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68-438: PEAR employed electronic random event generators (REGs) to explore the ability of test subjects to use psychokinesis to influence the random output distribution of these devices to conform to their pre-recorded intentions to produce higher numbers, lower numbers, or nominal baselines. Most of these experiments utilized a microelectronic REG, but experiments were also conducted with "a giant, wall-mounted pachinko -like machine with

136-500: A {\displaystyle E_{a}} and E b {\displaystyle E_{b}} are non-zero, and using these two results we obtain where " ∗ {\displaystyle ^{\ast }} " indicates the complex conjugate. It is now easy to show that τ † τ = I {\displaystyle \tau ^{\dagger }\tau =\mathbf {I} } where I {\displaystyle \mathbf {I} }

204-434: A ^ a † , a ^ b † , a ^ c † {\displaystyle {\hat {a}}_{a}^{\dagger },{\hat {a}}_{b}^{\dagger },{\hat {a}}_{c}^{\dagger }} , and a ^ d † {\displaystyle {\hat {a}}_{d}^{\dagger }} , so that where

272-669: A ^ d † {\displaystyle {\hat {a}}_{c}^{\dagger }{\hat {a}}_{d}^{\dagger }} term has cancelled. Therefore the output states always have even numbers of photons in each arm. A famous example of this is the Hong–Ou–Mandel effect , in which the input has n = m = 1 {\displaystyle n=m=1} , the output is always | 20 ⟩ c d {\displaystyle |20\rangle _{cd}} or | 02 ⟩ c d {\displaystyle |02\rangle _{cd}} , i.e.

340-561: A b {\displaystyle |00\rangle _{ab}} and add a photon in port a to produce then the beam splitter creates a superposition on the outputs of The probabilities for the photon to exit at ports c and d are therefore | r a c | 2 {\displaystyle |r_{ac}|^{2}} and | t a d | 2 {\displaystyle |t_{ad}|^{2}} , as might be expected. Likewise, for any input state | n m ⟩

408-401: A b {\displaystyle |nm\rangle _{ab}} and the output is Using the multi-binomial theorem , this can be written where M = n + m − N {\displaystyle M=n+m-N} and the ( n j ) {\displaystyle {\tbinom {n}{j}}} is a binomial coefficient and it is to be understood that

476-514: A c = ϕ 0 + ϕ R {\displaystyle \phi _{ad}=\phi _{0}+\phi _{T},\phi _{bc}=\phi _{0}-\phi _{T},\phi _{ac}=\phi _{0}+\phi _{R}} (and from the constraint ϕ b d = ϕ 0 − ϕ R − π {\displaystyle \phi _{bd}=\phi _{0}-\phi _{R}-\pi } ), so that where 2 ϕ T {\displaystyle 2\phi _{T}}

544-542: A d − ϕ b d + ϕ b c − ϕ a c = π {\displaystyle \phi _{ad}-\phi _{bd}+\phi _{bc}-\phi _{ac}=\pi } . To include the constraints and simplify to 4 independent parameters, we may write ϕ a d = ϕ 0 + ϕ T , ϕ b c = ϕ 0 − ϕ T , ϕ

612-434: A and E b each incident at one of the inputs, the two output fields E c and E d are linearly related to the inputs through where the 2×2 element τ {\displaystyle \tau } is the beam-splitter transfer matrix and r and t are the reflectance and transmittance along a particular path through the beam splitter, that path being indicated by the subscripts. (The values depend on

680-417: A Mach–Zehnder interferometer . In this case there are two incoming beams, and potentially two outgoing beams. But the amplitudes of the two outgoing beams are the sums of the (complex) amplitudes calculated from each of the incoming beams, and it may result that one of the two outgoing beams has amplitude zero. In order for energy to be conserved (see next section), there must be a phase shift in at least one of

748-523: A comparator . If the voltage is above threshold, the comparator output is 1, otherwise 0. The random bit value is latched using a flip-flop. Sources of noise vary and include: The drawbacks of using noise sources for an RNG design are: The idea of chaos-based noise stems from the use of a complex system that is hard to characterize by observing its behavior over time. For example, lasers can be put into (undesirable in other applications) chaos mode with chaotically fluctuating power, with power detected using

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816-401: A deterministic algorithm and non-physical nondeterministic random bit generators that do not include hardware dedicated to generation of entropy. Many natural phenomena generate low-level, statistically random " noise " signals, including thermal and shot noise, jitter and metastability of electronic circuits, Brownian motion , and atmospheric noise . Researchers also used

884-475: A hardware random number generator ( HRNG ), true random number generator ( TRNG ), non-deterministic random bit generator ( NRBG ), or physical random number generator is a device that generates random numbers from a physical process capable of producing entropy (in other words, the device always has access to a physical entropy source ), unlike the pseudorandom number generator (PRNG, a.k.a. "deterministic random bit generator", DRBG) that utilizes

952-462: A photodiode and sampled by a comparator. The design can be quite small, as all photonics elements can be integrated on-chip. Stipčević & Koç characterize this technique as "most objectionable", mostly due to the fact that chaotic behavior is usually controlled by a differential equation and no new randomness is introduced, thus there is a possibility of the chaos-based TRNG producing a limited subset of possible output strings. The TRNGs based on

1020-402: A physical vapor deposition method. The thickness of the deposit is controlled so that part (typically half) of the light, which is incident at a 45-degree angle and not absorbed by the coating or substrate material, is transmitted and the remainder is reflected. A very thin half-silvered mirror used in photography is often called a pellicle mirror . To reduce loss of light due to absorption by

1088-407: A TRNG (when compared with pseudo random number generators) provide no meaningful benefits. TRNGs have additional drawbacks for data science and statistical applications: impossibility to re-run a series of numbers unless they are stored, reliance on an analog physical entity can obscure the failure of the source. The TRNGs therefore are primarily used in the applications where their unpredictability and

1156-453: A beam as it reflects or transmits at that surface. Then we obtain Further simplifying, the relationship becomes which is true when ϕ a d − ϕ b d + ϕ b c − ϕ a c = π {\displaystyle \phi _{ad}-\phi _{bd}+\phi _{bc}-\phi _{ac}=\pi } and

1224-529: A beam-combiner in three- LCD projectors , in which light from three separate monochrome LCD displays is combined into a single full-color image for projection. Beam splitters with single-mode fiber for PON networks use the single-mode behavior to split the beam. The splitter is done by physically splicing two fibers "together" as an X. Arrangements of mirrors or prisms used as camera attachments to photograph stereoscopic image pairs with one lens and one exposure are sometimes called "beam splitters", but that

1292-562: A cascade of bouncing balls". In 1986 associates of PEAR published data collected over the course of seven years from a group of subjects attempting to influence random number generators across millions of trials. In all cases, the observed effects were very small (between one and about 0.1%), and although the statistical significance of the results at the P<;0.05 level is not generally disputed, detractors point to potential ethical violations and flaws in experiment procedures, as well as questioning

1360-415: A cube, a beam splitter is made from two triangular glass prisms which are glued together at their base using polyester, epoxy , or urethane-based adhesives. (Before these synthetic resins , natural ones were used, e.g. Canada balsam .) The thickness of the resin layer is adjusted such that (for a certain wavelength ) half of the light incident through one "port" (i.e., face of the cube) is reflected and

1428-515: A fast-rotating 10-sector disk that was illuminated by periodic bursts of light. The sampling was done by a human who wrote the number under the light beam onto a pad. The device was utilized to produce a 100,000-digit random number table (at the time such tables were used for statistical experiments, like PRNG nowadays). On 29 April 1947, the RAND Corporation began generating random digits with an "electronic roulette wheel", consisting of

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1496-559: A free-running oscillator (FRO) typically utilize one or more ring oscillators (ROs), outputs of which are sampled using yet another oscillator. Since inverters forming the RO can be thought of as amplifiers with a very large gain, an FRO output exhibits very fast oscillations in phase in frequency domains. The FRO-based TRNGs are very popular due to their use of the standard digital logic despite issues with randomness proofs and chip-to-chip variability. Quantum random number generation technology

1564-450: A free-running oscillator-based TRNG can be attacked using a frequency injection . There are mathematical techniques for estimating the entropy of a sequence of symbols. None are so reliable that their estimates can be fully relied upon; there are always assumptions which may be very difficult to confirm. These are useful for determining if there is enough entropy in a seed pool, for example, but they cannot, in general, distinguish between

1632-515: A large and carefully prepared table had never before been available. It has been a useful source for simulations, modeling, and for deriving the arbitrary constants in cryptographic algorithms to demonstrate that the constants had not been selected maliciously (" nothing up my sleeve numbers "). Since the early 1950s, research into TRNGs has been highly active, with thousands of research works published and about 2000 patents granted by 2017. A lot of different TRNG designs were proposed over time with

1700-402: A large variety of noise sources and digitization techniques ("harvesting"). However, practical considerations (size, power, cost, performance, robustness) dictate the following desirable traits: Stipčević & Koç in 2014 classified the physical phenomena used to implement TRNG into four groups: Noise-based RNGs generally follow the same outline: the source of a noise generator is fed into

1768-510: A medium with a lower refractive index. The behavior is dictated by the Fresnel equations . This does not apply to partial reflection by conductive (metallic) coatings, where other phase shifts occur in all paths (reflected and transmitted). In any case, the details of the phase shifts depend on the type and geometry of the beam splitter. For beam splitters with two incoming beams, using a classical, lossless beam splitter with electric fields E

1836-456: A prior resource only (this setting hence shares certain similarities with a Gaussian counterpart of the KLM protocol ). The building block of this simulation procedure is the fact that a beam splitter is equivalent to a squeezing transformation under partial time reversal . Reflection beam splitters reflect parts of the incident radiation in different directions. These partial beams show exactly

1904-465: A random bit) dates at least to the times of ancient Rome . The first documented use of a physical random number generator for scientific purposes was by Francis Galton (1890). He devised a way to sample a probability distribution using a common gambling dice. In addition to the top digit, Galton also looked at the face of a dice closest to him, thus creating 6*4 = 24 outcomes (about 4.6 bits of randomness). Kendall and Babington-Smith (1938) used

1972-446: A random frequency pulse source of about 100,000 pulses per second gated once per second with a constant frequency pulse and fed into a five-bit binary counter. Douglas Aircraft built the equipment, implementing Cecil Hasting's suggestion (RAND P-113) for a noise source (most likely the well known behavior of the 6D4 miniature gas thyratron tube, when placed in a magnetic field ). Twenty of the 32 possible counter values were mapped onto

2040-594: A sense that they can only operate in a fully controlled, trusted environment. The failure of a TRNG can be quite complex and subtle, necessitating validation of not just the results (the output bit stream), but of the unpredictability of the entropy source. Hardware random number generators should be constantly monitored for proper operation to protect against the entropy source degradation due to natural causes and deliberate attacks. FIPS Pub 140-2 and NIST Special Publication 800-90B define tests which can be used for this. The minimal set of real-time tests mandated by

2108-474: A single test subject (presumed to be a member of PEAR's staff) participated in 15% of PEAR's trials, and was responsible for half of the total observed effect. James Alcock in a review mentioned various problems with the PEAR experiments such as poor controls and documentation with the possibility of fraud, data selection and optional stopping not being ruled out. Alcock concluded there was no reason to believe

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2176-494: A symmetric beam splitter ϕ 0 = ϕ T = 0 , ϕ R = π / 2 {\displaystyle \phi _{0}=\phi _{T}=0,\phi _{R}=\pi /2} ), and for other phases where the output goes to one arm (e.g. the dielectric beam splitter ϕ 0 = ϕ T = ϕ R = 0 {\displaystyle \phi _{0}=\phi _{T}=\phi _{R}=0} )

2244-486: A true random source and a pseudorandom generator. This problem is avoided by the conservative use of hardware entropy sources. Beam splitter A beam splitter or beamsplitter is an optical device that splits a beam of light into a transmitted and a reflected beam. It is a crucial part of many optical experimental and measurement systems, such as interferometers , also finding widespread application in fibre optic telecommunications . In its most common form,

2312-477: A very literally "half-silvered" surface. Instead of a metallic coating, a dichroic optical coating may be used. Depending on its characteristics ( thin-film interference ), the ratio of reflection to transmission will vary as a function of the wavelength of the incident light. Dichroic mirrors are used in some ellipsoidal reflector spotlights to split off unwanted infrared (heat) radiation, and as output couplers in laser construction . A third version of

2380-563: Is a misnomer, as they are effectively a pair of periscopes redirecting rays of light which are already non-coincident. In some very uncommon attachments for stereoscopic photography, mirrors or prism blocks similar to beam splitters perform the opposite function, superimposing views of the subject from two different perspectives through color filters to allow the direct production of an anaglyph 3D image, or through rapidly alternating shutters to record sequential field 3D video. Beam splitters are sometimes used to recombine beams of light, as in

2448-430: Is a simplified version of Ref. The relation between the classical field amplitudes E a , E b , E c {\displaystyle {E}_{a},{E}_{b},{E}_{c}} , and E d {\displaystyle {E}_{d}} produced by the beam splitter is translated into the same relation of the corresponding quantum creation (or annihilation) operators

2516-487: Is an essential component in this scheme since it is the only one that creates entanglement between the Fock states . Similar settings exist for continuous-variable quantum information processing . In fact, it is possible to simulate arbitrary Gaussian (Bogoliubov) transformations of a quantum state of light by means of beam splitters, phase shifters and photodetectors, given two-mode squeezed vacuum states are available as

2584-423: Is expected to output near-perfect random numbers (" full entropy "). A physical process usually does not have this property, and a practical TRNG typically includes a few blocks: Hardware random number generators generally produce only a limited number of random bits per second. In order to increase the available output data rate, they are often used to generate the " seed " for a faster PRNG. DRBG also helps with

2652-852: Is given in the Fearn–Loudon 1987 paper and extended in Ref to include statistical mixtures with the density matrix . In general, for a non-symmetric beam-splitter, namely a beam-splitter for which the transmission and reflection coefficients are not equal, one can define an angle θ {\displaystyle \theta } such that { | R | = sin ⁡ ( θ ) | T | = cos ⁡ ( θ ) {\displaystyle {\begin{cases}|R|=\sin(\theta )\\|T|=\cos(\theta )\end{cases}}} where R {\displaystyle R} and T {\displaystyle T} are

2720-446: Is possible to create a universal quantum computer solely with beam splitters, phase shifters, photodetectors and single photon sources. The states that form a qubit in this protocol are the one-photon states of two modes, i.e. the states |01⟩ and |10⟩ in the occupation number representation ( Fock state ) of two modes. Using these resources it is possible to implement any single qubit gate and 2-qubit probabilistic gates. The beam splitter

2788-765: Is produced when θ = π / 4 {\displaystyle \theta =\pi /4} . The dielectric beam splitter above, for example, has i.e. ϕ T = ϕ R = ϕ 0 = 0 {\displaystyle \phi _{T}=\phi _{R}=\phi _{0}=0} , while the "symmetric" beam splitter of Loudon has i.e. ϕ T = 0 , ϕ R = − π / 2 , ϕ 0 = π / 2 {\displaystyle \phi _{T}=0,\phi _{R}=-\pi /2,\phi _{0}=\pi /2} . Beam splitters have been used in both thought experiments and real-world experiments in

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2856-433: Is the identity, i.e. the beam-splitter transfer matrix is a unitary matrix . Each r and t can be written as a complex number having an amplitude and phase factor; for instance, r a c = | r a c | e i ϕ a c {\displaystyle r_{ac}=|r_{ac}|e^{i\phi _{ac}}} . The phase factor accounts for possible shifts in phase of

2924-734: Is the phase difference between the transmitted beams and similarly for 2 ϕ R {\displaystyle 2\phi _{R}} , and ϕ 0 {\displaystyle \phi _{0}} is a global phase. Lastly using the other constraint that R 2 + T 2 = 1 {\displaystyle R^{2}+T^{2}=1} we define θ = arctan ⁡ ( R / T ) {\displaystyle \theta =\arctan(R/T)} so that T = cos ⁡ θ , R = sin ⁡ θ {\displaystyle T=\cos \theta ,R=\sin \theta } , hence A 50:50 beam splitter

2992-505: Is well established with 8 commercial quantum random number generator ( QRNG ) products offered before 2017. Herrero-Collantes & Garcia-Escartin list the following stochastic processes as "quantum": To reduce costs and increase robustness of quantum random number generators, online services have been implemented. A plurality of quantum random number generators designs are inherently untestable and thus can be manipulated by adversaries. Mannalath et al. call these designs "trusted" in

3060-465: The photoelectric effect , involving a beam splitter , other quantum phenomena, and even the nuclear decay (due to practical considerations the latter, as well as the atmospheric noise, is not viable). While "classical" (non-quantum) phenomena are not truly random, an unpredictable physical system is usually acceptable as a source of randomness, so the qualifiers "true" and "physical" are used interchangeably. A hardware random number generator

3128-506: The 10 decimal digits and the other 12 counter values were discarded. The results of a long run from the RAND machine, filtered and tested, were converted into a table, which originally existed only as a deck of punched cards , but was later published in 1955 as a book, 50 rows of 50 digits on each page ( A Million Random Digits with 100,000 Normal Deviates ). The RAND table was a significant breakthrough in delivering random numbers because such

3196-455: The area of quantum theory and relativity theory and other fields of physics . These include: In quantum mechanics, the electric fields are operators as explained by second quantization and Fock states . Each electrical field operator can further be expressed in terms of modes representing the wave behavior and amplitude operators, which are typically represented by the dimensionless creation and annihilation operators . In this theory,

3264-402: The beam splitter is a dichroic mirrored prism assembly which uses dichroic optical coatings to divide an incoming light beam into a number of spectrally distinct output beams. Such a device was used in three-pickup-tube color television cameras and the three-strip Technicolor movie camera. It is currently used in modern three-CCD cameras. An optically similar system is used in reverse as

3332-519: The certification bodies is not large; for example, NIST in SP 800-90B requires just two continuous health tests : Just as with other components of a cryptography system, a cryptographic random number generator should be designed to resist certain attacks . Defending against these attacks is difficult without a hardware entropy source. The physical processes in HRNG introduce new attack surfaces. For example,

3400-437: The coefficient is zero if j ∉ { 0 , n } {\displaystyle j\notin \{0,n\}} etc. The transmission/reflection coefficient factor in the last equation may be written in terms of the reduced parameters that ensure unitarity: where it can be seen that if the beam splitter is 50:50 then tan ⁡ θ = 1 {\displaystyle \tan \theta =1} and

3468-475: The constraints describing a lossless beam splitter, the initial expression can be rewritten as Applying different values for the amplitudes and phases can account for many different forms of the beam splitter that can be seen widely used. The transfer matrix appears to have 6 amplitude and phase parameters, but it also has 2 constraints: R 2 + T 2 = 1 {\displaystyle R^{2}+T^{2}=1} and ϕ

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3536-415: The cryptographic applications: A typical way to fulfill these requirements is to use a TRNG to seed a cryptographically secure pseudorandom number generator . Physical devices were used to generate random numbers for thousands of years, primarily for gambling . Dice in particular have been known for more than 5000 years (found on locations in modern Iraq and Iran), and flipping a coin (thus producing

3604-527: The exponential term reduces to -1. Applying this new condition and squaring both sides, it becomes where substitutions of the form | r a c | 2 = 1 − | t a d | 2 {\displaystyle |r_{ac}|^{2}=1-|t_{ad}|^{2}} were made. This leads to the result and similarly, It follows that R 2 + T 2 = 1 {\displaystyle R^{2}+T^{2}=1} . Having determined

3672-450: The four ports of the beam splitter are represented by a photon number state | n ⟩ {\displaystyle |n\rangle } and the action of a creation operation is a ^ † | n ⟩ = n + 1 | n + 1 ⟩ {\displaystyle {\hat {a}}^{\dagger }|n\rangle ={\sqrt {n+1}}|n+1\rangle } . The following

3740-405: The importance of large-sample studies that only marginally clear the p<0.05 significance threshold. The baseline for chance behavior used did not vary as statistically appropriate (baseline bind). Two PEAR researchers attributed this baseline bind to the motivation of the operators to achieve a good baseline and indicates that the random number generator used was not random. It has been noted that

3808-418: The impossibility to re-run the sequence of numbers are crucial to the success of the implementation: in cryptography and gambling machines. The major use for hardware random number generators is in the field of data encryption , for example to create random cryptographic keys and nonces needed to encrypt and sign data. In addition to randomness, there are at least two additional requirements imposed by

3876-515: The noise source "anonymization" (whitening out the noise source identifying characteristics) and entropy extraction . With a proper DRBG algorithm selected ( cryptographically secure pseudorandom number generator , CSPRNG), the combination can satisfy the requirements of Federal Information Processing Standards and Common Criteria standards. Hardware random number generators can be used in any application that needs randomness. However, in many scientific applications additional cost and complexity of

3944-383: The only factor that depends on j is the ( − 1 ) j {\displaystyle (-1)^{j}} term. This factor causes interesting interference cancellations. For example, if n = m {\displaystyle n=m} and the beam splitter is 50:50, then where the a ^ c †

4012-563: The other half is transmitted due to FTIR (frustrated total internal reflection) . Polarizing beam splitters , such as the Wollaston prism , use birefringent materials to split light into two beams of orthogonal polarization states. Another design is the use of a half-silvered mirror. This is composed of an optical substrate, which is often a sheet of glass or plastic, with a partially transparent thin coating of metal. The thin coating can be aluminium deposited from aluminium vapor using

4080-409: The outgoing beams. For example (see red arrows in picture on the right), if a polarized light wave in air hits a dielectric surface such as glass, and the electric field of the light wave is in the plane of the surface, then the reflected wave will have a phase shift of π, while the transmitted wave will not have a phase shift; the blue arrow does not pick up a phase-shift, because it is reflected from

4148-570: The output is always in the same arm, not random in either arm as is the case here. From the correspondence principle we might expect the quantum results to tend to the classical one in the limits of large n , but the appearance of large numbers of indistinguishable photons at the input is a non-classical state that does not correspond to a classical field pattern, which instead produces a statistical mixture of different | n , m ⟩ {\displaystyle |n,m\rangle } known as Poissonian light . Rigorous derivation

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4216-445: The polarization of the light.) If the beam splitter removes no energy from the light beams, the total output energy can be equated with the total input energy, reading Inserting the results from the transfer equation above with E b = 0 {\displaystyle E_{b}=0} produces and similarly for then E a = 0 {\displaystyle E_{a}=0} When both E

4284-405: The probability of output with a photon in each mode (a coincidence event) is zero. Note that this is true for all types of 50:50 beam splitter irrespective of the details of the phases, and the photons need only be indistinguishable. This contrasts with the classical result, in which equal output in both arms for equal inputs on a 50:50 beam splitter does appear for specific beam splitter phases (e.g.

4352-683: The procedure adopted. Details are not given about the subjects, the times they were tested, or the precise conditions under which they were tested." Physicist professor Milton Rothman has noted that Jahn's experiments at PEAR started from an idealistic assumption, ignored the laws of physics and had no basis in reality. PEAR's results have been criticized for deficient reproducibility . In one instance two German organizations failed to reproduce PEAR's results, while PEAR similarly failed to reproduce their own results. An attempt by York University's Stan Jeffers also failed to replicate PEAR's results. Hardware random number generator In computing ,

4420-642: The reflection and transmission coefficients. Then the unitary operation associated with the beam-splitter is then U ^ = e i θ ( a ^ a † a ^ b + a ^ a a ^ b † ) . {\displaystyle {\hat {U}}=e^{i\theta \left({\hat {a}}_{a}^{\dagger }{\hat {a}}_{b}+{\hat {a}}_{a}{\hat {a}}_{b}^{\dagger }\right)}.} In 2000 Knill, Laflamme and Milburn ( KLM protocol ) proved that it

4488-399: The reflective coating, so-called " Swiss-cheese " beam-splitter mirrors have been used. Originally, these were sheets of highly polished metal perforated with holes to obtain the desired ratio of reflection to transmission. Later, metal was sputtered onto glass so as to form a discontinuous coating, or small areas of a continuous coating were removed by chemical or mechanical action to produce

4556-412: The results were from paranormal origin. The psychologist C. E. M. Hansel , who evaluated Jahn's early psychokinesis experiments at the PEAR laboratory, wrote that a satisfactory control series had not been employed, that they had not been independently replicated, and that the reports lacked detail. Hansel noted that "very little information is provided about the design of the experiment, the subjects, or

4624-418: The transfer matrix is given in classical lossless beam splitter section above: Since τ {\displaystyle \tau } is unitary, τ − 1 = τ † {\displaystyle \tau ^{-1}=\tau ^{\dagger }} , i.e. This is equivalent to saying that if we start from the vacuum state | 00 ⟩

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