Abstract The Heisenberg limit is acknowledged as the ultimate precision limit for parameter estimation in quantum metrology. Recent studies have demonstrated super-Heisenberg-scaling limits, where the root mean square errors of estimation decrease faster-than-linearly with the number of probes or evolution processes. This casts doubt on what the ultimate precision limit of quantum metrology really is and, in particular, whether there is a deeper connection with Heisenberg's uncertainty principle. By linking the dynamics of the metrological process with properties in parameter space, we construct a Heisenberg's uncertainty relation between the unknown parameter and its canonical momentum. We propose a novel quantum metrological scheme that utilizes a generating process with indefinite time direction to increase the canonical momentum nonlinearly. This approach achieves a nonlinear-scaling precision limit that improves quadratically with the increase in evolution time length or the number of processes, in accordance with Heisenberg's uncertainty relation. Then we experimentally demonstrate in quantum optical systems that this nonlinear-scaling enhancement can be achieved with a fixed amount of energy in the probes. Consequently, our results provide a deeper insight into the Heisenberg limit in quantum metrology, and shed new light on improving the precision of practical metrological and sensing tasks in realistic quantum systems.

In the macroscopic world of classical theory, a wave is a wave and a particle is a particle. One cannot and will not ever be the other. However, in the microscopic quantum world, this isn’t true. Electrons of atoms and photons of light aren’t necessarily particles or waves. In fact, physicists are having a hard time determining just what they even are since they have the properties of both waves and particles. Another important idea in the field of quantum mechanics is the Heisenberg uncertainty principle. From a broad perspective, the uncertainty principle states that the position and momentum of a particle can never be precisely measured simultaneously. If one is known, the other cannot be determined accurately. This principle is a consequence of wave-particle duality and therefore leads physicists to embrace a modern description of atoms. References - Petrucci, Ralph H., William S. Harwood, F. Geoffrey Herring, and Jeffry D. Madura. General Chemistry. Principles and Modern Applications. 9th ed. Upper Saddle River, NJ: Pearson Prentice Hall, 2007. 298-299. Print. - Laidler, Jeith J., and John H. Meiser. Physical Chemistry. 2nd Ed. 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There were only a few questions that remained unanswered so physicists of the time thought that they were ready to say they knew all about the field of physics. However, as they tried to answer these questions, they came to realize that they were not generating answers but more questions that required different kinds of answers. It was then that physicists came to see that these unanswered questions would not mark the end of physics, but rather the beginning of a new field: quantum theory. However, it was a breakthrough that led physicists to discover more about the world of physics and to understand our own world better, starting from the tiny particles of matter that are its building blocks. Blackbody Radiation and Planck's Equation One of the first ideas proposed to set quantum mechanics apart from classical physics was Max Planck’s idea that energy, like matter, was discontinuous. This revolutionary idea stemmed from blackbody radiation. A blackbody is an object that absorbs all the radiation falling on it. An object that absorbs all the radiation can also perfectly emit all radiation therefore a blackbody will radiate maximum energy when heated to a given temperature. Under classical This experiment was one of several that gave rise to Quantum Mechanics This is an example of another separate rule for quantum mechanics. In the macroscopic world of classical theory, a wave is a wave and a particle is a particle. One cannot and will not ever be the other. However, in the microscopic quantum world, this isn’t true. Electrons of atoms and photons of light aren’t necessarily particles or waves. In fact, physicists are having a hard time determining just what they even are since they have the properties of both waves and particles. Another important idea in the field of quantum mechanics is the Heisenberg uncertainty principle .

simultaneously. However, this possibility is absent in the quantum world. In 1927 the German physicist Werner Heisenberg described such limitations as the Heisenberg Uncertainty Principle, or simply the Uncertainty Principle, stating that it is not possible to measure both the momentum and position of a particle simultaneously. The Nature of Measurement In order to understand the conceptual background of the Heisenberg Uncertainty Principle it is important to understand how physical values are measured. In almost any measurement that is made, light is reflected off the object that is being measured and processed. The shorter the wavelength of light used, or the higher its frequency and energy, the more accurate the results. For example, when attempting to measure the speed of a tennis ball as it is dropped off of a ledge, photons(measurement of light) are shot off the tennis ball, reflected, and then processed by certain equipment. Because the tennis ball is so large compared to the photons, it is unaffected by the efforts of the observer to measure its physical quantities.

Debates about artificial intelligence capabilities and risks are often conducted without quantitative grounding. This paper applies the methodology of MacKay [2009]who reframed energy policy as arithmetic-to the economy of AI computation. We define the token, the elementary unit of large language model input and output, as a physical quantity with measurable thermodynamic cost. Using Landauer's principle, Shannon's channel capacity, and current infrastructure data, we construct a supply-and-demand balance sheet for global token production. We then derive a finite question budget: the number of meaningful queries humanity can direct at AI systems under physical, information-theoretic, and economic constraints. We apply Coase's theory of the firm and the durable-goods monopoly problem to the AI value chain-from photon to atom to chip to power to token to question-to identify where economic value concentrates and where regulatory intervention is warranted. We argue that the expansion of the token budget does not resolve a deeper constraint: under structural uncertainty, the decisive variable is not how many questions can be answered but which questions are worth asking-a problem of agency and direction that computation alone cannot solve. We connect limits of measurement in the token economy to a structural parallel between Goodhart's law and the Heisenberg uncertainty principle, and to Arrow's impossibility result for efficient information pricing. The framework yields order-of

Barcoding photons can provide a host of functionalities that could benefit future quantum communication systems and networks beyond today's imagination. As a significant application of barcoding photons, we introduce code division multiple-access (CDMA) communication systems for various applications. In this context, we introduce and discuss the fundamental principles of a novel quantum CDMA (QCDMA) technique based on spectrally encoding and decoding of continuous-mode quantum light pulses. In particular, we present the mathematical models of various QCDMA modules that are fundamental in describing an ideal and typical QCDMA system, such as quantum signal sources, quantum spectral encoding phase operators, M$\times$M quantum broadcasting star-coupler, quantum spectral phase decoding operators, and the quantum receivers. In describing a QCDMA system, this paper considers a unified approach where the input continuous-mode quantum light pulses can take on any form of pure states such as Glauber states and quantum number states. For input number states, one can observe features like entanglement and quantum interference. More interestingly, due to Heisenberg's uncertainty principle, the quantum signals sent by photon number states obtain complete phase uncertainty at the time of measurement. Therefore, at the receiver output, the multiaccess inter-signal interference vanishes. Due to Heisenberg's uncertainty principle, the received signal intensity at the photodetector's output c The mathematical presentation is independent of the form of the input pure quantum states. We show that the spectrally encoded quantum states of the light at the quantum star-coupler output are not, in general, factorized states, except for input Glauber states. For input number states, one can observe features like entanglement and quantum interference. Moreover and interestingly, as a consequence of Heisenberg’s uncertainty principle, the quantum signals sent by photon number states obtain complete phase uncertainty at the time of measurement. Therefore, at the receiver output, the multiaccess inter-signal interference vanishes. Due to Heisenberg’s uncertainty principle, the received signal intensity at the photodetector’s output, right at the time of measurement, changes from coherent detection scheme for input Glauber states to incoherent detection scheme for input number states. We also would like to highlight that the mathematical models and tools developed in this paper, in the context of QCDMA, become very useful for developing and analyzing other quantum multiple-access techniques based on wavelength, space and time domain. In incoherent detection, there is no inter-signal interference amongst the input signals into the photodetector as if the signals were initially incoherent, which is not the case at all. As a matter of fact, the number states generated by the quantum transmitters are assumed to be entirely coherent, and that is why we can apply encoding/decoding phase operators to alter the phase of their frequency components and thereby changing their temporal shape from a peaked pulse into a random-looking time spread signal and vice versa. This funda- mental difference in the detection scheme between the number states and Glauber states is a consequence of Heisenberg’s uncertainty principle. Therefore, we can conclude that due to Heisenberg’s uncertainty principle, the quantum phase fluctuation at the time of measurement in number states alters the behavior or the functionality of the receiver structure from a coherent detection scheme for Glauber states into the incoherent detection scheme for number states. Consider the simulation of QCDMA via number states|nξs⟩ for the corresponding OOK’s sequences shown in Fig. 5a and Fig. 5d. For binary one, transmitters send a single-photon state, ns = 1, and for binary zero, a vacuum state, ns = 0, is transmitted. We discussed that the above difference in the received intensities at the time of measurement is due to Heisenberg’s uncertainty principle. In other words, because of the complete quantum phase uncer- tainty of particle-like single-photons, there is no inter-signal interference; however, the inter-signal interference emerges for wave-like Glauber states. It is worth noting that in this paper, the encoding and decod- ing operations apply to the spectrum of photons; however, the mathematical model is easily extendable to temporal encoding and decoding of quantum light states, namely direct sequence QCDMA. Again, Schr¨odinger and Heisenberg pictures are equivalent. This equivalency for the star-coupler’ssth input port indicates that ⟨Ψ|ˆa′ The interference term between signals of ports s and s′ corresponds to E⋆ s (t)Es′ (t). This inter-signal interference term disappears for some quantum states, such as number states and squeezed coherent states, due to Heisenberg’s uncertainty principle. From Heisenberg’s uncertainty principle, the quantum phase for number state (see appendix D-B) is uniformly distributed between [0, 2π], then the electric field expectation value E (t) becomes time-independent and zero. On the other hand, since the field operation on number state |n⟩, i.e., ˆa(t)|n⟩, corresponds to number state |n− 1⟩; therefore, the expectation value E(t) = ⟨n|ˆa(t)|n⟩ = 0 vanishes. To put it another way, a single-photon state with sub-Poissonian photon statistics and zero photon-number uncertainty is subject to Heisenberg’s uncertainty principle and has a complete phase uncertainty. ACKNOWLEDGMENT M. Rezai acknowledges the funding and the support from the Iran National Elite Foundation. REFERENCES [1] C. W. Helstrom, Quantum detection and estimation theory , ser. Math. Sci. Eng. New York, NY: Academic Press, 1976. [Online]. Available: http://cds.cern.ch/record/110988 [2] G. Cariolaro, Quantum Communications . Springer Publishing Com- pany, Incorporated, 2015. [3] M. M. Wilde, Quantum Information Theory . Cambridge University Press, 2013. [4] M.