The simulation of quantum effects requires certain classical resources, and quantifying them is an important step in order to characterize the difference between quantum and classical physics. For a simulation of the phenomenon of state-independent quantum contextuality, we show that the minimal amount of memory used by the simulation is the critical resource. We derive optimal simulation strategies for important cases and prove that reproducing the results of sequential measurements on a two-qubit system requires more memory than the information carrying capacity of the system.
Contextuality has been conjectured to be a super-classical resource for quantum computation, analogous to the role of non-locality as a super-classical resource for communication. We show that the presence of contextuality places a lower bound on the amount of classical memory required to simulate any quantum sub-theory, thereby establishing a quantitative connection between contextuality and classical simulability. We apply our result to the qubit stabilizer sub-theory, where the presence of state-independent contextuality has been an obstacle in establishing contextuality as a quantum comput…
Simulating quantum contextuality with classical systems requires memory. A fundamental yet open question is what is the minimum memory needed and, therefore, the precise sense in which quantum systems outperform classical ones. Here, we make rigorous the notion of classically simulating quantum state-independent contextuality (QSIC) in the case of a single quantum system submitted to an infinite sequence of measurements randomly chosen from a finite QSIC set. We obtain the minimum memory needed to simulate arbitrary QSIC sets via classical systems under the assumption that the simulation shoul…
In Itamar Pitowsky's reading of the Gleason and the Kochen-Specker theorems, in particular, his Logical Indeterminacy Principle, the emphasis is on the value indefiniteness of observables which are not within the preparation context. This is in stark contrast to the prevalent term {\em contextuality} used by many researchers in informal, heuristic yet omni-realistic and potentially misleading ways. This paper discusses both concepts and argues in favor of value indefiniteness in all but a continuum of contexts intertwining in the vector representing a single pure (prepared) state. Even more re…
We unify the resource-theoretic and the cohomological perspective on quantum contextuality. At the center of this unification stands the notion of the contextual fraction. For both symmetry and parity based contextuality proofs, we establish cohomological invariants which are witnesses of state-dependent contextuality. We provide two results invoking the contextual fraction, namely (i) refinements of logical contextuality inequalities, and (ii) upper bounds on the classical cost of Boolean function evaluation, given the contextual fraction of the corresponding measurement-based quantum computa…
Quantum systems show contextuality. More precisely, it is impossible to reproduce the quantum-mechanical predictions using a non-contextual realist model, i.e., a model where the outcome of one measurement is independent of the choice of compatible measurements performed in the measurement context. There has been several attempts to quantify the amount of contextuality for specific quantum systems, for example, in the number of rays needed in a KS proof, or the number of terms in certain inequalities, or in the violation, noise sensitivity, and other measures. This paper is about another appro…
Bell non-locality and Kochen-Specker (KS) contextuality are logically independent concepts, fuel different protocols with quantum vs classical advantage, and have distinct classical simulation costs. A natural question is what are the relations between these concepts, advantages, and costs. To address this question, it is useful to have a map that captures all the connections between Bell non-locality and KS contextuality in quantum theory. The aim of this work is to introduce such a map. After defining the theory-independent notions of Bell non-locality and KS contextuality for ideal measurem…
Probe of Generic Quantum Contextuality and Nonlocal Resources for Qubits
Hebei Normal University（河北师范大学）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；South China Normal University（华南师范大学）；The University of Hong Kong（香港大学）；Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）
We introduce a hierarchy of semidefinite relaxations of the set of quantum correlations in generalised contextuality scenarios. This constitutes a simple and versatile tool for bounding the magnitude of quantum contextuality. To illustrate its utility, we use it to determine the maximal quantum violation of several noncontextuality inequalities whose maximum violations were previously unknown. We then go further and use it to prove that certain preparation-contextual correlations cannot be explained with pure states, thereby showing that mixed states are an indispensable resource for contextua…
Quantum computers promise dramatic advantages over their classical counterparts, but the answer to the most basic question "What is the source of the power in quantum computing?" has remained elusive. Here we prove a remarkable equivalence between the onset of contextuality and the possibility of universal quantum computation via magic state distillation. This is a conceptually satisfying link because contextuality provides one of the fundamental characterizations of uniquely quantum phenomena and, moreover, magic state distillation is the leading model for experimentally realizing fault-toler…
Resource theories constitute a powerful theoretical framework and a tool that captures, in an abstract structure, pragmatic aspects of the most varied theories and processes. For physical theories, while this framework deals directly with questions about the concrete possibilities of carrying out tasks and processes, resource theories also make it possible to recast these already established theories on a new language, providing not only new perspectives on the potential of physical phenomena as valuable resources for technological development, for example, but they also provide insights into …
The most well-known tool for studying contextuality in quantum computation is the n-qubit stabilizer state tableau representation. We provide an extension that describes not only the quantum state, but is also outcome deterministic. The extension enables a value assignment to exponentially many Pauli observables, yet remains quadratic in both memory and computational complexity. Furthermore, we show that the mechanisms employed for contextuality and measurement disturbance are wholly separate. The model will be useful for investigating the role of contextuality in $n$-qubit quantum computation…
We present a unified axiomatic approach to contextuality and non-locality based on the fact that both are resource theories. In those theories the main objects are consistent boxes, which can be transformed by certain operations to achieve certain tasks. The amount of resource is quantified by appropriate measures of the resource. Following recent paper [J.I. de Vicente, J. Phys. A: Math. Theor. {\bf 47}, 424017 (2014)], and recent development of abstract approach to resource theories, such as entanglement theory, we propose axioms and welcome properties for operations and measures of resource…
We construct zero-error quantum finite automata (QFAs) for promise problems which cannot be solved by bounded-error probabilistic finite automata (PFAs). Here is a summary of our results: - There is a promise problem solvable by an exact two-way QFA in exponential expected time, but not by any bounded-error sublogarithmic space probabilistic Turing machine (PTM). - There is a promise problem solvable by an exact two-way QFA in quadratic expected time, but not by any bounded-error $ o(\log \log n) $-space PTMs in polynomial expected time. The same problem can be solvable by a one-way Las Vegas …
Quantum Cognition has delivered a number of models for semantic memory, but to date these have tended to assume pure states and projective measurement. Here we relax these assumptions. A quantum inspired model of human word association experiments will be extended using a density matrix representation of human memory and a POVM based upon non-ideal measurements. Our formulation allows for a consideration of key terms like measurement and contextuality within a rigorous modern approach. This approach both provides new conceptual advances and suggests new experimental protocols.
In this paper we consider the problem of minimizing the ancillary systems required to realize an arbitrary strategy of a quantum protocol, with the assistance of classical memory. For this purpose we introduce the notion of memory cost of a strategy, which measures the resources required in terms of ancillary dimension. We provide a condition for the cost to be equal to a given value, and we use this result to evaluate the cost in some special cases. As an example we show that any covariant protocol for the cloning of a unitary transformation requires at most one ancillary qubit. We also prove…
A possible notion of nonclassicality for single systems can be defined on the basis of the notion of memory cost of classically simulating probabilities observed in a temporal sequence of measurements. We further explore this idea in a theory-independent framework, namely, from the perspective of general probability theories (GPTs), which includes classical and quantum theory as special examples. Under the assumption that each system has a finite memory capacity, identified with the maximal number of states perfectly distinguishable with a single measurement, we investigate what are the tempor…
Sequences of compatible quantum measurements can be contextual and any simulation with a classical model accounting for the quantum observations, needs to use some internal memory. In Ref. [New J. Phys. 13, 113011 (2011)], it was shown that simulating the sequences of compatible observables from the Peres-Mermin scenario requires at least three different memory states in order to be not in contradiction with the deterministic predictions of quantum theory. We extend this analysis to the probabilistic quantum predictions and ask how much memory is required to simulate the quantum correlations g…
We consider memoryless quantum communication protocols, where the two parties do not possess any memory besides their classical input and they take turns performing unitary operations on a pure quantum state that they exchange between them. Most known quantum protocols are of this type and recently a deep connection between memoryless protocols and Bell inequality violations has been explored recently by Buhrman et al. We study the information cost of memoryless quantum protocols by looking at a canonical problem: bounded-round quantum communication protocols for the one-bit AND function. We p…
According to Landauer's principle, erasing one bit of information incurs a minimum energy cost. Recently, Vaccaro and Barnett (VB) explored information erasure within the context of generalized Gibbs ensembles and demonstrated that for energy-degenerate spin reservoirs, the cost of erasure can be solely in terms of a minimum amount of spin angular momentum and no energy. As opposed to the Landauer case, the cost of erasure in this case is associated with the discrete variable. Here we study the {\it discrete} fluctuations in this cost and the probability of violation of the VB bound. We also o…
Landauer argued that the process of erasing the information stored in a memory device incurs an energy cost in the form of a minimum amount of mechanical work. We find, however, that this energy cost can be reduced to zero by paying a cost in angular momentum or any other conserved quantity. Erasing the memory of Maxwell's demon in this way implies that work can be extracted from a single thermal reservoir at a cost of angular momentum and an increase in total entropy. The implications of this for the second law of thermodynamics are assessed.
Loading data in a quantum device is required in several quantum computing applications. Without an efficient loading procedure, the cost to initialize the algorithms can dominate the overall computational cost. A circuit-based quantum random access memory named FF-QRAM can load M n-bit patterns with computational cost O(CMn) to load continuous data where C depends on the data distribution. In this work, we propose a strategy to load continuous data without post-selection with computational cost O(Mn). The proposed method is based on the probabilistic quantum memory, a strategy to load binary d…
In 1961, Rolf Landauer pointed out that resetting a binary memory requires a minimum energy of $k_BT \ln(2)$. However, once written, any memory is doomed to loose its content if no action is taken. To avoid memory losses, a refresh procedure is periodically performed. In this paper we present a theoretical model and an experiment on a micro-electro-mechanical system to evaluate the minimum energy required to preserve one bit of information over time. Two main conclusions are drawn: i) in principle the energetic cost to preserve information for a fixed time duration with a given error probabili…
Quantum coherence allows for reduced-memory simulators of classical processes. Using recent results in single-shot quantum thermodynamics, we derive a minimal work cost rate for quantum simulators that is quasistatically attainable in the limit of asymptotically-infinite parallel simulation. Comparing this cost with the classical regime reveals that quantizing classical simulators not only results in memory compression but also in reduced dissipation. We explore this advantage across a suite of representative examples.
Thermodynamics with multiple-conserved quantities offers a promising direction for designing novel devices. For example, Vaccaro and Barnett's [J. A. Vaccaro and S. M. Barnett, Proc. R. Soc. A 467, 1770 (2011); S. M. Barnett and J. A. Vaccaro, Entropy 15, 4956 (2013)] proposed information erasure scheme, where the cost of erasure is solely in terms of a conserved quantity other than energy, allows for new kinds of heat engines. In recent work, we studied the discrete fluctuations and average bounds of the erasure cost in spin angular momentum. Here we clarify the costs in terms of the spin equ…
Landauer's erasure principle puts a fundamental constraint on the amount of work required to erase information using thermal reservoirs. Recently this bound was improved to include corrections for finite-sized thermal reservoirs. In conventional information-erasure schemes, conservation of energy plays a key role with the cost of erasure. However, it has been shown that erasure can be achieved through the manipulation of spin angular momentum rather than energy, using a reservoir composed of energy-degenerate spin particles under the constraint of the conservation of spin angular momentum, in …
Modularity dissipation identifies how locally-implemented computation entails costs beyond those required by Landauer's bound on thermodynamic computing. We establish a general theorem for efficient local computation, giving the necessary and sufficient conditions for a local operation to have zero modularity cost. Applied to thermodynamically-generating stochastic processes it confirms a conjecture that classical generators are efficient if and only if they satisfy retrodiction, which places minimal memory requirements on the generator. This extends immediately to quantum computation: Any qua…
