Quanta
type: channel
description: A peer-reviewed, open access journal in quantum mechanics, foundations of physics, mathematical physics & philosophy of science.
publisher: Danko Georgiev
language: en
publication name: Quanta
issn: 1314-7374
items: The Enigma of Delayed Choice Quantum Eraser
items: Geometric Algebras and Fermion Quantum Field Theory
items: On the Physical Basis for the Incomparability of NP and BQP
items: Uncertainty of Quantum States
items: Single and Entangled Atomic Systems in Thermal Bath and the Fulling-Davies-Unruh Effect
items: Truncated Modular Exponentiation Operators: A Strategy for Quantum Factoring
items: Penrose Dodecahedron, Witting Configuration and Quantum Entanglement
items: Josephson Oscillations of Two Weakly Coupled Bose-Einstein Condensates
items: On Classical Simulation of Quantum Circuits Composed of Clifford Gates
items: Tunneling Probability of Quantum Wavepacket in Time-Dependent Potential Well
items: Conditional Effects, Observables and Instruments
items: The Enigmas of Fluctuations of the Universal Quantum Fields
items: On Weak Values and Feynman's Blind Alley
items: Monte Carlo Simulation for the Frequency Comb Spectrum of an Atom Laser
items: Quantum Mechanics with Real Numbers: Entanglement, Superselection Rules and Gauges
items: Digital Simulation of Single Qubit Markovian Open Quantum Systems: A Tutorial
items: Shor's Factoring Algorithm and Modular Exponentiation Operators
items: A Theory of Quantum Instruments
items: Evaluation of the Feynman Propagator by Means of the Quantum Hamilton-Jacobi Equation
items: Role of Steering Inequality in Quantum Key Distribution Protocol
items: Can Decoherence Solve the Measurement Problem?
items: Clifford Algebras, Spin Groups and Qubit Trees
items: The Physical Meaning of the Holographic Principle
items: Steady State in Ultrastrong Coupling Regime: Expansion and First Orders
items: The Quantum Hamilton–Jacobi Equation and the Link Between Classical and Quantum Mechanics
items: On the Role of Inconsistency in Quantum Foundational Debate and Hilbert Space Formulation
items: Dual Instruments and Sequential Products of Observables
items: "Mysteries" of modern physics and the fundamental constants c, h, and G
items: Jordan in the Church of the Higher Hilbert Space: Entanglement and Thermal Fluctuations
items: George Sudarshan: Perspectives and Legacy

The Enigma of Delayed Choice Quantum Eraser
type: item
description: The delayed-choice quantum eraser represents an interesting experiment that exemplifies Bohr's principle of complementarity in a beautiful way. According to the complementarity principle, in a two-path interference experiment, the knowledge of which path was taken by the particle and the appearance of interference are mutually exclusive. Even when the which-path information is merely retained in specific quantum path-markers, without being actually read, it suffices to eliminate interference. Nevertheless, if this path information is erased in some manner, the interference re-emerges, a phenomenon referred to as the quantum eraser. An intriguing aspect of this experiment is that if the path information is erased after the particle has been detected on the screen, the interference still reappears, a phenomenon known as the delayed-choice quantum eraser. This observation has led to the interpretation that the particle can be influenced to exhibit characteristics of either a particle or a wave based on a decision made long after it has been registered on the screen. This idea has sparked considerable debate and discussions surrounding retrocausality. This controversy is reviewed here, and a detailed resolution provided. Quanta 2025; 14: 66–74.
subject: Articles
  topic: section
creator: Tabish Qureshi
rights: Copyright (c) 2025 Tabish Qureshi (Author) https://creativecommons.org/licenses/by/4.0
date: 2025-11-23
publication date: 2025-11-23
starting page: 66
ending page: 74
doi: 10.12743/quanta.93

Geometric Algebras and Fermion Quantum Field Theory
type: item
description: Corresponding to a finite dimensional Hilbert space $H$ with $\dim H=n$, we define a geometric algebra $\mathcal{G}(H)$ with $\dim\left[\mathcal{G}(H)\right]=2^n$. The algebra $\mathcal{G}(H)$ is a Hilbert space that contains $H$ as a subspace. We interpret the unit vectors of $H$ as states of individual fermions of the same type and $\mathcal{G}(H)$ as a fermion quantum field whose unit vectors represent states of collections of interacting fermions. We discuss creation operators on $\mathcal{G}(H)$ and provide their matrix representations. Evolution operators provided by self-adjoint Hamiltonians on $H$ and $\mathcal{G}(H)$ are considered. Boson-Fermion quantum fields are constructed. Extensions of operators from $H$ to $\mathcal{G}(H)$ are studied. Finally, we present a generalization of our work to infinite dimensional separable Hilbert spaces. Quanta 2025; 14: 48–65.
subject: Articles
  topic: section
creator: Stan Gudder
rights: Copyright (c) 2025 Stan Gudder (Author) https://creativecommons.org/licenses/by/4.0
date: 2025-07-26
publication date: 2025-07-26
starting page: 48
ending page: 65
doi: 10.12743/quanta.92

On the Physical Basis for the Incomparability of NP and BQP
type: item
description: What does it mean for the physically motivated computational complexity class $\mathcal{BQP}$ and the logically motivated class $\mathcal{NP}$ to be incomparable, from a physical perspective? Although it is unlikely that quantum computers can solve $\mathcal{NP}$-complete problems in polynomial time, yet it may be instructive to see how a physical theory (denoted $\Theta^{(\mathcal{NP})}$ or $\Theta^{(\mathcal{PSPACE})}$) for which $\mathcal{NP}$ or even $\mathcal{PSPACE}$, respectively, constitutes an efficient model of computation, would compare with quantum theory. The present work explores this question, after briefly reviewing some complexity results indicative of the incomparability of $\mathcal{BQP}$ and $\mathcal{NP}$. We argue that theories $\Theta^{(\mathcal{NP})}$ or $\Theta^{(\mathcal{PSPACE})}$ effectively feature degrees of nonlinearity. Thus, the linearity of quantum theory prevents complete problems in these complexity classes to be efficiently solvable on a quantum computer. On the other hand, the exponentiality of quantum theory endows quantum computers with a certain strength going beyond $\mathcal{NP}$ or even $\mathcal{PH}$. Thus the contrasting roles of the empowerment provided by quantum mechanical exponentiality and the limitation imposed by quantum mechanical linearity seem to be at the heart of the incomparability of the classes $\mathcal{BQP}$ and $\mathcal{NP}$. Quanta 2025; 14: 38–47.
subject: Articles
  topic: section
creator: Vivek Kumar
creator: Maheshwari Prasad Singh
creator: Srikanth Radhakrishna
rights: Copyright (c) 2025 Vivek Kumar, Maheshwari Prasad Singh, Srikanth Radhakrishna (Author) https://creativecommons.org/licenses/by/4.0
date: 2025-06-22
publication date: 2025-06-22
starting page: 38
ending page: 47
doi: 10.12743/quanta.91

Uncertainty of Quantum States
type: item
description: The 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.
subject: Articles
  topic: section
creator: Stan Gudder
rights: Copyright (c) 2025 Stan Gudder (Author) https://creativecommons.org/licenses/by/4.0
date: 2025-04-06
publication date: 2025-04-06
starting page: 28
ending page: 37
doi: 10.12743/quanta.v14i1.295

Single and Entangled Atomic Systems in Thermal Bath and the Fulling-Davies-Unruh Effect
type: item
description: We 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.
subject: Articles
  topic: section
creator: Arnab Mukherjee
creator: Sunandan Gangopadhyay
creator: Archan S. Majumdar
rights: Copyright (c) 2025 Arnab Mukherjee, Sunandan Gangopadhyay, Archan S. Majumdar (Author) https://creativecommons.org/licenses/by/4.0
date: 2025-01-08
publication date: 2025-01-08
starting page: 1
ending page: 27
doi: 10.12743/quanta.v14i1.280

Truncated Modular Exponentiation Operators: A Strategy for Quantum Factoring
type: item
description: Modular 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 $3n + 1$ qubits with no ancillary qubits. The method relies upon the simple observation that the work register starts in state $\vert 1 \rangle$. Therefore, we do not have to create an ME operator $U$ that accepts a general input, but rather, one that takes an input from the periodic sequence of states $\vert f(x) \rangle$ for $x \in \{0, 1, \cdots, r-1\}$. Here, the ME function with base $a$ is defined by $f(x) = a^x ~({\textrm mod}~N)$ and has a period of $r$. For an $n$-bit number $N$, the operator $U$ can be partitioned into $r$ levels, where the gates in level $x \in \{0, 1, \cdots, r-1\}$ increment the state $\vert f(x) \rangle = \vert w_{n-1} \cdots w_1 w_0 \rangle$ to the state $\vert f(x+1) \rangle = \vert w_{n-1}^\prime \cdots w_1^\prime w_0^\prime\rangle$. The gates below $x$ do not affect the state $\vert f(x+1) \rangle$. This amounts to transforming an $n$-bit binary number $w_{n-1} \cdots w_1 w_0$ into another binary number $w_{n-1}^\prime \cdots w_1^\prime w_0^\prime$, 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 $U$, then we would know the period $r$ 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 …
subject: Articles
  topic: section
creator: Robert L. Singleton Jr
rights: Copyright (c) 2024 Robert L. Singleton Jr (Author) https://creativecommons.org/licenses/by/4.0
date: 2024-12-30
publication date: 2024-12-30
starting page: 47
ending page: 82
doi: 10.12743/quanta.v13i1.277

Penrose Dodecahedron, Witting Configuration and Quantum Entanglement
type: item
description: A 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.
subject: Articles
  topic: section
creator: Alexander Yurievich Vlasov
rights: Copyright (c) 2024 Alexander Yurievich Vlasov (Author) https://creativecommons.org/licenses/by/4.0
date: 2024-12-15
publication date: 2024-12-15
starting page: 38
ending page: 46
doi: 10.12743/quanta.v13i1.276

Josephson Oscillations of Two Weakly Coupled Bose-Einstein Condensates
type: item
description: A 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.
subject: Articles
  topic: section
creator: Alexej Schelle
rights: Copyright (c) 2024 Alexej Schelle (Author) https://creativecommons.org/licenses/by/4.0
date: 2024-08-15
publication date: 2024-08-15
starting page: 28
ending page: 37
doi: 10.12743/quanta.v13i1.266

On Classical Simulation of Quantum Circuits Composed of Clifford Gates
type: item
description: The 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.
subject: Articles
  topic: section
creator: George Biswas
rights: Copyright (c) 2024 George Biswas (Author) https://creativecommons.org/licenses/by/4.0
date: 2024-07-29
publication date: 2024-07-29
starting page: 20
ending page: 27
doi: 10.12743/quanta.v13i1.265

Tunneling Probability of Quantum Wavepacket in Time-Dependent Potential Well
type: item
description: Quantum 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.
subject: Articles
  topic: section
creator: Aarouj Ahmed
rights: Copyright (c) 2024 Aarouj Ahmed (Author) https://creativecommons.org/licenses/by/4.0
date: 2024-05-03
publication date: 2024-05-03
starting page: 11
ending page: 19
doi: 10.12743/quanta.v13i1.246

Conditional Effects, Observables and Instruments
type: item
description: We begin with a study of operations and the effects they measure. We define the probability that an effect $a$ occurs when the system is in a state $\rho$ by $P_{\rho}(a)=\textrm{Tr}(\rho a)$. If $P_{\rho}(a)\ne0$ and $\mathcal{I}$ is an operation that measures $a$, we define the conditional probability of an effect $b$ given $a$ relative to $\mathcal{I}$ by $P_{\rho}(b\mid a)=\textrm{Tr}[\mathcal{I}(\rho)b]/P_{\rho}(a)$. We characterize when Bayes' quantum second rule $P_{\rho}(b\mid a)=\frac{P_{\rho}(b)}{P_{\rho}(a)}\,P_{\rho}(a\mid b)$ holds. We then consider Lüders and Holevo operations. We next discuss instruments and the observables they measure. If $A$ and $B$ are observables and an instrument $\mathcal{I}$ measures $A$, we define the observable $B$ conditioned on $A$ relative to $\mathcal{I}$ and denote it by $(B\mid A)$. 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 $A$. We then extend this to Bayes' quantum first rule for expectations. We show that two observables $B$ and $C$ are jointly commuting if and only if there exists an atomic observable $A$ such that $B=(B\mid A)$ and $C=(C\mid A)$. 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.
subject: Articles
  topic: section
creator: Stan Gudder
rights: Copyright (c) 2025 Stan Gudder (Author) https://creativecommons.org/licenses/by/4.0
date: 2024-02-06
publication date: 2024-02-06
starting page: 1
ending page: 10
doi: 10.12743/quanta.v13i1.259

The Enigmas of Fluctuations of the Universal Quantum Fields
type: item
description: The 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.
subject: Articles
  topic: section
creator: Mani Lal Bhaumik
rights: Copyright (c) 2023 Mani Lal Bhaumik (Author) https://creativecommons.org/licenses/by/4.0
date: 2023-12-10
publication date: 2023-12-10
starting page: 190
ending page: 201
doi: 10.12743/quanta.v12i1.250

On Weak Values and Feynman's Blind Alley
type: item
description: Feynman famously recommended accepting the basic principles of quantum mechanics without trying to guess the machinery behind the law. One of the corollaries of the Uncertainty Principle is that the knowledge of probability amplitudes does not allow one to make meaningful statements about the past of an unobserved quantum system. A particular type of reasoning, based on weak values, appears to do just that. Has Feynman been proven wrong by the more recent developments? Most likely not. Quanta 2023; 12: 180–189.
subject: Articles
  topic: section
creator: Dmitri Sokolovski
rights: Copyright (c) 2023 Dmitri Sokolovski (Author) https://creativecommons.org/licenses/by/4.0
date: 2023-11-19
publication date: 2023-11-19
starting page: 180
ending page: 189
doi: 10.12743/quanta.v12i1.249

Monte Carlo Simulation for the Frequency Comb Spectrum of an Atom Laser
type: item
description: A theoretical particle-number conserving quantum field theory based on the concept of imaginary time is presented and applied to the scenario of a coherent atomic laser field at ultra-cold temperatures. The proposed theoretical model describes the analytical derivation of the frequency comb spectrum for an atomic laser realized from modeling a coherent atomic beam of condensate and non-condensate quantum field components released from a trapped Bose–Einstein condensate at a given repetition phase and frequency. The condensate part of the atomic vapor is assumed to be subjected to thermal noise induced by the temperature of the surrounding thermal atomic cloud. This new quantum approach uses time periodicity and an orthogonal decomposition of the quantum field in a complex-valued quantum field representation to derive and model the quantum field's forward- and backward-propagating components as a standing wave field in the same unique time and temperature domain without quantitative singularities at finite temperatures. The complex-valued atom laser field, the resulting frequency comb, and the repetition frequency distribution with the varying shape of envelopes are numerically monitored within a Monte Carlo sampling method, as a function of temperature and trap frequency of the external confinement. Quanta 2023; 12: 171–179.
subject: Articles
  topic: section
creator: Alexej Schelle
rights: Copyright (c) 2023 Alexej Schelle (Author) https://creativecommons.org/licenses/by/4.0
date: 2023-11-16
publication date: 2023-11-16
starting page: 171
ending page: 179
doi: 10.12743/quanta.v12i1.243

Quantum Mechanics with Real Numbers: Entanglement, Superselection Rules and Gauges
type: item
description: We show how imaginary numbers in quantum physics can be eliminated by enlarging the Hilbert space followed by an imposition of—what effectively amounts to—a superselection rule. We illustrate this procedure with a qubit and apply it to the Mach–Zehnder interferometer. The procedure is somewhat reminiscent of the constrained quantization of the electromagnetic field, where, in order to manifestly comply with relativity, one enlarges the Hilbert Space by quantizing the longitudinal and scalar modes, only to subsequently introduce a constraint to make sure that they are actually not directly observable. Quanta 2023; 12: 164–170.
subject: Articles
  topic: section
creator: Vlatko Vedral
rights: Copyright (c) 2023 Vlatko Vedral (Author) https://creativecommons.org/licenses/by/4.0
date: 2023-09-24
publication date: 2023-09-24
starting page: 164
ending page: 170
doi: 10.12743/quanta.v12i1.241

Digital Simulation of Single Qubit Markovian Open Quantum Systems: A Tutorial
type: item
description: One of the first proposals for the use of quantum computers was the simulation of quantum systems. Over the past three decades, great strides have been made in the development of algorithms for simulating closed quantum systems and the more complex open quantum systems. In this tutorial, we introduce the methods used in the simulation of single qubit Markovian open quantum systems. It combines various existing notations into a common framework that can be extended to more complex open system simulation problems. The only currently available algorithm for the digital simulation of single qubit open quantum systems is discussed in detail. A modification to the implementation of the simpler channels is made that removes the need for classical random sampling, thus making the modified algorithm a strictly quantum algorithm. The modified algorithm makes use of quantum forking to implement the simpler channels that approximate the total channel. This circumvents the need for quantum circuits with a large number of CNOT gates. Quanta 2023; 12: 131–163.
subject: Articles
  topic: section
creator: Ian Joel David
creator: Ilya Sinayskiy
creator: Francesco Petruccione
rights: Copyright (c) 2023 Ian Joel David, Ilya Sinayskiy, Francesco Petruccione (Author) https://creativecommons.org/licenses/by/4.0
date: 2023-09-18
publication date: 2023-09-18
starting page: 131
ending page: 163
doi: 10.12743/quanta.v12i1.226

Shor's Factoring Algorithm and Modular Exponentiation Operators
type: item
description: We 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 larg…
subject: Articles
  topic: section
creator: Robert L. Singleton Jr
rights: Copyright (c) 2023 Robert L. Singleton Jr (Author) https://creativecommons.org/licenses/by/4.0
date: 2023-09-15
publication date: 2023-09-15
starting page: 41
ending page: 130
doi: 10.12743/quanta.v12i1.235

A Theory of Quantum Instruments
type: item
description: Until recently, a quantum instrument was defined to be a completely positive operation-valued measure from the set of states on a Hilbert space to itself. In the last few years, this definition has been generalized to such measures between sets of states from different Hilbert spaces called the input and output Hilbert spaces. This article presents a theory of such instruments. Ways that instruments can be combined such as convex combinations, post-processing, sequential products, tensor products and conditioning are studied. We also consider marginal, reduced instruments and how these are used to define coexistence (compatibility) of instruments. Finally, we present a brief introduction to quantum measurement models where the generalization of instruments is essential. Many of the concepts of the theory are illustrated by examples. In particular, we discuss Holevo and Kraus instruments. Quanta 2023; 12: 27–40.
subject: Articles
  topic: section
creator: Stan Gudder
rights: Copyright (c) 2023 Stan Gudder (Author) https://creativecommons.org/licenses/by/4.0
date: 2023-06-06
publication date: 2023-06-06
starting page: 27
ending page: 40
doi: 10.12743/quanta.v12i1.233

Evaluation of the Feynman Propagator by Means of the Quantum Hamilton-Jacobi Equation
type: item
description: It is shown that the complex phase of the Feynman propagator is a solution of the quantum Hamilton–Jacobi equation, namely, it is the quantum Hamilton's principal function (or quantum action). Therefore, the Feynman propagator can be computed either by means of the path integration, or by the way of the Hamilton–Jacobi equation. This is analogous to what happens in classical mechanics, where the Hamilton's principal function can be computed either by integrating the Lagrangian along the extremal paths, or as a solution of partial differential equation, namely the classical Hamilton–Jacobi equation. If the path is decomposed in the classical one and quantum fluctuations, the contribution of these quantum fluctuations satisfies a non-linear partial differential equation, whose coefficients depend on the classical action. When the contribution of the quantum fluctuations depend only on the time, it can be computed by means of a simple integration. The final results for the propagators in this case are equal to the Van Vleck–Pauli–Morette expressions, even though the two derivations are quite different. Quanta 2023; 12: 22–26.
subject: Articles
  topic: section
creator: Mario Fusco Girard
rights: Copyright (c) 2023 Mario Fusco Girard (Author) https://creativecommons.org/licenses/by/4.0
date: 2023-04-24
publication date: 2023-04-24
starting page: 22
ending page: 26
doi: 10.12743/quanta.v12i1.223

Role of Steering Inequality in Quantum Key Distribution Protocol
type: item
description: Violation of Bell's inequality has been the mainspring for secure key generation in an entanglement assisted Quantum Key Distribution (QKD) protocol. Various contributions have relied on the violation of Bell inequalities to build an appropriate QKD protocol. Residing between Bell nonlocality and entanglement, there exists a hybrid trait of correlations, namely correlations exhibited through the violation of steering inequalities. However, such correlations have not been put to use in QKD protocols as much as their stronger counterpart, the Bell violations. In the present work, we show that the violations of the Cavalcanti–Jones–Wiseman–Reid (CJWR) steering inequalities can act as key ingredients in an entanglement assisted QKD protocol. We work with arbitrary two-qubit entangled states, and characterize them by their utility in such protocols. The characterization is based on the quantum bit error rate and violation of the CJWR inequality. Furthermore, we show that subsequent applications of local filtering operations on initially entangled states exhibiting no violation, lead to violations necessary for the successful implementation of the protocol. An additional vindication of our protocol is provided by the use of absolutely Bell–Clauser–Horne–Shimony–Holt (Bell–CHSH) local states, states which remain Bell–CHSH local even under global unitary operations. Quanta 2023; 12: 1–21.
subject: Articles
  topic: section
creator: Kaushiki Mukherjee
creator: Tapaswini Patro
creator: Nirman Ganguly
rights: Copyright (c) 2025 Kaushiki Mukherjee, Tapaswini Patro, Nirman Ganguly (Author) https://creativecommons.org/licenses/by/4.0
date: 2023-04-18
publication date: 2023-04-18
starting page: 1
ending page: 21
doi: 10.12743/quanta.v12i1.210

Can Decoherence Solve the Measurement Problem?
type: item
description: The quantum decoherence program has become more attractive in providing an acceptable solution for the long-standing quantum measurement problem. Decoherence by quantum entanglement happens very quickly to entangle the quantum system with the environment including the detector. But in the final stage of measurement, acquiring the unentangled pointer states poses some problems. Recent experimental observations of the effect of the ubiquitous quantum vacuum fluctuations in destroying quantum entanglement appears to provide a solution. Quanta 2022; 11: 115–123.
subject: Articles
  topic: section
creator: Mani Lal Bhaumik
rights: Copyright (c) 2022 Mani Lal Bhaumik (Author) https://creativecommons.org/licenses/by/4.0
date: 2022-12-04
publication date: 2022-12-04
starting page: 115
ending page: 123
doi: 10.12743/quanta.v11i1.208

Clifford Algebras, Spin Groups and Qubit Trees
type: item
description: Representations of Spin groups and Clifford algebras derived from the structure of qubit trees are introduced in this work. For ternary trees the construction is more general and reduction to binary trees is formally defined by deletion of superfluous branches. The usual Jordan–Wigner construction also may be formally obtained in this approach by bringing the process up to trivial qubit chain (trunk). The methods can also be used for effective simulation of some quantum circuits corresponding to the binary tree structure. The modeling of more general qubit trees, as well as the relationship with the mapping used in the Bravyi–Kitaev transformation, are also briefly discussed. Quanta 2022; 11: 97–114.
subject: Articles
  topic: section
creator: Alexander Yurievich Vlasov
rights: Copyright (c) 2022 Alexander Yurievich Vlasov (Author) https://creativecommons.org/licenses/by/4.0
date: 2022-12-01
publication date: 2022-12-01
starting page: 97
ending page: 114
doi: 10.12743/quanta.v11i1.199

The Physical Meaning of the Holographic Principle
type: item
description: We show in this pedagogical review that far from being an apparent law of physics that stands by itself, the holographic principle is a straightforward consequence of the quantum information theory of separable systems. It provides a basis for the theories of measurement, time, and scattering. Utilizing the notion of holographic screens, which are information encoding boundaries between physical subsystems, we demonstrate that the physical interaction is an information exchange during which information is strictly conserved. Then we use generalized holographic principle in order to flesh out a fully-general quantum theory of measurement in which the measurement produces finite-resolution, classical outcomes. Further, we show that the measurements are given meaning by quantum reference frames and sequential measurements induce topological quantum field theories. Finally, we discuss principles equivalent to the holographic principle, including Markov blankets and the free-energy principle in biology, multiple realizability and virtual machines in computer science, and active inference and interface theories in cognitive science. This appearance in multiple disciplines suggests that the holographic principle is not just a fundamental principle of physics, but of all of science. Quanta 2022; 11: 72–96.
subject: Articles
  topic: section
creator: Chris Fields
creator: James F. Glazebrook
creator: Antonino Marcianò
rights: Copyright (c) 2022 Chris Fields, James F. Glazebrook, Antonino Marcianò (Author) https://creativecommons.org/licenses/by/4.0
date: 2022-11-21
publication date: 2022-11-21
starting page: 72
ending page: 96
doi: 10.12743/quanta.v11i1.206

Steady State in Ultrastrong Coupling Regime: Expansion and First Orders
type: item
description: Understanding better the dynamics and steady states of systems strongly coupled to thermal baths is a great theoretical challenge with promising applications in several fields of quantum technologies. Among several strategies to gain access to the steady state, one consists in obtaining approximate expressions of the mean force Gibbs state, the reduced state of the global system-bath thermal state, largely credited to be the steady state. Here, we present analytical expressions of corrective terms to the ultrastrong coupling limit of the mean force Gibbs state, which has been recently derived. We find that the first order term precisely coincides with the first order correction obtained from a dynamical approach—master equation in the strong-decoherence regime. This strengthens the identification of the reduced steady state with the mean force Gibbs state. Additionally, we also compare our expressions with another recent result obtained from a high temperature expansion of the mean force Gibbs state. We observe numerically a good agreement for ultra strong coupling as well as for high temperatures. This confirms the validity of all these results. In particular, we show that, in term of coherences, all three results allow one to sketch the transition from ultrastrong coupling to weak coupling. Quanta 2022; 11: 53–71.
subject: Articles
  topic: section
creator: Camille Lombard Latune
rights: Copyright (c) 2022 Camille Lombard Latune (Author) https://creativecommons.org/licenses/by/4.0
date: 2022-11-20
publication date: 2022-11-20
starting page: 53
ending page: 71
doi: 10.12743/quanta.v11i1.167

The Quantum Hamilton–Jacobi Equation and the Link Between Classical and Quantum Mechanics
type: item
description: We study how the classical Hamilton's principal and characteristic functions are generated from the solutions of the quantum Hamilton–Jacobi equation. While in the classically forbidden regions these quantum quantities directly tend to the classical ones, this is not the case in the allowed regions. There, the limit is reached only if the quantum fluctuations are eliminated by means of coarse-graining averages. Analogously, the classical Hamilton–Jacobi scheme bringing to the motion's equations arises from a similar formal quantum procedure. Quanta 2022; 11: 42–52.
subject: Articles
  topic: section
creator: Mario Fusco Girard
rights: Copyright (c) 2022 Mario Fusco Girard (Author) https://creativecommons.org/licenses/by/4.0
date: 2022-11-03
publication date: 2022-11-03
starting page: 42
ending page: 52
doi: 10.12743/quanta.v11i1.202

On the Role of Inconsistency in Quantum Foundational Debate and Hilbert Space Formulation
type: item
description: This article is intended mainly to develop an expository outline of an inherently inconsistent reasoning in the development of quantum mechanics during 1920s, which set up the background of proposing different variants of quantum logic a bit later. We will discuss here two of the quantum logical variants with reference to Hilbert space formulation, based on the proposals of Bohr and Schrödinger as a result of addressing the same kernel of difficulties and will give a relative comparison. Our presentation is fairly informal, as our goal here is to simply sketch the central ideas leaving further details for other occasions. Quanta 2022; 11: 28–41.
subject: Articles
  topic: section
creator: Debajyoti Gangopadhyay
rights: Copyright (c) 2022 Debajyoti Gangopadhyay (Author) https://creativecommons.org/licenses/by/4.0
date: 2022-10-05
publication date: 2022-10-05
starting page: 28
ending page: 41
doi: 10.12743/quanta.v11i1.195

Dual Instruments and Sequential Products of Observables
type: item
description: We first show that every operation possesses an unique dual operation and measures an unique effect. If a and b are effects and J is an operation that measures a, we define the sequential product of a then b relative to J. Properties of the sequential product are derived and are illustrated in terms of Lüders and Holevo operations. We next extend this work to the theory of instruments and observables. We also define the concept of an instrument (observable) conditioned by another instrument (observable). Identity, state-constant and repeatable instruments are considered. Sequential products of finite observables relative to Lüders and Holevo instruments are studied. Quanta 2022; 11: 15–27.
subject: Articles
  topic: section
creator: Stan Gudder
rights: Copyright (c) 2022 Stan Gudder (Author) https://creativecommons.org/licenses/by/4.0
date: 2022-08-30
publication date: 2022-08-30
starting page: 15
ending page: 27
doi: 10.12743/quanta.v11i1.197

"Mysteries" of modern physics and the fundamental constants c, h, and G
type: item
description: We review how the kinematic structures of special relativity and quantum mechanics both stem from the relativity principle, i.e., "no preferred reference frame" (NPRF). Essentially, NPRF applied to the measurement of the speed of light c gives the light postulate and leads to the geometry of Minkowski space, while NPRF applied to the measurement of Planck's constant h gives "average-only" projection and leads to the denumerable-dimensional Hilbert space of quantum mechanics. These kinematic structures contain the counterintuitive aspects ("mysteries") of time dilation, length contraction, and quantum entanglement. In this essay, we extend the application of NPRF to the gravitational constant G and show that it leads to the "mystery" of the contextuality of mass in general relativity. Thus, we see an underlying coherence and integrity in modern physics via its "mysteries" and the fundamental constants c, h, and G. It is well known that Minkowski and Einstein were greatly influenced by David Hilbert in their development of special relativity and general relativity, respectively, but relating those theories to quantum mechanics via its non-Boolean Hilbert space kinematics is perhaps surprising. Quanta 2022; 11: 5–14.
subject: Articles
  topic: section
creator: W. M. Stuckey
creator: Timothy McDevitt
creator: Michael Silberstein
rights: Copyright (c) 2022 W. M. Stuckey, Timothy McDevitt, Michael Silberstein (Author) https://creativecommons.org/licenses/by/4.0
date: 2022-06-15
publication date: 2022-06-15
starting page: 5
ending page: 14
doi: 10.12743/quanta.v11i1.189

Jordan in the Church of the Higher Hilbert Space: Entanglement and Thermal Fluctuations
type: item
description: I revisit Jordan's derivation of Einstein's formula for energy fluctuations in the black body in thermal equilibrium. This formula is usually taken to represent the unification of the wave and the particle aspects of the electromagnetic field since the fluctuations can be shown to be the sum of wave-like and particle-like contributions. However, in Jordan's treatment there is no mention of the Planck distribution and all averages are performed with respect to pure number states of radiation (mixed states had not yet been discovered!). The chief reason why Jordan does reproduce Einstein's result despite not using thermal states of radiation is that he focuses on fluctuations in a small (compared to the whole) volume of the black body. The state of radiation in a small volume is highly entangled to the rest of the black body which leads to the correct fluctuations even though the overall state might, in fact, be assumed to be pure (i.e. at zero temperature). I present a simple derivation of the fluctuations formula as an instance of mixed states being reductions of higher level pure states, a representation that is affectionately known as "Church of the Higher Hilbert Space". According to this view of mixed states, temperature is nothing but the amount of entanglement between the system and its environment. Quanta 2022; 11: 1–4.
subject: Articles
  topic: section
creator: Vlatko Vedral
rights: Copyright (c) 2025 Vlatko Vedral (Author) https://creativecommons.org/licenses/by/4.0
date: 2022-03-20
publication date: 2022-03-20
starting page: 1
ending page: 4
doi: 10.12743/quanta.v11i1.180

George Sudarshan: Perspectives and Legacy
type: item
description: George Sudarshan has been hailed as a titan in physics and as one who has made some of the most significant contributions in several areas of physics. This article is an attempt to highlight the seminal contributions he has made in physics and the significant developments that arose from his work. Quanta 2021; 10: 75–104.
subject: Articles
  topic: section
creator: Gopalakrishnan Bhamathi
rights: Copyright (c) 2021 Gopalakrishnan Bhamathi (Author) https://creativecommons.org/licenses/by/4.0
date: 2021-12-24
publication date: 2021-12-24
starting page: 75
ending page: 104
doi: 10.12743/quanta.v10i1.174