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Single and Entangled Atomic Systems in Thermal Bath and the Fulling-Davies-Unruh Effect
Full text linkWe revisit the Fulling-Davies-Unruh effect in the context of two-level single and entangled atomic systems that are static in a thermal bath. We consider the interaction between the systems and a massless scalar field, covering the scenarios of free space as well as within a cavity. Through the calculation of atomic transition rates it is found that in free space there is an equivalence between the upward and downward transition rates of an uniformly accelerated atom with respect to an observer with that of a single atom which is static with respect to the observer and immersed in a thermal bath, as long as the temperature of the thermal bath matches the Unruh temperature. This equivalence between the upward and downward transition rates breaks down in the presence of a cavity. For two-atom systems, considering the initial state to be in a general pure entangled form, we find that in this case the equivalence between the upward and downward transition rates of the accelerated and static thermal bath scenarios holds only under specific limiting conditions in free space, but breaks down completely in a cavity setup.Quanta 2025; 14: 1–27
Uncertainty of Quantum States
No full textThe uncertainty of a quantum state is given by the composition of two components. The first is called the quantum component and is given by the probability distribution of an observable relative to the state. The second is the classical component which is an uncertainty function that is applied to the first component. We characterize uncertainty functions in terms of four axioms. We then study four examples called variance, entropy, geometric and sine uncertainty functions. The final section presents the general theory of state uncertainty.Quanta 2025; 14: 28–37
Tunneling Probability of Quantum Wavepacket in Time-Dependent Potential Well
Full text linkQuantum tunneling of particles plays an important role in many chemical reactions. Studying quantum tunneling in time-dependent potential wells is tricky since most of the available solutions of time-dependent potential wells are coupled to specific properties of the Hamiltonian. Here, we investigate the tunneling probability of a quantum wavepacket in time-dependent potential well by using the split operator method. This numerical method can give us an overview of the tunneling probability of a quantum wavepacket for any temporal change in the shape of the potential well. We study a time-dependent potential well model evolving from symmetric to asymmetric quartic double well since quartic potential wells resemble certain practically available potential well models.Quanta 2024; 13: 11–19
Conditional Effects, Observables and Instruments
Full text linkWe begin with a study of operations and the effects they measure. We define the probability that an effect occurs when the system is in a state by . If and is an operation that measures , we define the conditional probability of an effect given relative to by . We characterize when Bayes' quantum second rule holds. We then consider Lüders and Holevo operations. We next discuss instruments and the observables they measure. If and are observables and an instrument measures , we define the observable conditioned on relative to and denote it by . Using these concepts, we introduce Bayes' quantum first rule. We observe that this is the same as the classical Bayes' first rule, except it depends on the instrument used to measure . We then extend this to Bayes' quantum first rule for expectations. We show that two observables and are jointly commuting if and only if there exists an atomic observable such that and . We next obtain a general uncertainty principle for conditioned observables. Finally, we discuss observable conditioned quantum entropies. The theory is illustrated with many examples.Quanta 2024; 13: 1–10
On Classical Simulation of Quantum Circuits Composed of Clifford Gates
Full text linkThe Gottesman–Knill theorem asserts that quantum circuits composed solely of Clifford gates can be efficiently simulated classically. This theorem hinges on the fact that Clifford gates map Pauli strings to other Pauli strings, thereby allowing for a structured simulation process using classical computations. In this work, we break down the step-by-step procedure of the Gottesman–Knill theorem in a beginner-friendly manner, leveraging concepts such as matrix products, tensor products, commutation, anti-commutation, eigenvalues, and eigenvectors of quantum mechanical operators. Through detailed examples illustrating superposition and entanglement phenomena, we aim to provide a clear understanding of the classical simulation of Clifford gate-based quantum circuits. While we do not provide a formal proof of the theorem, we offer intuitive physical insights at each stage where necessary, empowering readers to grasp the fundamental principles underpinning this intriguing aspect of quantum computation.Quanta 2024; 13: 20–27
Penrose Dodecahedron, Witting Configuration and Quantum Entanglement
Full text linkA model with two entangled spin-3/2 particles based on geometry of dodecahedron was suggested by Roger Penrose for formulation of analogue of Bell theorem without probabilities. The model was later reformulated using so-called Witting configuration with 40 rays in 4D Hilbert space. However, such reformulation needs for some subtleties related with entanglement of two such configurations essential for consideration of non-locality and some other questions. Two entangled systems with quantum states described by Witting configurations are discussed in the present work. Duplication of points with respect to vertices of dodecahedron produces rather significant increase with number of symmetries in 25920/60=432 times. Quantum circuits model is a natural language for description of operations with different states and measurements of such systems.Quanta 2024; 13: 38–46
Truncated Modular Exponentiation Operators: A Strategy for Quantum Factoring
Full text linkModular exponentiation (ME) operators are one of the fundamental components of Shor's algorithm, and the place where most of the quantum resources are deployed. These operators are often referred to as the bottleneck of the algorithm. I propose a method for constructing the ME operators that requires only qubits with no ancillary qubits. The method relies upon the simple observation that the work register starts in state . Therefore, we do not have to create an ME operator that accepts a general input, but rather, one that takes an input from the periodic sequence of states for . Here, the ME function with base is defined by and has a period of . For an -bit number , the operator can be partitioned into levels, where the gates in level increment the state to the state . The gates below do not affect the state . This amounts to transforming an -bit binary number into another binary number , without altering the previous states, which can be accomplished by a set of formal rules involving multi-control-NOT gates and single-qubit NOT gates. The process of gate construction can therefore be automated, which is essential for factoring larger numbers. The obvious problem with this method is that it is self-defeating: If we knew the operator , then we would know the period of the ME function, and there would be no need for Shor's algorithm. I show, however, that the ME operators are very forgiving, and truncated approximate forms in which levels have been omitted are able to extract factors just as well as the exact operators. I demonstrate this by factoring the numbers by using less than half the requisite number of levels in the ME operators. This procedure works because the method of continued fractions only requires an approximate phase value, which suggests that implementing Shor's algorithm might not be as difficult as first suspected. This is the basis for a factorization strategy in which one level at a time is iterated over using an automated script. In this way, we fill the circuits for the ME operators with more and more gates, and the correlations between the various composite operators (where is a power of two) compensate for the missing levels.Quanta 2024; 13: 47–82
Josephson Oscillations of Two Weakly Coupled Bose-Einstein Condensates
Full text linkA numerical experiment based on a particle number-conserving quantum field theory is performed for two initially independent Bose–Einstein condensates that are coherently coupled at two temperatures. The present model illustrates ab initio that the initial phase of each of the two condensates does not remain random at the Boltzmann equilibrium, but is distributed around integer multiple values of 2π from the interference and thermalization of forward and backward propagating matter waves. The thermalization inside the atomic vapors can be understood as an intrinsic measurement process that defines a temperature for the two condensates and projects the quantum states to an average wave field with zero (relative) phases. Following this approach, focus is put on the original thought experiment of Anderson on whether a Josephson current between two initially separated Bose–Einstein condensates occurs in a deterministic way or not, depending on the initial phase distribution.Quanta 2024; 13: 28–37
The Enigmas of Fluctuations of the Universal Quantum Fields
Full text linkThe primary ingredients of reality are the universal quantum fields, which fluctuate persistently, spontaneously, and randomly. The general perception of the scientific community is that these quantum fluctuations are due to the uncertainty principle. Here, we present cogent arguments to show that the uncertainty principle is a consequence of the quantum fluctuations, but not their cause. This poses a conspicuous enigma as to how the universal fields remain immutable with an expectation value so accurate that it leads to experimental results, which are precise to one part in a trillion. We discuss some reasonable possibilities in the absence of a satisfactory solution to this enigma.Quanta 2023; 12: 190–201
Shor's Factoring Algorithm and Modular Exponentiation Operators
Full text linkWe provide a pedagogical presentation of Shor's factoring algorithm, which is a quantum algorithm for factoring very large numbers (of order of hundreds to thousands of bits) in polynomial time. In contrast, all known classical algorithms for the factoring problem take an exponential time to factor such large numbers. Shor's algorithm therefore has profound implication for public-key encryption such as RSA and Diffie–Hellman key exchange. We assume no prior knowledge of Shor's algorithm beyond a basic familiarity with the circuit model of quantum computing. Shor's algorithm contains a number of moving parts, and can be rather daunting at first. The literature is replete with derivations and expositions of Shor's algorithm, but most of them seem to be lacking in essential details, and none of them provide a pedagogical presentation. They require a thicket of appendices and assume a knowledge of quantum algorithms and classical mathematics with which the reader might not be familiar. We therefore start with first principle derivations of the quantum Fourier transform (QFT) and quantum phase estimation (QPE), which are the essential building blocks of Shor's algorithm. We then go on to develop the theory of modular exponentiation (ME) operators, one of the fundamental components of Shor's algorithm, and the place where most of the quantum resources are deployed. We also delve into the number theory that establishes the link between factorization and the period of the modular exponential function. We then apply the QPE algorithm to obtain Shor's factoring algorithm. We also discuss the post-quantum processing and the method of continued fractions, which is used to extract the exact period of the modular exponential function from the approximately measured phase angles of the ME operator. The manuscript then moves on to a series of examples. We first verify the formalism by factoring N=15, the smallest number accessible to Shor's algorithm. We then proceed to factor larger integers, developing a systematic procedure that will find the ME operators for any semi-prime N=p×q (where q and p are prime). Finally, we factor the composite numbers N=21, 33, 35, 143, 247 using the Qiskit simulator. It is observed that the ME operators are somewhat forgiving, and truncated approximate forms are able to extract factors just as well as the exact operators. This is because the method of continued fractions only requires an approximate phase value for its input, which suggests that implementing Shor's algorithm might not be as difficult as first suspected.Quanta 2023; 12: 41–130