Seminars

This talk is a brief summary of my Habilitation thesis, tailored at the members of the Rada Dyscypliny, and hopefully interesting for other quantum scientists as well. I will walk you through 12 of my main publications from the last six years, touching on almost-quantum correlations, causal structures, quantum and post-quantum steering, and a resource theory of Bell-nonclassicality.

In this presentation I will focus on some numerical approaches to certain types of linear Klein Gordon equations. More particularly, I will present the numerical approach based on the Duhamel formula, where we obtain second order approximation. Modulated Fourier expansion based method will be presented as the one useful in case of highly oscillatory forcing term. Finally, I will present some splitting methods of fourth order. I will present plenty of numerical examples and comparisons.

References:
[1] Bader P., Blanes S., Casas F., Kopylov N., Novel symplectic integrators for the Klein–Gordon equation with space- and time-dependent mass, Journal of Computational and Applied Mathematics, Volume 350, 2019, 130-138 (2019).
[2] M. Condon, K. Kropielnicka, K. Lademann, R. Perczyński, Asymptotic numerical solver for the linear Klein–Gordon equation with space- and time-dependent mass, Applied Mathematics Letters,115,106935, (2021)
[3] Bauke H., Ruf M, Keitel Ch., A real space split operator method for the Klein-Gordon equation, J. Comput. Phys., 228, 24, 9092–9106, (2009).
[4] Mostafazadeh A., Quantum mechanics of Klein–Gordon-type fields and quantum cosmology, Ann. Physics 309, no. 1, 1–48 (2004).
[5] Mostafazadeh A.,Hilbert space structures on the solution space of Klein-Gordon-type evolution equations, Classical Quantum Gravity 20, no. 1, 155–171 (2003).
[6] Znojil M., Klein-Gordon equation with the time- and space-dependent mass: Unitary evolution picture, arXiv:1702.08493v1
[7] Znojil M., Non-Hermitian interaction representation and its use in relativistic quantum mechanics, Ann. Physics 385, 162–179 (2017).

“The unambiguous account of proper quantum phenomena must, in principle, include a description of all relevant features of experimental arrangement” (Bohr). The measurement process is composed of pre-measurement (quantum correlation of the system with the pointer variable), and an irreversible decoherence via interaction with an environment. The system ends up in a probabilistic mixture of the eigenstates of the measured observable. For pre-measurement stage, any attempt to introduce an `outcome’ leads, as we show, to a logical contradiction, $1=i$. This nullifies claims that a modified concept of Wigner’s Friend, who just pre-measures, can lead to valid results concerning quantum theory.

Quantum key distribution (QKD) is a method that distributes a secret key to a sender and receiver by the transmission of quantum particles (e.g. photons). Device-independent quantum key distribution (DIQKD) is a version of QKD with a stronger notion of security, in that the sender and receiver base their protocol only on the statistics of input and outputs of their devices as inspired by Bell’s theorem. We study the rate at which DIQKD can be carried out for a given bipartite quantum state distributed between the sender and receiver or a quantum channel connecting them.We provide upper bounds on the achievable rate going beyond upper bounds possible for QKD. In particular, we construct states and channels where the QKD rate is significant while the DIQKD rateis negligible. This gap is illustrated for a practical case arising when using standard post-processing techniques for entangled two-qubit state.
Based on: https://arxiv.org/abs/2005.13511

Abstract: The amplification of light in quantum theory has a long history, with research devoted mostly to the linear (i.e. one-photon) regime due to its simplicity [1]. However, nonlinear amplification has also received some interest since the 1960s. One obvious generalisation of linear/one-photon amplification is to allow for two photons (and only two) to be emitted or lost during amplification [2,3]. It was shown in the early days of amplifier research that such a two-photon device contained an operating regime where one-photon amplification was seemingly possible despite being prohibited at the outset by the model [2]. This apparent paradox has remained unresolved till now [4].

In this talk I will show how this apparent contradiction can be resolved and what the resolution implies for the physics of linear amplifiers. In particular, we will see how additive noise and multiplicative noise in a linear amplifier can be understood in terms of elementary atom-photon interactions. This understanding of amplifier noise also sheds light on the status of the parametric amplifier where it has been claimed to be a universal model for any phase-preserving linear amplifier [1,4].

I will try to explain my results with minimal background knowledge in amplifiers and theoretical techniques with emphasis on the physics and concepts.

[1] C. Caves+, Quantum limits on phase-preserving linear amplifiers, Physical Review A, 2012.
[2] P. Lambropoulos, Quantum statistics of a two-photon amplifier, Physical Review, 1967.
[3] K. J. McNeil and D. F. Walls, A master equation approach to nonlinear optics, Journal of Physics A, 1974.
[4] A. Chia+, Phase-preserving linear amplifiers not simulable by the parametric amplifier, Physical Review Letters, 2020.

While the standard formulation of quantum theory relies on a fixed background causal structure, one can consider a more general framework – based on quantum theory – where the causal structure is indefinite (the “process matrix formalism”). Interestingly, the tools introduced to identify causal indefiniteness – causal non-separability and causal inequalities – have been developed by analogy with quantum entanglement and Bell inequalities. An explicit example of a physical process with indefinite causal structure is the so-called ‘quantum switch’.
In this work, we take a bottom-up approach. We identify two new types of circuits that naturally generalise the fixed-order case which we fully characterise. We first introduce “quantum circuits with classical control of causal order ”, in which the order of operations is still well-defined, but not necessarily fixed in advance: it can in particular be established dynamically, in a classically-controlled manner, as the circuit is being used. We then consider “ quantum circuits with quantum control of causal order ”, in which the order of operations is controlled coherently. The latter encompasses all known examples of physically realisable processes with indefinite causal order, including the celebrated quantum switch. Interestingly, it also contains new examples arising from the combination of dynamical and coherent control of causal order, and we detail explicitly one such process. Nevertheless, we show that quantum circuits with quantum control of causal order can only generate “causal” correlations, compatible with a well-defined causal order. We furthermore extend our considerations to probabilistic circuits that produce also classical outcomes, and we demonstrate by an example how the characterisations derived in this work allow us to identify new advantages for quantum information processing tasks that could be demonstrated in practice.

We discuss schemes to certify local quantum systems via self-testing and dimension witnessing. This work leverages the graph-theoretic framework for contextuality introduced by Cabello, Severini, and Winter, combined with tools from combinatorial optimisation that guarantee the unicity of optimal solutions. We first show that the celebrated Klyachko-Can-Binicioglu-Shumovsky inequality and its generalisation to contextuality scenarios with odd cycle compatibility relations admit robust self-testing. We extend this robust self-testing result to the class of contextuality scenarios with odd anti-cycle compatibility structure which enables us to self-test arbitrary high dimensional quantum systems. Dimension witnessing is another tool for device certification where the goal is to certify the underlying dimensions of the quantum systems just from the measurement statistics. We present the concepts and tools needed for graph-theoretic quantum dimension witnessing and illustrate their use by identifying quantum dimension witnesses, including a family that can certify arbitrarily high quantum dimensions with few events.

Bell’s theorem is supposed to exclude all local hidden-variable models of quantum correlations. However, an explicit counterexample shows that a new class of local realistic models, based on generalized arithmetic and calculus, can exactly reconstruct rotationally symmetric quantum probabilities typical of two-electron singlet states. Observable probabilities are consistent with the usual arithmetic employed by macroscopic observers, but counterfactual aspects of Bell’s theorem are sensitive to the choice of hidden-variable arithmetic and calculus. The model is classical in the sense of Einstein, Podolsky, Rosen, and Bell: elements of reality exist and probabilities are modeled by integrals of hidden-variable probability densities. Probability densities have a Clauser-Horne product form typical of local realistic theories. However, neither the product nor the integral nor the representation of rotations are the usual ones. The integral has all the standard properties but only with respect to the arithmetic that defines the product. Certain formal transformations of integral expressions one finds in the usual proofs à la Bell do not work, so standard Bell-type inequalities cannot be proved. The system we consider is deterministic, local-realistic, rotationally invariant, observers have free will, detectors are perfect, so is free of all the canonical loopholes discussed in the literature.

Quantum Darwinism posits that the emergence of a classical reality relies on the spreading of classical information from a quantum system to many parts of its environment. But what are the essential physical principles of quantum theory that make this mechanism possible? We address this question by formulating the simplest instance of Darwinism — CNOT-like fan-out interactions — in a class of probabilistic theories that contain classical and quantum theory as special cases. We determine necessary and sufficient conditions for any theory to admit such interactions. We find that every non-classical theory that admits this spreading of classical information must have both entangled states and entangled measurements. Furthermore, we show that Spekkens’ toy theory admits this form of Darwinism, and so do all probabilistic theories that satisfy principles like strong symmetry, or contain a certain type of decoherence processes. Our result suggests the counterintuitive general principle that in the presence of local non-classicality, a classical world can only emerge if this non-classicality can be “amplified” to a form of entanglement.

Nonlocality, as demonstrated by the violation of Bell inequalities, enables device-independent cryptographic tasks that do not require users to trust their apparatus. In this presentation, we consider devices whose inputs are spatiotemporal degrees of freedom, e.g. orientations or time durations. Without assuming the validity of quantum theory, we prove that the devices’ statistical response must respect their input’s symmetries, with profound foundational and technological implications. We exactly characterize the bipartite binary quantum correlations in terms of local symmetries, indicating a fundamental relation between spacetime and quantum theory. For Bell experiments characterized by two input angles, we show that the correlations are accounted for by a local hidden variable model if they contain enough noise, but conversely must be nonlocal if they are pure enough. This allows us to construct a “Bell witness” that certifies nonlocality with fewer measurements than possible without such spatiotemporal symmetries, suggesting a new class of semi-device-independent protocols for quantum technologies.