterize the ground state of the critical 1D Ising (resp. XX) model and find th…
This paper is a comment on the paper "Quantum Mechanics and Algorithmic Randomness" was written by Ulvi Yurtsever \cite{Yurtsever} and the briefly explanation of the algorithmic randomness of quantum measurements results. There are differences between the computability of probability sources, ( which means there is an algorithm that can define the way that random process or probability source generates the numbers ) and the algorithmic randomness of the sequences or strings which are produced by a source. We may have the source without a computable algorithm for that but it can produce compres…
This paper explores the problem of quantum measurement complexity. In computability theory, the complexity of a problem is determined by how long it takes an effective algorithm to solve it. This complexity may be compared to the difficulty for a hypothetical oracle machine, the output of which may be verified by a computable function but cannot be simulated on a physical machine. We define a quantum oracle machine for measurements as one that can determine the state by examining a single copy. The complexity of measurement for a realizable machine will then be respect to the number of copies …
We investigate what a snapshot of a quantum evolution - a quantum channel reflecting open system dynamics - reveals about the underlying continuous time evolution. Remarkably, from such a snapshot, and without imposing additional assumptions, it can be decided whether or not a channel is consistent with a time (in)dependent Markovian evolution, for which we provide computable necessary and sufficient criteria. Based on these, a computable measure of `Markovianity' is introduced. We discuss how the consistency with Markovian dynamics can be checked in quantum process tomography. The results als…
The maximum observable correlation between the two components of a bipartite quantum system is a property of the joint density operator, and is achieved by making particular measurements on the respective components. For pure states it corresponds to making measurements diagonal in a corresponding Schmidt basis. More generally, it is shown that the maximum correlation may be characterised in terms of a `correlation basis' for the joint density operator, which defines the corresponding (nondegenerate) optimal measurements. The maximum coincidence rate for spin measurements on two-qubit systems …
Quantum coherence is a fundamental resource that quantum technologies exploit to achieve performance beyond that of classical devices. A necessary prerequisite to achieve this advantage is the ability of measurement devices to detect coherence from the measurement statistics. Based on a recently developed resource theory of quantum operations, here we quantify experimentally the ability of a typical quantum-optical detector, the weak-field homodyne detector, to detect coherence. We derive an improved algorithm for quantum detector tomography and apply it to reconstruct the positive-operator-va…
We introduce a novel measure for the quantum property of nonstabilizerness - commonly known as "magic" - by considering the Rényi entropy of the probability distribution associated to a pure quantum state given by the square of the expectation value of Pauli strings in that state. We show that this is a good measure of nonstabilizerness from the point of view of resource theory and show bounds with other known measures. The stabilizer Rényi entropy has the advantage of being easily computable because it does not need a minimization procedure. We present a protocol for an experimental measureme…
Is the universe computable? If yes, is it computationally a polynomial place? In standard quantum mechanics, which permits infinite parallelism and the infinitely precise specification of states, a negative answer to both questions is not ruled out. On the other hand, empirical evidence suggests that NP-complete problems are intractable in the physical world. Likewise, computational problems known to be algorithmically uncomputable do not seem to be computable by any physical means. We suggest that this close correspondence between the efficiency and power of abstract algorithms on the one han…
Quantum simulation can beat current classical computers with minimally a few tens of qubits and will likely become the first practical use of a quantum computer. One promising application of quantum simulation is to attack challenging quantum chemistry problems. Here we report an experimental demonstration that a small nuclear-magnetic-resonance (NMR) quantum computer is already able to simulate the dynamics of a prototype chemical reaction. The experimental results agree well with classical simulations. We conclude that the quantum simulation of chemical reaction dynamics not computable on cu…
There has been a concerted effort to identify problems computable with quantum technology which are intractable with classical technology or require far fewer resources to compute. Recently, randomness processing in a Bernoulli factory has been identified as one such task. Here, we report two quantum photonic implementations of a Bernoulli factory, one utilising quantum coherence and single-qubit measurements and the other which uses quantum coherence and entangling measurements of two qubits. We show that the former consumes three orders of magnitude fewer resources than the best known classi…
In this study the determinant of the average quadratic error matrix is used as the measure of state estimation efficiency. This quantity is easily computable in some cases, so it gives us a reasonable tool to find optimal measurement setup for different quantum tomography problems. We present numerous applications for a single qubit when von Neumann measurements or a single POVM is used and a part of the parameters of the state is given. Under some restriction the optimality is found for $n$-level system as well. The optimal measurements have some complementary relation to each other or to the…
We study the dynamics of genuine multipartite entanglement for quantum systems upto four qubits interacting with general collective dephasing process. Using a computable entanglement monotone for multipartite systems, we observe the feature of freezing dynamics of genuine entanglement for three and four qubits entangled states. We compare the dynamics with that of random states and find that most states exibit this feature. We then study the effects of collective dephasing on genuine nonlocality and find out that although quantum states remain genuinely entangled yet their genuine nonlocality …
Pivotal within quantum physics, the concept of quantum incompatibility is generally related to algebraic aspects of the formalism, such as commutation relations and unbiasedness of bases. Recently, the concept was identified as a resource in tasks involving quantum state discrimination and quantum programmability. Here we link quantum incompatibility with the amount of information that can be extracted from a system upon successive measurements of noncommuting observables, a scenario related to communication tasks. This approach leads us to characterize incompatibility as a resource encoded in…
We introduce, for any bipartite Bell scenario, a measure that quantifies both the amount of nonlocality and the efficiency in device-independent quantum key distribution of a set of measurement outcomes probabilities. It is a proper measure of nonlocality as it vanishes when this set is Bell local and does not increase under the allowed transformations of the nonlocality resource theory. This device-independent key rate $R$ is defined by optimizing over a class of protocols, to generate the raw keys, in which each legitimate party does not use just one preselected measurement but randomly choo…
We demonstrate how quantum optimal control can be used to enhance quantum resources for bipartite one-way protocols, specifically EPR-steering with qubit measurements. Steering is relevant for one-sided device-independent key distribution, the realistic implementations of which necessitate the study of noisy scenarios. So far mainly the case of imperfect detection efficiency has been considered; here we look at the effect of dynamical noise responsible for decoherence and dissipation. In order to set up the optimisation, we map the steering problem into the equivalent joint measurability probl…
This paper considers distributed computing on an anonymous quantum network, a network in which no party has a unique identifier and quantum communication and computation are available. It is proved that the leader election problem can exactly (i.e., without error in bounded time) be solved with at most the same complexity up to a constant factor as that of exactly computing symmetric functions (without intermediate measurements for a distributed and superposed input), if the number of parties is given to every party. A corollary of this result is a more efficient quantum leader election algori…
Invariant Set (IS) theory is a locally causal ontic theory of physics based on the Cosmological Invariant Set postulate that the universe $U$ can be considered a deterministic dynamical system evolving precisely on a (suitably constructed) fractal dynamically invariant set in $U$'s state space. IS theory violates the Bell inequalities by violating Measurement Independence. Despite this, IS theory is not fine tuned, is not conspiratorial, does not constrain experimenter free will and does not invoke retrocausality. The reasons behind these claims are discussed in this paper. These arise from pr…
The Everett Box is a device in which an observer and a lethal quantum apparatus are isolated from the rest of the universe. On a regular basis, successive trials occur, in each of which an automatic measurement of a quantum superposition inside the apparatus either causes instant death or does nothing to the observer. From the observer's perspective, the chances of surviving $m$ trials monotonically decreases with increasing $m$. As a result, if the observer is still alive for sufficiently large $m$ she must reject any interpretation of quantum mechanics which is not the many-worlds interpreta…
