Bell's theorem,inference, and quantum transactions

Bell's theorem is expounded as an analysis in Bayesian inference. Assuming the result of a spin measurement on a particle is governed by a causal variable internal (hidden, local) to the particle, one learns about it by making a spin measurement; thence about the internal variable of a second particle correlated with the first; and from there predicts the probabilistic result of spin measurements on the second particle. Such predictions are violated by experiment: locality/causality fails. The statistical nature of the observations rules out signalling; acausal, superluminal, or otherwise. Quantum mechanics is irrelevant to this reasoning, although its correct predictions of experiment imply that it has a nonlocal/acausal interpretation. Cramer's newtransactional interpretation, which incorporates this feature by adapting the Wheeler-Feynman idea of advanced and retarded processes to the quantum laws, is advocated. It leads to an invaluable way of envisaging quantum processes. The usual paradoxes melt before this, and one, the delayed choice experiment, is chosen for detailed inspection. Nonlocality implies practical difficulties in influencing hidden variables, which provides a very plausible explanation for why they have not yet been found; from this standpoint, Bell's theoremreinforces arguments in favor of hidden variables.     

Locality,Bell's theorem,and quantum mechanics

Classical relativistic physics assumes that spatially separated events cannot influence one another (locality) and that values may be assigned to quantities independently of whether or not they are actually measured (realism). These assumptions have consequences—the Bell inequalities—that are sometimes in disagreement with experiment and with the predictions of quantum mechanics. It has been argued that, even if realism is not assumed, the violation of the Bell inequalities implies nonlocality—and hence that radical changes are necessary in the foundations of physics. We show that this conclusion does not follow unless the locality hypothesis is strengthened in an implausible manner.     

Neumark's theorem and quantum inseparability

The most efficient way of obtaining information about the state of a quantum system is not always a direct measurement. It is sometimes preferable to extend the original Hilbert space of states into a larger space, and then to perform a quantum measurement in the enlarged space. Such an extension is always possible, by virtue of Neumark's theorem. The physical interpretation usually given to that theorem is the introduction of an auxiliary quantum system, prepared in a standard state, and the execution of a quantum measurement on both systems together. However, this widespread interpretation is unacceptable, because the statistical properties of the supposedly standard auxiliary system are inseparably entangled with those of the original, unknown system. A different method of preparing the auxiliary system is proposed, and shown to be physically acceptable.Dedicated to John S. Bell, whose profound insight helped us to overcome deeply ingrained prejudices.     

Entropy majorization, thermal adiabatic theorem, and quantum phase transitions

Let a general quantum many-body system at a low temperature adiabatically cross through the vicinity of the system’s quantum critical point. We show that the system’s temperature is significantly suppressed due to both the entropy majorization theorem in quantum information science and the entropy conservation law in reversible adiabatic processes. We take the one-dimensional transverse-field Ising model and the spinless fermion system as concrete examples to show that the inverse temperature might become divergent around the systems’ critical points. Since the temperature is a measurable quantity in experiments, it can be used, via reversible adiabatic processes at low temperatures, to detect quantum phase transitions in the perspectives of quantum information science and quantum statistical mechanics.     

Interference of quantum channels

We show how interferometry can be used to characterize certain aspects of general quantum processes and, in particular, the coherence of completely positive maps. We derive a measure of coherent fidelity, the maximum interference visibility, and the closest unitary operator to a given physical process under this measure.     

Random unitary qubit channels: entropy relations,private quantum channels and non-malleability

Channels encrypting quantum bits by the application of randomly chosen unitary operators are studied. Quantities based on averages of linear entropies which characterize certain aspects of the encoding quality and the non-malleability of the channels are introduced. The relation between the entropy of the classical key and the choice of the encryption operators with the behaviour of these properties is discussed. The extension of exact private quantum channels in order to improve non-malleability via additional encryption operators is considered.     

Comments on “locality,Bell's theorem,and quantum mechanics”

Two different ideas of locality are described. Both are due essentially to einstein. Quantum theory is compatible with the first but not the second. The problems encountered in the article cited in the title arise from trying to use only the first idea of locality, whereas Bell's-theorem considerations pertain to the second.     

On the discrimination of qubit channels

We discuss the discrimination of arbitrary qubit channels and the discrimination of qubit channels with the aid of entanglement. Nevertheless, the maximally entangled states might decrease the distinguishability, even if the channel is not entanglement-breaking one.     

Supersymmetric quantum mechanics and the index theorem

We consider the classical mechanics of the spinning particle and investigate which Abelian interactions can be added without breaking supersymmetry. A quantum theory is presented. The well known index theorem for the Dirac operator is extended to take into account the effect of anti-symmetric Abelian tensor fields. Furthermore interactions with non-Abelian anti-symmetric tensor fields are investigated. It turns out in both cases that these fields do not give any non-trivial contributions to the index.     

Unambiguous discrimination of mixed quantum states

The problem of unambiguous discrimination between mixed quantum states is addressed by isolating the part of each mixed state which has no contribution to discrimination and by employing the strategy of set discrimination of pure states. A necessary and sufficient condition of unambiguous mixed state discrimination is presented. An upper bound of the efficiency is also derived.     

Interpretation of the quantum formalism and Bell's theorem

It is argued that quantum mechanics must be interpreted according to the Copenhagen interpretation. Consequently the formalism must be used in a purely operational way. The relation between realism, hidden variables, and the Bell inequalities is discussed. The proof of impossibility of local hidden-variables theories (Bell's theorem) is criticized on the basis that the quantum mechanical states violating local realism are not physically realizable states.Einstein had great difficulty in reaching a sharp formulation of Bohr's meaning. What hope then for the rest of us.—John S. Bell (Ref. 1, p. 189).     

Detecting entanglement of quantum channels

Entanglement is the crucial resource for different quantum information processing tasks. While conventional studies focus on the entanglement of bipartite or multipartite quantum states, recent works have extended the scenario to the entanglement of quantum channels, an operational quantification of the channel entanglement manipulation capability. Based on the recently proposed channel entanglement resource framework, here we study a further task of resource detection—witnessing entanglement of quantum channels. We first introduce the general framework and show how channel entanglement detection is related to the Choi state of the channel, enabling channel entanglement detection via conventional state entanglement detection methods. We also consider entanglement of multipartite quantum channels and use the stabilizer formalism to construct entanglement witnesses for circuits consisting of controlled-Z gates. We study the effectiveness of the proposed detection methods and compare their performance for several typical channels. Our work paves the way for systematic theoretical studies of channel entanglement and practical benchmarking of noisy intermediate scaled quantum devices.     

A link between quantum and classical Potts models

We study ground states of quantum Potts models. We construct ground states of certaind-dimensional quantum models as Gibbs measures of ad-dimensional classical spin system. Our results imply that various phenomena of classical spin systems can also be found in quantum ground states.     

Experimental test of the quantum no-hiding theorem

The no-hiding theorem says that if any physical process leads to bleaching of quantum information from the original system, then it must reside in the rest of the Universe with no information being hidden in the correlation between these two subsystems. Here, we report an experimental test of the no-hiding theorem with the technique of nuclear magnetic resonance. We use the quantum state randomization of a qubit as one example of the bleaching process and show that the missing information can be fully recovered up to local unitary transformations in the ancilla qubits.     

Hidden variables and Bell's theorem in quantum mechanics

In the present paper we give a precise definition of a hidden-variable theory for quantum mechanics, whereby we adopt the weakest possible definition of a hidden-variable theory, which is compatible with the assumption that the bounded observables of a quantum mechanical system are represented by the elements of the real part A_r of a W^*-algebra A (of the most general type) and the states are represented by the normal states (in the mathematical sense) of A. We then go on to show that an example put forward by Bell in 1966 satisfies our definition (Sec. 2). Finally we make use of Bell's famous theorem to show that for a sufficiently non-commutative W^*-algebra A no hidden-variable theory in our sense exists (Theorem 3.3 and its corollaries).     

Probabilistically perfect quantum cloning and unambiguous state discrimination

We consider the N → M probabilistically perfect quantum cloning machine that perfectly produces M faithful copies from N identical input states, where the input states are selected, with prior probabilities η₁and η₂ = 1 − η₁, from a given set of the two linearly independent states |ψ₁〉^⊗ N = (cosθ|0〉 + sinθ|1〉)^⊗ N and |ψ₂〉^⊗ N = (sinθ|0〉 + cosθ|1〉)^⊗ N (θ∈(0,π/2)). We derive the optimal distribution of the success probabilities. When M approaches infinite, the probabilistically perfect quantum cloning can be regarded as a kind of the unambiguous state discrimination, and theoretically provides the upper bound of the unambiguous state discrimination. By using the optimal distribution of the success probabilities of the optimal asymmetric 1 → M probabilistically perfect quantum cloning, we can derive the maximum average success probability of the unambiguous discrimination of two nonorthogonal quantum states |ψ₁〉and|ψ₂〉. As an example, we give the explicit transformation of the optimal symmetric 1 → M probabilistically perfect quantum cloning to copy the two input states |ψ₁〉 and |ψ₂〉.     

Optimal processing of reversible quantum channels

We consider the general problem of the optimal transformation of N uses of (possibly different) unitary channels to a single use of another unitary channel in any finite dimension. We show how the optimal transformation can be fully parallelized, consisting in a preprocessing channel followed by a parallel action of all the N unitaries and a final postprocessing channel. Our techniques allow to achieve an exponential reduction in the number of the free parameters of the optimization problem making it amenable to an efficient numerical treatment. Finally, we apply our general results to find the analytical solution for special cases of interest like the cloning of qubit phase gates.     

Instruments and channels in quantum information theory

While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities and the a posteriori states. By interpreting the instrument as a quantum channel and by using the typical inequalities for the quantum and classical relative entropies, many bounds on the classical information extracted in a quantum measurement, of the type of the Holevo bound, are obtained in a unified manner.