An introduction to quantum measurements with a historical motivation

Carlos Alexandre Brasil; Leonardo Andreta de Castro; Ol\'impio Pereira de S\'a Neto

An introduction to quantum measurements with a historical motivation

T0 review · 0 major / 8 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This review argues that von Neumann's 1932 unitary measurement model, in which apparatus and system are coupled by an interaction Hamiltonian, unifies the measurement problem, weak measurements, POVMs, Kraus operators, and the Quantum…

desk verdict A solid, historically motivated review centered on von Neumann's measurement model—not new research, but a genuinely useful teaching resource with a few fixable slips. read the letter →

arxiv 1908.03949 v1 pith:RIFR5VC7 submitted 2019-08-11 quant-ph physics.hist-ph

Leonardo Andreta de Castro , Olímpio Pereira de Sá Neto , Carlos Alexandre Brasil This is my paper · ORCID

classification quant-phphysics.hist-ph MSC 81P1581P05 PACS 03.65.Ta03.65.-w

keywords quantummeasurementvonNeumannmodelweakmeasurementsvaluesPOVMKrausoperatorsmasterequationZenoeffect

topics The Measurement Problem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that von Neumann's 1932 model of quantum measurement—in which the apparatus is treated as a quantum system that becomes entangled with the one being measured—is the single most useful thread for teaching and applying the subject today. It retells how the old quantum theory and the orthodox interpretation progressively restricted what could be measured, then shows how von Neumann's unitary interaction overcomes those restrictions by producing correlated pointer states. The same model, with the interaction strength turned down, yields weak measurements and anomalous weak values; with ideal projectors replaced by POVMs and Kraus operators, it yields generalized measurements and a master equation for continuous observation. The paper closes by using that master equation to derive the Quantum Zeno Effect, demonstrating that one physical picture connects foundational puzzles with current research tools.

What carries the argument

The central machinery is von Neumann's measurement model and its modern descendants. The model itself is a unitary coupling $\exp(-i \varepsilon \tau \hat S^{(S)} \hat M^{(M)}/\hbar)$ between the measured observable $\hat S^{(S)}$ and a Hermitian apparatus operator $\hat M^{(M)}$; with $\hat M^{(M)} = \hat P^{(M)}$ conjugate to a pointer position $\hat Q^{(M)}$, it translates the apparatus wave packet by $\varepsilon \tau s_n$, creating a one-to-one correlation between outcome and pointer reading. The later machinery generalizes this same translation: weak values come from keeping only the first-order expansion of the coupling; POVMs replace projectors by positive operators that need not commute; Kraus operators implement the resulting state changes; and the measurement master equation $\dot{\hat\rho}^{(S)} = -\frac{i}{\hbar}[\hat H^{(S)},\hat\rho^{(S)}] + \lambda\big(\sum_i \hat K_i \hat\rho^{(S)} \hat K_i^\dagger - \hat\rho^{(S)}\big)$ describes measurements occurring at random times with rate $\lambda$. Each tool is introduced as an extension or weakening of the original unitary pointer interaction.

What would settle it

A concrete test comes from the continuous-measurement master equation: it predicts that the survival probability of a monitored two-level system depends only on the current average rate $\lambda$, so a setup whose detection events are time-clustered or otherwise correlated should still follow the Zeno curve if the Markov assumption holds. Recording an appreciable faster decay in clustered measurement schedules, or any dependence on the temporal correlations of detections, would falsify the constant-rate memoryless model on which Eq. (108) rests.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that von Neumann's measurement model is a unifying model whose consequences are still being worked out. The model uses an impulsive interaction Hamiltonian $\hat H^{(S+M)} = \varepsilon \hat S^{(S)} \hat P^{(M)}$, so that if the system starts in $\sum_n c_n |s_n\rangle$ and the apparatus in a wave packet $\varphi_0(q)$, the joint state becomes $\sum_n c_n |s_n\rangle \varphi_0(q - \varepsilon\tau s_n)$. When the displaced packets no longer overlap, the apparatus states are orthogonal and the measurement is projective; when the displacement is small, expanding the same exponential gives the weak value $S_w = \langle \Phi_{\rm out} | \hat S | \varphi_0\rangle / \langle \Phi_{\rm out} | \varphi_0\rangle$ and the surprising result that a spin component can effectively read 100. The paper also shows that the model's limitations motivate POVMs and Kraus operators, whose completeness condition $\sum_i \hat K_i^\dagger \hat K_i = \hat 1$ guarantees normalized outcomes, and that a Lindblad-type measurement master equation follows when the system undergoes such generalized measurements at a constant rate $\lambda$. Solving that equation for a two-level system under continuous monitoring yields the Quantum Zeno Effect: the survival probability tends to one as the measurement rate grows.

Load-bearing premise

The load-bearing premise is that repeated measurements are memoryless events with a constant probability rate $\lambda$ and that the system–apparatus coupling is weak at every instant; if the apparatus remembers earlier outcomes or the rate varies in time, the continuous-measurement master equation and the Zeno dynamics built on it do not follow.

Editorial extensions

If this is right

Mastering the von Neumann model lets a reader derive the projective measurement rule from pointer-state orthogonality rather than taking collapse as an unexplained postulate.
Weak measurements and their anomalous results, including the apparent measurement of a spin-1/2 component as 100, are direct consequences of the same interaction Hamiltonian in the small-coupling limit.
POVMs and Kraus operators extend the framework to imperfect, non-orthogonal, or open-system measurements, with the completeness condition $\sum_i \hat K_i^\dagger \hat K_i = \hat 1$ replacing unitarity.
The measurement master equation shows that continuous observation at rate $\lambda$ freezes a monitored two-level system in its initial state, recovering the Quantum Zeno Effect as a quantitative prediction.
Because the model unifies projective, weak, and continuous measurements, it supplies one vocabulary for foundational questions such as entanglement, Wigner's friend, and decoherence, and for applied topics such as measurement-based quantum computation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The authors leave implicit that the same master-equation machinery could be used to design measurement sequences that steer a system into a target state, since the Zeno calculation already exhibits the measurement rate as a control knob.
The classical coin-toss analogue of weak values suggests that post-selection anomalies are not inherently quantum; a natural test is to engineer a classical optical or electronic experiment with the same pre/post-selection structure and see whether the anomalous values persist.
A reader could extend the paper's didactic claim by asking how the pointer-displacement picture survives when the apparatus has internal dynamics or when successive measurements are correlated in time, since the memoryless weak-coupling approximation is the point where the Zeno prediction could break down.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The short version: this is a pedagogical review, not a research paper, and it is a rather good one. The reader's report largely matches my own read. If you teach quantum measurement theory or advise someone entering the field, this is worth knowing about. Don't cite it for a new result—there isn't one—but do keep it in mind as a historically motivated map of the subject centered on von Neumann's measurement model.

What the paper actually does well is build a coherent narrative: from classical measurement, through the old quantum theory, Born rule, complementarity, and uncertainty, to von Neumann's unitary system-apparatus interaction, entanglement, decoherence, weak values, POVMs, Kraus operators, and the Lindblad/measurement master equation, ending with the quantum Zeno effect. The derivations are standard but mostly clean, and the historical thread is not decoration; it explains why the measurement problem looks the way it does. The inclusion of the Ferrie–Combes classical weak-value analog is a nice touch that keeps the exposition honest.

The soft spots are real but minor. Table II's weak value for sigma_y is missing a factor of i on the second numerator term; the final quoted result is correct, so it's a typographical slip, but it will confuse anyone working through the section. The continuous-measurement master equation is explicitly derived under Born–Markov and constant-rate-λ assumptions; the paper states this, but it never quantifies when those approximations break down for a realistic measurement process. Two smaller presentational gaps: 'quantum algorithms' appears in the introduction as motivation but is never developed, and the classical weak-value reduction in Section XI has a few unstated normalizations. None of this undermines the central pedagogical claim.

This is a paper for students and non-specialists, and for teachers looking for a single, readable introduction that gives the tools to go further. It deserves serious peer review—send it to referees, not to the desk reject pile—with a request to fix the Table II typo and add a sentence about the domain of validity of the rate λ. It is not a research contribution, but as a review it is solid and useful.

Referee Report

0 major / 8 minor

Summary. This is a pedagogical review, not an original research paper. It argues that John von Neumann's unitary model of measurement provides a historically motivated and technically effective organizing principle for teaching quantum measurement theory. The paper first surveys the historical development from Planck through Bohr, Born, Heisenberg, and the Copenhagen orthodoxy, then presents the von Neumann/Bohm interaction model, density matrices, entanglement, and decoherence. It uses the same framework to introduce weak measurements and their seemingly anomalous values, generalizes projective measurements to POVMs and Kraus operators, derives a Markovian measurement master equation, and applies the formalism to the Quantum Zeno Effect. The stated contribution is expository: equipping students and non-specialists with the tools needed to apply von Neumann's model to modern problems.

Significance. Judged as a pedagogical review, the paper is largely successful. The central derivations—von Neumann's pointer displacement, the weak-value expansion, POVM construction, Kraus decomposition, the Lindblad-form measurement master equation, and the Zeno freezing result—are standard and internally consistent. The paper is honest about its scope: it explicitly states that it is not a complete treatise, notes that backaction effects are omitted, and lists alternative approaches it does not cover. Its main strength is the coherent progression from a historical motivation to modern tools in a single accessible narrative. I found no load-bearing technical error; the issues listed below are local corrections.

minor comments (8)

[Table II] The weak-value entry for σ_y is missing the factor i on the second numerator term; it should read i(a_+ b_-^* - a_- b_+^*)/(a_+ b_+^* + a_- b_-^*). The final result (σ_y)_w = i tan θ in Eq. (55) is correct, so this is a local typo, but it should be corrected because Table II is a reference point for readers.
[Eq. (102)] The sentence that 'terms with m>2 will vanish' excludes m=2; the correct statement is that all terms of order n^{-2} or higher (m≥2) do not contribute to the limit. The conclusion P_∞(T)=1 remains correct.
[Eq. (18)] The summation index r on the right-hand side should be n, so that the equation reads E = Σ_n |c_n|^2 E_n.
[Fig. 3 caption] The references to 'Fig. 2(a)' and 'Fig. 2(b)' in the caption should be 'Fig. 3(a)' and 'Fig. 3(b)'.
[Fig. 4 caption] A Gaussian cannot have compact support; rephrase to something like 'a Gaussian effectively confined to [−b,b]'.
[Sec. XIV] The Born-Markov and constant-rate assumptions for Eq. (85) are stated, but a quantitative statement about the required separation of timescales and the constancy of λ would help readers apply Eq. (108) safely.
[Appendix B] The reference to 'Eq. (C)' is unclear; it should cite the numbered equation defining the density operator or Eq. (34).
[Sec. I / Summary] The abstract promises quantum algorithms as a motivation, but only Quantum Phase Estimation is mentioned in Sec. XVI; either add a few sentences on measurement-based quantum computation or soften the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the pedagogical derivations are self-contained textbook arguments, and the paper's self-citations are confined to background or motivation.

full rationale

The paper is an exposition of established quantum measurement theory, and its central claim is pedagogical: that von Neumann's measurement model provides a unifying thread. Walking the derivation chain, each loaded step is derived in the text from stated assumptions rather than from a fitted value or from the paper's own prior conclusions. The von Neumann measurement state, Eq. (25), follows from the unitary operator in Eq. (24) applied to the pre-measurement state in Eq. (23). The weak-measurement displacement, Eqs. (46)-(51), follows from Taylor expansion of the same interaction Hamiltonian and from the definition of the weak value in Eq. (48); the weak value is a defined quantity, not a fit. The POVM and Kraus operator formalism, Eqs. (64)-(77), is derived from positivity, normalization, the spectral decomposition of the environment state, and the completeness identity; no parameter is later renamed as a prediction. The measurement master equation, Eq. (85), is explicitly derived in Section XIV B from the Cresser et al. ansatz (probability lambda*Delta-t of a POVM Kraus-map per time step) and the Delta-t -> 0 limit, with the derivation reproduced in the text. The Quantum Zeno effect is then obtained either from the projective-measurement argument, Eq. (104), proven from unitarity, or from solving the linear system in Eq. (108) with the explicit Kraus operators of Eq. (106); the Zeno plateau in Fig. 5 is the solution, not an input. Self-citations such as refs. [131], [165], [172], [173], [178], and [179] appear as background, motivation, or side references, but the load-bearing derivations cite and reproduce independent sources (refs. [149], [153], [174], [180]) and are not replaced by self-citation. The paper also explicitly states the limitation that Eq. (85) excludes reduction and backaction effects and relies on the Born-Markov and constant-rate assumptions, so those restrictions are not silently overclaimed. No derivation reduces by construction to its own inputs or to the authors' prior results.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The review introduces no new parameters, entities, or fitted constants. All quantities are standard observables and operators from quantum mechanics. The only invented element is the pedagogical framing around von Neumann's model, which is not a scientific entity.

assumptions (6)

domain assumption Born's rule: the probability of an outcome is given by the squared modulus of the amplitude.
Used throughout Sections V, VIII, and XII to define measurement probabilities and expectation values.
domain assumption Two-process evolution: unitary evolution (Process 2) and wave-packet reduction (Process 1).
Central to von Neumann's model and the measurement problem; introduced in Section VIII.
domain assumption The measuring apparatus can be described quantum mechanically and prepared in a known initial state.
Bohm's Assumption 2 in Section VIII.A; required for constructing the pre-measurement state.
domain assumption Pointer states of the apparatus are orthonormal or become approximately orthogonal.
Assumption 3 in Section VIII.A; needed for distinguishable measurement outcomes.
domain assumption Born-Markov approximation: the environment is memoryless and the system-environment state factorizes at all times.
Used to derive the master equation in Section XIV.A, Eq. (83).
domain assumption The measurement master equation from Cresser et al. is valid.
Adopted in Section XIV.B to describe continuous measurements; assumes a constant measurement rate lambda.

how reviews work

0 comments

Cite this review

Pith. "Pith review of An introduction to quantum measurements with a historical motivation." pith.science (2026). https://pith.science/paper/RIFR5VC7

@misc{pith2026190803949,
  author       = {Pith},
  title        = {Pith review of: An introduction to quantum measurements with a historical motivation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RIFR5VC7}},
  note         = {Machine review of arXiv:1908.03949}
}

read the original abstract

We provide an introduction to the theory of quantum measurements that is centered on the pivotal role played by John von Neumann's model. This introduction is accessible to students and researchers from outside the field of foundations of quantum mechanics and presented within a historical context. We first explain the origins and the meaning of the measurement problem in quantum theory, and why it is not present in classical physics. We perform a chronological review of the quantization of action and explain how this led to successive restrictions on what could be measured in atomic phenomena, until the consolidation of the orthodox interpretation of quantum mechanics. The clear separation between quantum system and classical apparatus that causes these restrictions is subverted in von Neumann's paradigmatic model of quantum measurements, a subject whose concepts we explain, while also providing the mathematical tools necessary to apply it to new problems. We show how this model was important in discussing the interpretations of quantum mechanics and how it is still relevant in modern applications. In particular, we explain in detail how it can be used to describe weak measurements and the surprising results they entail. We also discuss the limitations of von Neumann's model of measurements, and explain how they can be overcome with POVMs and Kraus operators. We provide the mathematical tools necessary to work with these generalized measurements and to derive master equations from them. Finally, we demonstrate how these can be applied in research problems by calculating the Quantum Zeno Effect.

Figures

Figures reproduced from arXiv: 1908.03949 by the authors.

**Figure 2.** FIG. 2. Evolution of the wave function of the measurement apparatus, originally with a compact support of [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗

**Figure 3.** FIG. 3. How a probability distribution [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Wave function of an apparatus measuring [PITH_FULL_IMAGE:figures/full_fig_p039_4.png]

**Figure 5.** Figure 5: FIG. 5. Probability [PITH_FULL_IMAGE:figures/full_fig_p060_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

208 extracted references · 76 canonical work pages

[1]

de Laplace

P.-S. de Laplace. A philosophical essay on probabilities. John Wiley & Sons, New York, 1902. 67

1902
[2]

A. Pais. Subtle is the Lord... – The science and the life of Albert Einstein . Oxford University Press, Oxford, 1982

1982
[3]

P. W. Shor. Scheme for reducing decoherence in quantum computer memory. Physical Review A, 52, 1995

1995
[4]

H. J. Briegel and R. Raussendorf. Persistent entanglement in arrays of interacting particles. Physical Review Letters, 86:910–913, 2001

2001
[5]

H. J. Briegel, D. E. Browner, W. Dür, R. Raussendorf, and M. V . D. Nest. Measurement-based quantum computation. Nature Physics, 5:19–26, 2009

2009
[6]

Eisberg and R

R. Eisberg and R. Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles . John Wiley & Sons, New York, 1985

1985
[7]

A. Messiah. Quantum mechanics, vol. 1. North-Holland Publishing Company, Amsterdam, 1961

1961
[8]

L. I. Schiff. Quantum mechanics. McGraw-Hill Book Company, New York, 1968

1968

Show all 208 references

[9]

Eisberg and R

R. Eisberg and R. Resnick. Fundamentals of Modern Physics. John Wiley & Sons, New York, 1961

1961
[10]

Merzbacher

E. Merzbacher. Quantum mechanics. John Wiley & Sons Inc, New York, 1998

1998
[11]

L. E. Ballentine. Quantum mechanics: A modern development. World Scientiﬁc Publishing, Singa- pore, 1988

1988
[12]

A. Peres. Quantum theory: Concepts and methods. Kluwer Academic Publishers, New York, 2002

2002
[13]

R. B. Grifﬁths. Consistent Quantum Theory. Cambridge University Press, Cambridge, 2002

2002
[14]

E. M. Purcell. Electricity and magnetism – Berkeley Physics Course vol. 2 . McGraw–Hill Book Company, New York, 1985

1985
[15]

F. W. Sears and G. L. Salinger. Thermodynamics, kinetic theory, and statistical thermodynamics . Addison-Wesley Publishing Company, Reading, 1982

1982
[16]

A. P. French and E. F. Taylor. Special relativity. W. W. Norton and Company Inc, New York, 1968

1968
[17]

M. Bucher. Rise and premature fall of the old quantum theory, 2008. arXiv:0802.1366

2008 arXiv
[18]

A. Hermann. The genesis of quantum theory 1899–1913. MIT Press, Cambridge, 1971

1913
[19]

M. Jammer. The conceptual development of quantum mechanics . McGraw–Hill Book Company, New York, 1966

1966
[20]

Feldens, P

B. Feldens, P. M. C. Dias, and W. M. S. Santos. E assim se fez o quantum... / let there be quantum... Revista Brasileira de Ensino de Física, 22:2602, 2010

2010
[21]

Tomonaga

S.-I. Tomonaga. Quantum mechanics vol. 1. North-Holland Publishing Company, Amsterdam, 1968. 68

1968
[22]

L. A. Castro, C. A. Brasil, and R. d. J. Napolitano. Elliptical orbits in the phase-space quantization. Revista Brasileira de Ensino de Física, 38:e3318, 2016

2016
[23]

A. d’Abro. The rise of the new physics. Dover Publications Inc, New York, 1951

1951
[24]

Duncan and M

A. Duncan and M. Janssen. The trouble with the orbits: The Stark effect in the old and the new quantum theory. Studies in History and Philosophy of Science, 48:68–83, 2014

2014
[25]

Duncan and M

A. Duncan and M. Janssen. The Stark effect in the Bohr–Sommerfeld theory and in Schrödinger’s wave mechanics, 2014. arXiv:1404.5341

2014 arXiv
[26]

F. A. G. Parente, A. C. F. dos Santos, and A. C. Tort. Os 100 anos do átomo de Bohr / one hundred years of Bohr’s atom. Revista Brasileira de Ensino de Física, 35:4301, 2013

2013
[27]

N. Studart. A invenÃ ˘gÃˇco do conceito de quantum de energia segundo Planck. Revista Brasileira de Ensino de Física, 22:523–535, 2000

2000
[28]

M. Planck. Über eine verbesserung der wienschen spektralgleichung. Verhandlungen der Deutschen Physicalishen Gesellschaft, 2:202–204, 1900

1900
[29]

M. Planck. Ueber das gesetz der energieverteilung im normalspectrum. Annalen der Physik , 309:553–563, 1901

1901
[30]

ter Haar

D. ter Haar. The old quantum theory. Pergamon Press Ltd, Oxford, 1967

1967
[31]

T. S. Kuhn. Black-body theory and the quantum discontinuity, 1894–1912 . University of Chicago Press, Chicago, 1987

1912
[32]

Beck and P

A. Beck and P. Havas. The collected papers of Albert Einstein vol. 2 - The Swiss Years: Writings, 1900-1909. Princeton University Press, Princeton, 1989

1900
[33]

Arons and M

A. Arons and M. B. Peppard. Einstein’s proposal of the photon concept – a translation of the Annalen der Physik paper of 1905. American Journal of Physics, 33:367–374, 1965

1905
[34]

J. J. Thomson. The magnetic properties of systems of corpuscles describing circular orbits. Philo- sophical Magazine Series 6, 6:673–693, 1903

1903
[35]

J. J. Thomson. On the structure of the atom: An investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure. Philosophica...

1904
[36]

Rutherford

E. Rutherford. The structure of the atom. Philosophical Magazine and Journal of Science, 27:488– 498, 1914. 69

1914
[37]

H. Nagaoka. Motion of particles in an ideal atom illustrating the line and band spectra and the phenomena of radioactivity. Bulletin of the Mathematical and Physical Society of Tokyo, 2:140–141, 1904

1904
[38]

H. Nagaoka. On a dynamical system illustrating the spectrum lines and the phenomena of radio- activity. Nature, 69:392–393, 1904

1904
[39]

H. Nagaoka. Kinetics of a system of particles illustrating the line and band spectrum and the phe- nomena of radio-activity. Philosophical Magazine, 7:445–455, 1904

1904
[40]

Heisenberg

W. Heisenberg. The physical principles of the quantum theory. Dover, Mineola, 1949

1949
[41]

F. Laloë. Do we really understand quantum mechanics? Strange correlations, paradoxes, and theo- rems. American Journal of Physics, 69:655–701, 2001

2001
[42]

d’Espagnat

B. d’Espagnat. Conceptual foundations of quantum mechanics. Perseus Books, Reading, 1999

1999
[43]

R. L. Liboff. The correspondence principle revisited. Physics Today, 37:50–55, 1984

1984
[44]

A. J. Makowski. A brief survey of various formulations of the correspondence principle. European Journal of Physics, 27:1133–1139, 2006

2006
[45]

G. Q. Hassoun and D. H. Kobe. Synthesis of the Planck and Bohr formulations of the correspondence principle. American Journal of Physics, 57:658–662, 1969

1969
[46]

Bhattacharyya

B. Bhattacharyya. Looking back into Bohr’s atom. European Journal of Physics, 27:497–500, 2006

2006
[47]

N. Bohr. On the constitution of atoms and molecules. Philosophical Magazine, 26:1–25, 1913

1913
[48]

Pelogia and C

K. Pelogia and C. A. Brasil. Analysis of the Jun Ishiwara’s ‘The universal meaning of the quantum of action’. European Physical Journal H, 43:507–521, 2017

2017
[49]

M. Eckert. How Sommerfeld extended Bohr’s model of the atom (1913–1916). European Physical Journal H, 39:141–156, 2014

1913
[50]

Sommerfeld

A. Sommerfeld. On the theory of the Balmer series. European Physical Journal H , 39:157–177, 2014

2014
[51]

Sommerfeld

A. Sommerfeld. The ﬁne structure of hydrogen and hydrogen-like lines. European Physical Journal H, 39:179–204, 2014

2014
[52]

Sommerfeld

A. Sommerfeld. Zur quantentheorie der spektrallinien. Annalen der Physik, 51:1–94, 1916

1916
[53]

Sommerfeld

A. Sommerfeld. Zur quantentheorie der spektrallinien. Annalen der Physik, 51:125–167, 1916

1916
[54]

W. Wilson. The quantum-theory of radiation and line spectra. Philosophical Magazine, 29:795–802, 1915

1915
[55]

W. Wilson. The quantum of action. Philosophical Magazine, 31:156–162, 1916. 70

1916
[56]

Ishiwara

J. Ishiwara. Die universelle bedeutung des wirkungsquantums. Proceedings of Tokyo Mathematico– Physical Society, 8:106–116, 1915

1915
[57]

Ishiwara

J. Ishiwara. The universal meaning of the quantum of action. European Physical Journal H, 43:523– 536, 2017

2017
[58]

B. L. van der Waerden. Sources of quantum mechanics. Dover Publications Inc, Mineola, 2007

2007
[59]

M. Planck. The nobel prize in physics 1918. http://www.nobelprize.org/nobel_prizes/physics /laureates/1918/planck-lecture. Accessed on: September 17, 2015

1918
[60]

R. A. Martins and P. S. Rosa. História da teoria quântica – a dualidade onda partícula, de Einstein a de Broglie. Editora Livraria da Física, SÃ ˇco Paulo, 2014

2014
[61]

de Broglie

L. de Broglie. An introduction to the study of wave mechanics. Methuen & Co. Ltd, Essex, 1930

1930
[62]

de Broglie

L. de Broglie. Recherches sur la théorie des quanta. Annales de Physique, 10:22–128, 1925

1925
[63]

de Broglie

L. de Broglie. On the theory of quanta. Fondation Louis de Broglie, Paris, 2004

2004
[64]

MacKinnon

E. MacKinnon. De Broglie’s thesis: a critical retrospective. American Journal of Physics, 44:1047– 1055, 1976

1976
[65]

M. Jammer. The philosophy of quantum mechanics: the interpretations of quantum mechanics in historical perspective. Wiley, New York, 1974

1974
[66]

Einstein

A. Einstein. Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung. Sitzungsberichte der Preußischen Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse , pages 3–14, 1925

1925
[67]

N. Bohr, H. K. Kramers, and J. C. Slater. The quantum theory of radiation. Philosophical Magazine, 47:785–802, 1924

1924
[68]

Heisenberg

W. Heisenberg. Physics and philosophy – The revolution in modern science. Harper Perennial, New York, 2007

2007
[69]

Joas and C

C. Joas and C. Lehner. The classical roots of wave mechanics: Schrödinger’s transformations of the optical-mechanical analogy. Studies in History and Philosophy of Modern Physics, 40:338–351, 2009

2009
[70]

R. Köberle. Sobre a gÃłnese da mecânica ondulatória. Revista Brasileira de Física , 9:243–273, 1979

1979
[71]

L. Wessels. Schrödinger’s route to wave mechanics. Studies in History and Philosophy of Science , 10:311–340, 1977. 71

1977
[72]

A. L. Fetter and J. D. Walecka. Theoretical mechanics of particles and continua . Dover, Mineola, 2003

2003
[73]

Schrödinger

E. Schrödinger. Quantisierung als eigenwertproblem. Annalen der Physik, 384:489–527, 1926

1926
[74]

Schrödinger

E. Schrödinger. Quantisierung als eigenwertproblem. Annalen der Physik, 385:361–376, 1926

1926
[75]

Schrödinger

E. Schrödinger. Collected papers on wave mechanics. Chelsea Pub. Co., New York, 1978

1978
[76]

C. C. Cohen-Tannoudji, B. Diu, and F. Laloë. Quantum mechanics. Wiley, New York, 1977

1977
[77]

G. S. Im. Experimental constraints on formal quantum mechanics: The emergence of Born’s quantum theory of collision processes in Göttingen, 1924–1927.Archive for History of Exact Sciences, 50:73– 101, 1995

1924
[78]

M. Born. Zur quantenmechanik der stoßvorgänge. Zeitschrift für Physik, 37:863–867, 1926

1926
[79]

J. A. Wheeler and W. H. Zurek. Quantum theory and measurement . Princeton University Press, Princeton, 1983

1983
[80]

M. Born. Quantenmechanik der stoßvorgänge. Zeitschrift für Physik, 38:803–827, 1926

1926
[81]

G. Ludwig. Wave mechanics. Pergamon Press, Oxford, 1968

1968
[82]

E. Butkov. Mathematical physics. Addison–Wesley, Reading, 1968

1968
[83]

Morse and H

P. Morse and H. Feshbach. Methods of theoretical physics. McGraw–Hill, New York, 1953

1953
[84]

Gottfried

K. Gottfried. P. a. m. dirac and the discovery of quantum mechanics. American Journal of Physics, 79:261–266, 2011

2011
[85]

P. A. M. Dirac. On the theory of quantum mechanics. Proceedings of the Royal Society of London A, 112:661–677, 1926

1926
[86]

M. Bunge. Survey of the interpretations of quantum mechanics. American Journal of Physics , 24:272–286, 1956

1956
[87]

J. B. Hartle. Quantum mechanics of individual systems. American Journal of Physics, 36:704–712, 1968

1968
[88]

L. E. Ballentine. The statistical interpretation of quantum mechanics. Reviews of Modern Physics, 42:358–380, 1970

1970
[89]

N. R. Hanson. Copenhagen interpretation of quantum theory. American Journal of Physics, 27:1–15, 1959

1959
[90]

H. P. Stapp. The Copenhagen interpretation. American Journal of Physics, 40:1098–1116, 1972

1972
[91]

von Neumann

J. von Neumann. Mathematical foundations of quantum mechanics . Princeton University Press, Princeton, 1953. 72

1953
[92]

D. Howard. Who invented the ‘Copenhagen interpretation’? a study in mythology. Philosophy of Science, 71:669–682, 2004

2004
[93]

Pauli, L

W. Pauli, L. Rosenfeld, and V . Weisskopf. Niels Bohr and the development of physics – Essays dedicated to Niels Bohr on the occasion of his seventieth birthday. Pergamon Press, London, 1955

1955
[94]

N. Bohr. The quantum postulate and the recent development of atomic theory. Nature, 121:580–590, 1928

1928
[95]

Roldán-Charria

J. Roldán-Charria. Indivisibility, complementarity and ontology: A Bohrian interpretation of quan- tum mechanics. Foundations of Physics, 44:1336–1356, 2014

2014
[96]

R. Omnès. The interpretation of quantum mechanics. Princeton University Press, Princeton, 1994

1994
[97]

N. Bohr. Atomic physics and human knowledge. John Wiley & Sons, New York, 1958

1958
[98]

R. Omnès. Quantum philosophy. Princeton University Press, Princeton, 1994

1994
[99]

J. R. Henderson. Classes of Copenhagen interpretations: Mechanisms of collapse as typologically determinative. Studies in History and Philosophy of Modern Physics, 41:1–8, 2010

2010
[100]

Heisenberg

W. Heisenberg. Physics and beyond – encounters and conversations . Harper & Row, New York, 1971

1971
[101]

S. Seth. Crafting the quantum: Arnold Sommerfeld and the older quantum theory. Studies in History and Philosophy of Science, 39:335–349, 2008

2008
[102]

S. Seth. Zeideutigkeit about Zweideutigkeit: Sommerfeld, Pauli and the methodological origins of quantum mechanics. Studies in History and Philosophy of Science, 40:303–315, 2009

2009
[103]

Reichenbach

H. Reichenbach. Philosophic foundations of quantum mechanics . University of California Press, Berkeley, 1948

1948
[104]

Heisenberg

W. Heisenberg. Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik. Zeitschrift für Physik, 43:172–198, 1927

1927
[105]

R. W. Ditchburn. The uncertainty principle in quantum mechanics. Proceedings of the Royal Irish Academy, 39:73–81, 1930

1930
[106]

H. P. Robertson. The uncertainty principle. Physical Review, 34:163–164, 1929

1929
[107]

H. P. Robertson. Proceedings of the american physical society, minutes of the new york meeting, feb. 21 and 22 – A general formulation of the uncertainty principle and its classical interpretation. Physical Review, 35:667A, 1930

1930
[108]

H. P. Robertson. An indeterminacy relation for several observables and its classical interpretation. Physical Review, 46:794–801, 1934. 73

1934
[109]

Einstein

A. Einstein. Über einen die erzeugung und verwandlung des lichtes betreffenden heuristischen gesichtspunkt. Annalen der Physik, 17:132, 1905

1905
[110]

N. Bohr. On the constitution of atoms and molecules. Philosophical Magazine, 26:476–502, 1913

1913
[111]

N. Bohr. On the constitution of atoms and molecules. Philosophical Magazine, 26:857–875, 1913

1913
[112]

M. Planck. Scientiﬁc autobiography and other papers. Williams & Norgate Ltd, London, 1950

1950
[113]

D. Bohm. Quantum theory. Dover Publications Inc, New York, 1989

1989
[114]

F. D. Peat. Inﬁnity potential – The life and times of David Bohm. Basic Books, 1996

1996
[115]

L. D. Landau. Collected Papers of L. D. Landau. Pergamon Press Ltd, Oxford, 1965

1965
[116]

U. Fano. Description of states in quantum mechanics by density matrix and operator techniques. Reviews of Modern Physics, 29:74–93, 1957

1957
[117]

B. S. DeWitt and R. N. Graham. The many-worlds interpretation of quantum mechanics: a funda- mental exposition by Hugh Everett III. Princeton University Press, Princeton, 1973

1973
[118]

J. M. Jauch. Foundations of quantum mechanics. Addison–Wesley Publishing Company, Reading, 1968

1968
[119]

M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information . Cambridge University Press, Cambridge, 2000

2000
[120]

Einstein, B

A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Physical Review, 47:777–780, 1935

1935
[121]

N. Bohr. Can quantum-mechanical description of physical reality be considered complete? Physical Review, 48:696–702, 1935

1935
[122]

J. Mehra. The quantum principle: its interpretation and epistemology . D. Reidel Publishing Com- pany, Dordrecht, 1974

1974
[123]

M. Born. The Born–Einstein Letters. Macmillan, London, 1971

1971
[124]

Schrödinger

E. Schrödinger. Discussion of probability relations between separated systems. Proceedings of the Cambridge Philosophical Society, 31:555–563, 1935

1935
[125]

Schrödinger

E. Schrödinger. Probability relations between separated systems. Proceedings of the Cambridge Philosophical Society, 32:446–452, 1935

1935
[126]

J. D. Trimmer. The present situation in quantum mechanics: A translation of schrödinger’s ‘cat paradox’ paper. Proceedings of the American Philosophical Society, 124:323–338, 1980

1980
[127]

Everett III

H. Everett III. ‘relative state’ formulation of quantum mechanics. Reviews of Modern Physics , 29:454–492, 1957. 74

1957
[128]

J. A. Wheeler. Assessment of Everett’s ’relative state’ formulation of quantum theory. Reviews of Modern Physics, 29:463–465, 1957

1957
[129]

Freitas and O

F. Freitas and O. Freire Jr. A formulaÃ ˘gÃˇco dos ‘estados relativos’ da teoria quântica. Revista Brasileira de Ensino de Física, 30:1–15, 2008

2008
[130]

B. S. DeWitt. Bryce S. DeWitt letter to John Wheeler, Wheeler’s copy. http://ucispace.lib.uci.edu/handle/10575/1145
[131]

C. A. Brasil and L. A. Castro. Understanding the pointer states. European Journal of Physics , 36:065024, 2014

2014
[132]

Lombardi and L

O. Lombardi and L. Vanni. Medición cuántica y decoherencia: Â£qué medimos cuando medimos? Scientiae Studia, 8:273–291, 2010

2010
[133]

W. H. Zurek. Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse? Physical Review D, 24:1516–1525, 1981

1981
[134]

W. H. Zurek. Decoherence and the transition from quantum to classical. Physics Today, 44:36, 1991

1991
[135]

Fortin and L

S. Fortin and L. Vanni. Quantum decoherence: a logical perspective. Foundations of Physics , 44:1258–1268, 2014

2014
[136]

R. Omnès. Results and problems in decoherence theory. Brazilian Journal of Physics, 35:207–210, 2005

2005
[137]

Schlosshauer

M. Schlosshauer. Decoherence, the measurement problem, and interpretations of quantum mechan- ics. Reviews of Modern Physics, 76:1267–1305, 2004

2004
[138]

Schlosshauer

M. Schlosshauer. Decoherence and the quantum-to-classical transition. Springer, Berlin, 2007

2007
[139]

W. H. Zurek. Decoherence, einselection, and the quantum origins of the classical.Reviews of Modern Physics, 75:715–775, 2003

2003
[140]

W. H. Zurek. Decoherence and the transition from quantum to classical – revisited. Los Alamos Science, 27:2–25, 2002

2002
[141]

R. Omnès. Logical reformulation of quantum mechanics. i. Foundations. Journal of Statistical Physics, 53:893–932, 1988

1988
[142]

R. Omnès. Logical reformulation of quantum mechanics. ii. Interferences and the Einstein– Podolsky–Rosen experiment. Journal of Statistical Physics, 53:933–955, 1988

1988
[143]

R. Omnès. Logical reformulation of quantum mechanics. iii. Classical limit and irreversibility. Jour- nal of Statistical Physics, 53:957–975, 1988. 75

1988
[144]

R. Omnès. Logical reformulation of quantum mechanics. iv. Projectors in semiclassical physics. Journal of Statistical Physics, 57:357–382, 1989

1989
[145]

R. Omnès. Consistent interpretations of quantum mechanics. Reviews of Modern Physics, 64:329– 382, 1992

1992
[146]

R. B. Grifﬁths. Consistent histories and the interpretation of quantum mechanics. Journal of Statis- tical Physics, 36:212–272, 1984

1984
[147]

Dowker and A

F. Dowker and A. Kent. On the consistent histories approach to quantum mechanics. Journal of Statistical Physics, 82:1575–1646, 1996

1996
[148]

R. Kastner. ‘Einselection’ of point observables: The new H-theorem?, 2014. arXiv:1406.4126

2014 arXiv
[149]

Aharonov, D

Y . Aharonov, D. Z. Albert, and L. Vaidman. How the result of a measurement of a component of the spin of a spin- 1 2 particle can turn out to be 100. Physical Review Letters, 60:1351–1354, 1988

1988
[150]

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan. The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100.Physical Review D, 40:2112–2117, 1989

1989
[151]

Danan, D

A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman. Asking photons where they have been.Physical Review Letters, 111:240402, 2013

2013
[152]

L. A. Castro, C. A. Brasil, and R. d. J. Napolitano. Weak values in collision theory. Annals of Physics, 392:272–286, 2018

2018
[153]

Ferrie and J

C. Ferrie and J. Combes. How the result of a single coin toss can turn out to be 100 heads. Physical Review Letters, 113:120404, 2014

2014
[154]

S. K. Berberian. Notes on Spectral Theory. D. Van Nostrand Company, Inc., Princeton, 1966

1966
[155]

J. M. Jauch and C. Piron. Generalized localizability. Helvetica Physica Acta, 40:559, 1967

1967
[156]

H. E. Brandt. Positive operator valued measure in quantum information processing. American Jour- nal of Physics, 67:434–439, 1999

1999
[157]

Hellwig and K

K.-E. Hellwig and K. Kraus. Pure operations and measurements. Communications in Mathematical Physics, 11:214–220, 1969

1969
[158]

Hellwig and K

K.-E. Hellwig and K. Kraus. Operations and measurements ii. Communications in Mathematical Physics, 16:142–147, 1969

1969
[159]

K. Kraus. States, effects and operations – Fundamental notions of quantum theory. Springer–Verlag, Berlin, 1983

1983
[160]

K. Kraus. General state changes in quantum theory. Annals of Physics, 64:311–335, 1971. 76

1971
[161]

J. Maziero. A representaÃ ˘gÃˇco de kraus para a dinâmica de sistemas quânticos abertos – the kraus representation for the dynamics of open quantum systems. Revista Brasileira de Ensino de Física , 38:e2307, 2016

2016
[162]

Breuer and F

H.-P. Breuer and F. Petruccione. The theory of open quantum systems . Oxford University Press, Oxford, 2002

2002
[163]

Caldeira

A. Caldeira. An introduction to macroscopic quantum phenomena and quantum dissipation . Cam- bridge University Press, Cambridge, 2014

2014
[164]

F. Reif. Fundamentals of statistical and thermal physics. Waveland Press Inc, Long Grove, 1965

1965
[165]

C. A. Brasil, F. F. Fanchini, and R. d. J. Napolitano. A simple derivation of the Lindblad equation. Revista Brasileira de Ensino de Física, 35:1303, 2013

2013
[166]

Lindblad

G. Lindblad. On the generators of quantum dynamical semigroups. Communications in Mathemati- cal Physics, 48:119–130, 1976

1976
[167]

S. L. Adler. Derivation of the Lindblad generator structure by use of the it ¯o stochastic calculus. Physics Letters A, 265:58–61, 2000

2000
[168]

S. L. Adler. Corrigendum to: Derivation of the Lindblad generator structure by use of the it¯o stochas- tic calculus. Physics Letters A, 267:212, 2000

2000
[169]

A. Peres. Classical interventions in quantum systems. i. The measuring process. Physical Review A, 61:022116, 2000

2000
[170]

Weinberg

S. Weinberg. What happens in a measurement? Physical Review A, 93:032124, 2016

2016
[171]

Distler and S

J. Distler and S. Paban. von Neumann’s formula, measurements and the Lindblad equation, 2017. arXiv:1702.01724v1

2017 arXiv
[172]

C. A. Brasil and R. d. J. Napolitano. The master equation for the reduced open-system dynamics, including a Lindbladian description of the ﬁnite-duration measurement. European Physical Journal Plus, 126:91, 2011

2011
[173]

C. A. Brasil, L. A. Castro, and R. d. J. Napolitano. Protecting a quantum state from environmental noise by an incompatible ﬁnite-time measurement. Physical Review A, 84:022112, 2011

2011
[174]

J. D. Cresser, S. M. Barnett, J. Jeffers, and D. T. Pegg. Measurement master equation. Optics Communications, 264:352–361, 2006

2006
[175]

Jacobs and D

K. Jacobs and D. A. Steck. A straightforward introduction to continuous quantum measurement. Contemporary Physics, 47:279–303, 2006. 77

2006
[176]

D. N. Klyshko. Reduction of the wave function: an operational approach. Physics Letters A , 243:179–186, 1998

1998
[177]

H. W. Wiseman and G. Milburn. Quantum measurement and control. Cambridge University Press, Cambridge, 2009

2009
[178]

C. A. Brasil, L. A. Castro, and R. d. J. Napolitano. How much time does a measurement take? Foundations of Physics, 43:642–655, 2015

2015
[179]

C. A. Brasil, L. A. Castro, and R. d. J. Napolitano. Efﬁcient ﬁnite-time measurements under thermal regimes. European Physical Journal Plus, 129:206, 2014

2014
[180]

Misra and E

B. Misra and E. C. G. Sudarshan. The Zeno’s paradox in quantum theory. Journal of Mathematical Physics, 18:756, 1977

1977
[181]

Facchi and S

P. Facchi and S. Pascazio. Quantum Zeno phenomena: Pulsed versus continuous measurement. Fortschritte der Physik, 49:941, 2001

2001
[182]

W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland. Quantum Zeno effect. Physical Review A, 41:2295, 1990

1990
[183]

A. G. Kofman and G. Kurizki. Acceleration of quantum decay processes by frequent observations. Nature, 405:546–550, 2000

2000
[184]

McArdle, S

S. McArdle, S. Endo, A. Aspuru-Guzik, S. Benjamin, and X. Yuan. Quantum computational chem- istry, 2019. arXiv:quant-ph/1808.10402v2

2019 arXiv
[185]

V . B. Braginsky and F. Y . Khalili.Quantum measurement. Cambridge University Press, Cambridge, 1992

1992
[186]

Busch, P

P. Busch, P. J. Lahti, and P. Mittelstaedt. The quantum theory of measurement . Springer–Verlag, Berlin, 1996

1996
[187]

K. Jacobs. Quantum measurement theory and its applications . Cambridge University Press, Cam- bridge, 2014

2014
[188]

M. B. Mensky. Continuous quantum measurements and path integrals . IOP Publishing, London, 1993

1993
[189]

M. B. Mensky. Quantum measurements and decoherence – models and phenomenology . Springer- Science+Business, Dordrecht, 2000

2000
[190]

Bassi and G

A. Bassi and G. C. Ghirardi. Dynamical reduction models. Physics Reports, 379:257–426, 2003

2003
[191]

Namiki and S

M. Namiki and S. Pascazio. Quantum theory of measurement based on the many-hilbert-space ap- proach. Physics Reports, 232:201–411, 1993. 78

1993
[192]

Tombesi and D

P. Tombesi and D. F. Walls. Quantum measurements in optics – Proceedings of a NATO Advanced Research Workshop on Quantum Measurements in Optics . Springer Science+Business, New York, 1992

1992
[193]

Percival

I. Percival. Quantum state diffusion. Cambridge University Press, Cambridge, 1998

1998
[194]

Ghirardi, R

G. Ghirardi, R. Rimini, and T. Weber. Uniﬁed dynamics for microscopic and macroscopic systems. 34:470–491, 1986

1986
[195]

Ghirardi, P

G. Ghirardi, P. Pearle, and A. Rimini. Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles. Physical Review A, 42:78–89, 1990

1990
[196]

P. Pearle. Combining stochastic dynamical state-vector reduction with spontaneous localization. Physical Review A, 39:2277–2289, 1989

1989
[197]

G. C. Wick, A. S. Wightman, and E. P. Wigner. The intrinsic parity of elementary particles. Physical Review, 88:101–105, 1952

1952
[198]

G. C. Wick, A. S. Wightman, and E. P. Wigner. Superselection rule for charge. Physical Review D, 1:3267–3267, 1970

1970
[199]

J. Bub. How to solve the measurement problem of quantum mechanics. Foundations of Physics, 18:701–722, 1988

1988
[200]

N. P. Landsman. Observation and superselection in quantum mechanics, 1994. arXiv:hep- th/9411173

1994
[201]

D. Giulini. Superselection rules, 2007. arXiv:0710.1516

2007 arXiv
[202]

G. J. Milburn and D. F. Walls. Quantum nondemolition measurements via quadratic coupling. Phys- ical Review A, 28:2065–2070, 1983

1983
[203]

D. F. Walls, M. J. Collet, and G. J. Milburn. Analysis of a quantum measurement. Physical Review D, 32:3208, 1985

1985
[204]

Grangier, J

P. Grangier, J. A. Levenson, and J.-P. Poizat. Quantum non-demolition measurements in optics. Nature, 396:537–542, 1998

1998
[205]

O. P. Sá Neto, M. C. de Oliveira, and G. J. Milburn. Temperature measurement and phonon number statistics of a nanoelectromechanical resonator. New Journal of Physics, 17:093010, 2015

2015
[206]

C. S. Unnikrishnan. Quantum non-demolition measurements: concepts, theory and practice. Current Science, 109:2052, 2015

2015
[207]

V . B. Braginsky and F. Y . Khalili. Quantum nondemolition measurements. Reviews of Modern Physics, 68, 1996. 79

1996
[2005]

the number of available preparations and that of available outcomes may be different from each other, and also different from the dimensionality of Hilbert space

as corresponding to the situation where “ the number of available preparations and that of available outcomes may be different from each other, and also different from the dimensionality of Hilbert space ”[12]. In other words, POVMs allow us to generalize projective measuremen...

1938

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

de Laplace

P.-S. de Laplace. A philosophical essay on probabilities. John Wiley & Sons, New York, 1902. 67

1902

[2] [2]

A. Pais. Subtle is the Lord... – The science and the life of Albert Einstein . Oxford University Press, Oxford, 1982

1982

[3] [3]

P. W. Shor. Scheme for reducing decoherence in quantum computer memory. Physical Review A, 52, 1995

1995

[4] [4]

H. J. Briegel and R. Raussendorf. Persistent entanglement in arrays of interacting particles. Physical Review Letters, 86:910–913, 2001

2001

[5] [5]

H. J. Briegel, D. E. Browner, W. Dür, R. Raussendorf, and M. V . D. Nest. Measurement-based quantum computation. Nature Physics, 5:19–26, 2009

2009

[6] [6]

Eisberg and R

R. Eisberg and R. Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles . John Wiley & Sons, New York, 1985

1985

[7] [7]

A. Messiah. Quantum mechanics, vol. 1. North-Holland Publishing Company, Amsterdam, 1961

1961

[8] [8]

L. I. Schiff. Quantum mechanics. McGraw-Hill Book Company, New York, 1968

1968