IIIT Hyderabad Publications
Foundations of Quantum Mechanics, Non-locality and Randomness
Author: Anubhav Chaturvedi
Report no: IIIT/TH/2016/64
Advisor:Indranil Chakrabarty,Kannan Srinathan,Chandrasekher Mukku
Scientific method of finding a new law pertains to three sequential steps. First we guess a law, second we compute its consequences and third we compare the consequences with experiments. On top of these lies the pillar of science, if these consequences disagree with observation the guessed law is wrong. Science can never assert that any theory, or law explains nature. It can only tell which ones cannot. The twentieth century saw Humankind asking one of the most fundamental question regarding the nature of reality and determinism. EPR in 1935 questioned the completeness of Quantum Mechanics (a proba- bilistic theory as a fundamental theory). Bell’s framework developed to answer the question, introduced first notions of theory independent tests. While we cannot say whether there exists a deterministic ex- planation to every prediction of QM, but we can reject a subclass of deterministic explanation, local deterministic explanation. This experimentally verified departure of predictions of QM or any theory from any local deterministic explanation is referred to as non-locality. In first chapter origins and mathematical formalism of quantum mechanics is presented. Followed by presentation of EPR argument, Bell’s theorem, implications and applications in the second chapter. Fol- lowing chapters contain the main contributions from the author. In third chapter, we study the consequences of both answers set in a device (theory) independent frame- work, based only on observed statistics. We start with taking up post-selection as an assumption and model the same using independent devices governed by Boolean functions. We establish analogy be- tween the post selection functions and the general probabilistic games in a two party binary input-output scenario. As an observation, we categorize all possible post-selection functions based on the effect on a uniform input probability distribution. We find that post-selection can transform simple no signaling probability distributions to signaling. Similarly, solving NP complete problems is easy independent of classical or quantum computation (in particular we prove that Post RP = NP). Finally, we demonstrate an instance of the violation of the pigeon hole principle independent of underlying theory. As result of our theory independent modeling we conclude that post-selection as an assumption adds power to the underlying theory. In particular, quantum mechanics benefits more with the post-selection assumption, only because it admits a more general set of allowed probabilities as compared to the local hidden vari- able model. Without the assumption we associate a device independent efficiency factor to quantify the cost of post selection. Our study shows that in the real world post-selection is not efficient enough to be of any advantage. But from an adversarial perspective it is still of significance. As an application, we obtain robust bounds on faking the bell violation in terms of minimum efficiency required using post selection. In fourth chapter we study a semi device-independent protocol wherein the knowledge of success prob- ability of the associated dimension witness and the observed efficiency of the detectors (η a vg) are suf- ficient to determine the security. We consider practically possible, individual attacks with and without eavesdropper having access to quantum memory. We find critical detection efficiencies required for security in both cases. In fifth chapter we introduce generalizations to PR and (2 → 1) RAC and study their inter-convertibility. We introduce generalizations based on the number of inputs provided to Alice, B n -BOX and (n → 1) RAC. We show that a B n -BOX is equivalent to a no-signaling (n → 1) RACBOX (RB). Further we introduce a signaling (n → 1) RB which cannot simulate a B n -BOX. Finally to quantify the same we provide a resource inequality between (n → 1) RB and B n -BOX, and show that it is saturated. As an application we prove that one requires atleast (n − 1) PRs supplemented with a bit of communication to win a (n → 1) RAC. We further introduce generalizations based on the dimension of inputs provided to Alice and the mes- sage she sends, B n d (+)-BOX, B n d (−)-BOX and (n → 1, d) RAC (d > 2). We show that no-signaling condition is not enough to enforce strict equivalence in the case of d > 2. We introduce classes of no-signaling (n → 1, d) RB, one which can simulate B n d (+)-BOX, second which can simulate B n d (−)- BOX and third which cannot simulate either. Finally to quantify the same we provide a resource in- equality between (n → 1, d) RB and B n d (+)-BOX, and show that it is saturated. In sixth chapter we introduce the measurement-device-independent randomness certification task where one has trusted quantum state preparation device but the mesurement devices are completely unspec- ified. Interestingly we show that there exist entangled states, with local description, that are useful resource in such task which otherwise are useless in corresponding DI scenario. In seventh chapter we show that there exists a complementary relationship in terms of the genuine non-locality of a multipartite system vs. that of reduced subsystems. We further assert that the same is enforced due to no-signaling condition. To this end we consider Svetlichny games in a multiparty binary input and output scenario with a threshold value of the winning probability as a signature of genuine multiparty non locality. We analytically show that, in the Svetlichny games setup, there exists complementary relations between Svetlichny correlations of n party and Svetlichny correlations of any subset of k ≤ n parties within the no-signaling framework.
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