Planck Constant and the Deutsch-Jozsa Algorithm

Transcription

1 Adv. Studies Theor. Phys., Vol. 4, 010, no. 0, Planck Constant and the Deutsch-Jozsa Algorithm Koji agata Future University Hakodate ko mi Tadao akamura Keio University Science and Technology Abstract It is shown that there is a contradiction within conventional quantum mechanics [K. agata, Int J Theor Phys: Volume 48, Issue 1 (009), Page 5: DOI: /s z] based on Planck constant. We investigate whether Planck constant meets our physical world. We derive a proposition concerning a quantum expectation value under the assumption of the existence of the orientation of reference frames in spin-1/ systems. This assumption intuitively depictures our physical world. The quantum predictions within the formalism of Planck constant violate the proposition with a magnitude that grows exponentially with the number of particles. Planck constant cannot depicture our physical world with a violation factor that grows exponentially with the number of particles. Planck constant cannot meet our physical world. We formulate Planck constant. Our formulation is equivalent to changing Planck constant (h) to new constant (h/ Dimension). PACS: a 1 Introduction It is shown that there is a contradiction within conventional quantum mechanics [1] based on Planck constant []. Planck constant (cf. [, 4, 5, 6, 7, 8]) gives approximate and at times remarkably accurate numerical predictions. Much experimental data approximately fits to the quantum predictions of Planck constant for the past some 100 years.

2 974 K. agata and T. akamura Planck constant says new science with respect to information theory. The science is called the quantum information theory [8]. Planck constant gives us very useful another theory in order to create new information science and to explain the handing of raw experimental data in our physical world. As for the foundations of Planck constant, Leggett-type nonlocal variables theory [9] is experimentally investigated [10, 11, 1]. The experiments report that Planck constant does not accept Leggett-type nonlocal variables interpretation. As for the applications of Planck constant, there are several attempts to use single-photon two-qubit states for quantum computing. Oliveira et al. implement Deutsch s algorithm [1] with polarization and transverse spatial modes of the electromagnetic field as qubits [14]. Single-photon Bell states are prepared and measured [15]. The decoherence-free implementation of Deutsch s algorithm is reported by using such single-photon and by using two logical qubits [16]. A one-way based experimental implementation of Deutsch s algorithm is reported [17]. Planck constant seems successful formula and it looks no problem in order to use it experimentally. Several researches address [] the mathematical formulation of the quantum theory. Planck constant is accepted widely. It is desirable that Planck constant is mathematically successful because we predict unknown physical phenomena precisely. Sometimes such predictions are effective in the field of elementary particle physics. We endure much time in order to see the fact by using for example large-scale accelerator. Rolf Landauer says Information is Physical [8]. We cannot create any computer without physical phenomena. This fact motivates us to investigate the Hilbert space formalism of Planck constant. Here we ask. Does Planck constant depicture our physical world? Unfortunately it is not so even in both the macroscopic scale and the microscopic scale. The theoretical formalism of the implementation of the Deutsch-Jozsa algorithm [1, 18] relies on Planck constant. We cannot implement the Deutsch-Jozsa algorithm by using Planck constant. In this paper we investigate whether Planck constant meets our physical world. We derive a proposition concerning a quantum expectation value under the assumption of the existence of the orientation of reference frames in spin-1/ systems. This assumption intuitively depictures our physical world. The quantum predictions within the formalism of Planck constant violate the proposition with a magnitude that grows exponentially with the number of particles. Planck constant cannot depicture our physical world with a violation factor that grows exponentially with the number of particles. Planck constant cannot meet our physical world. We formulate Planck constant. Our formulation is equivalent to changing Planck constant (h) to new constant

3 Planck constant and the Deutsch-Jozsa algorithm 975 (h/ Dimension). We use an established mathematical method presented in Refs. [1, 19, 0, 1,,, 4, 5]. In what follows we confine ourselves to the two-level (e.g. electron spin, photon polarizations, and so on). In what follows we confine ourselves to the discrete eigenvalue case. Problem of Planck constant.1 Plane settings model.1.1 Original Planck constant Assume that we have a set of spins 1. Each of them is a spin-1/ pure state lying in the z-x plane. Assume that one source of uncorrelated spin-carrying particles emits them in a state which can be described as a multi spin-1/ pure uncorrelated state. Parameterize the settings of the jth observer with a unit vector n j (its direction along which the spin component is measured) with j = 1,...,. We can introduce the Planck correlation function which is the average of the product of the results of measurements E Planck ( n 1, n,..., n )= r( n 1, n,..., n ) avg (1) where r is the result. The value of r is ±1 (in ( /) unit) which is obtained if the measurement directions are set at n 1, n,..., n if we accept Planck constant. Also we can introduce a quantum correlation function with the system in a multi spin-1/ pure uncorrelated state E QM ( n 1, n,..., n )=Tr[ρ n 1 σ n σ n σ]. () denotes the tensor product. denotes the scalar product in R. σ =(σ z,σ x ) denotes a vector of two Pauli operators. ρ denotes a multi spin-1/ pure uncorrelated state ρ = ρ 1 ρ ρ () with ρ j = Ψ j Ψ j. Ψ j is a spin-1/ pure state lying in the z-x plane. We can write the observable (unit) vector n j in a plane coordinate system as follows n j (θ k j j ) = cos θk j j x(1) j + sin θ k j j x() j (4)

4 976 K. agata and T. akamura where x (1) j = z j and x () j values θj 1 =0,θj = π,...,θn 1 j = n = x j are the Cartesian axes. The angle θ k j j (n )π, and θj n = n takes n (n 1)π. (5) n We derive a necessary condition to be satisfied by the quantum correlation function with the system in a multi spin-1/ pure uncorrelated state given in (). In more detail we derive (E QM,E QM ) which is the value of the scalar product of the quantum correlation function E QM given in (). We use decomposition (4). We introduce simplified notations as and Then we have T i1 i...i =Tr[ρ x (i 1) 1 σ x (i ) σ x (i ) σ] (6) (E QM,E QM )= c j =(c 1 j,c j ) = (cos θk j j, sin θk j j ). (7) n k 1 =1 ( n k =1 i 1,...,i =1 T i1...i c i 1 1 c i ) ( n ) = i 1,...,i =1 T i 1...i ( n ) (8) where we use the orthogonality relation n k j =1 cα j cβ j = nδ α,β. The value of i 1,...,i =1 T i 1...i is bounded as i 1,...,i =1 T i 1...i 1. We have j=1 i j =1 (Tr[ρ j x (i j) j σ]) 1. (9) It is important that the inequality (8) is saturated iff ρ is a multi spin-1/ pure uncorrelated state such that ρ j = Ψ j Ψ j and Ψ j is a spin-1/ pure state lying in the z-x plane for every j. The reason of the inequality (8) is due to the following quantum inequality (Tr[ρ j x (i j) j σ]) 1. (10) i j =1 The inequality (10) is saturated iff ρ j = Ψ j Ψ j and Ψ j is a spin-1/ pure state lying in the z-x plane. The inequality (8) is saturated iff the inequality

5 Planck constant and the Deutsch-Jozsa algorithm 977 (10) is saturated for every j. Thus we have the maximal possible value of the scalar product as a quantum proposition concerning our physical world ( n ) (E QM,E QM ) max = (11) when the system is in a multi spin-1/ pure uncorrelated state. On the other hand a correlation function satisfies Planck constant if it can be written as E Planck ( n 1, n,..., n ) = lim r( n 1, n,..., n,l) m (1) where l denotes a label and r denotes the result of measurements of the dichotomic observables parameterized by directions of n 1, n,..., n. Assume the quantum correlation function with the system in a multi spin- 1/ pure uncorrelated state given in () admits Planck constant. We have the following proposition concerning Planck constant E QM ( n 1, n,..., n ) = lim r( n 1, n,..., n,l). (1) m In what follows we show that we cannot assign the truth value 1 for the proposition (1) concerning Planck constant. Assume the proposition (1) is true. We have same quantum correlation function by changing the label l into l and by changing the label m into m as follows l E QM ( n 1, n,..., n ) = lim =1 r( n 1, n,..., n,l ). (14) m m The value of the right-hand-side of (1) is equal to the value of the righthand-side of (14) because we only change the labels. We abbreviate r( n 1, n,..., n,l)tor(l). We abbreviate r( n 1, n,..., n,l ) to r(l ).

6 978 K. agata and T. akamura We have (E QM,E QM ) ( n n = = = k 1 =1 n k 1 =1 n k 1 =1 n k 1 =1 We use the following k =1 ( n k =1 ( n k =1 ( n k =1 lim lim lim lim r(l) ) m l lim =1 r(l ) m m m ) m m lim l =1 r(l)r(l ) m m m m ) lim l =1 r(l)r(l ) m m ) m lim l =1 = n. (15) m m r( n 1, n,..., n,l) r( n 1, n,..., n,l ) =1. (16) The inequality (15) is saturated since we have {l r( n 1, n,..., n,l)=1 l } = {l r( n 1, n,..., n,l )=1 l } and {l r( n 1, n,..., n,l)= 1 l } = {l r( n 1, n,..., n,l )= 1 l }.(17) denotes {1,,...,+ }. Hence we have the following proposition concerning Planck constant (E QM,E QM ) max = n. (18) We cannot assign the truth value 1 for two propositions (11) (concerning our physical world) and (18) (concerning Planck constant) simultaneously when the system is in a multi spin-1/ pure uncorrelated state. Each of them is a spin-1/ pure state lying in the z-x plane. We are in the contradiction when the system is in a multi spin-1/ pure uncorrelated state. We cannot accept the validity of the proposition (1) (concerning Planck constant) if we assign the truth value 1 for the proposition (11) (concerning our physical world). In other words Planck constant does not reveal our physical world..1. Reformulation of Planck constant We formulate Planck constant. We have the maximal possible value of the scalar product as a quantum proposition concerning our physical world ( n ) (E QM,E QM ) max = (19)

7 Planck constant and the Deutsch-Jozsa algorithm 979 when the system is in a multi spin-1/ pure uncorrelated state. On the other hand we have the following proposition concerning Planck constant (E QM,E QM ) max = n. (0) We cannot assign the truth value 1 for two propositions (19) (concerning our physical world) and (0) (concerning Planck constant) simultaneously when the system is in a multi spin-1/ pure uncorrelated state. Each of them is a spin-1/ pure state lying in the z-x plane. We are in the contradiction when the system is in a multi spin-1/ pure uncorrelated state. We formulate Planck constant as follows. Formulated Planck constant: h. (1) The value of r is ±( 1 ) (in ( /) unit) which is obtained if the measurement directions are set at n 1, n,..., n if we accept formulated Planck constant. The proposition (0) (concerning Planck constant) becomes the following new proposition concerning formulated Planck constant when we accept formulated Planck constant ( n ) (E QM,E QM ) max =. () We can assign the truth value 1 for both two propositions (19) (concerning our physical world) and () (concerning formulated Planck constant) simultaneously when the system is in a multi spin-1/ pure uncorrelated state. Each of them is a spin-1/ pure state lying in the z-x plane. We are not in the contradiction when the system is in a multi spin-1/ pure uncorrelated state. We solve the contradiction presented in the previous section by formulating Planck constant..1. ew type of the Deutsch-Jozsa algorithm The earliest quantum algorithm, the Deutsch-Jozsa algorithm, is representative to show that quantum computation is faster than classical counterpart with a magnitude that grows exponentially with the number of qubits. Let us follow the argumentation presented in [8]. The application, known as Deutsch s problem, may be described as the following game. Alice, in Amsterdam, selects a number x from 0 to 1, and mails it in a letter to Bob, in Boston. Bob calculates the value of some function f : {0,..., 1}

8 980 K. agata and T. akamura {0, 1} and replies with the result, which is either 0 or 1. ow, Bob has promised to use a function f which is of one of two kinds; either the value of f(x) is constant for all values of x, or else the value of f(x) is balanced, that is, equal to 1 for exactly half of all the possible x, and 0 for the other half. Alice s goal is to determine with certainty whether Bob has chosen a constant or a balanced function, corresponding with him as little as possible. How fast can she succeed? In the classical case, Alice may only send Bob one value of x in each letter. At worst, Alice will need to query Bob at least / + 1 times, since she may receive / 0s before finally getting a 1, telling her that Bob s function is balanced. The best deterministic classical algorithm she can use therefore requires / + 1 queries. ote that in each letter, Alice sends Bob bits of information. Furthermore, in this example, physical distance is being used to artificially elevate the cost of calculating f(x), but this is not needed in the general problem, where f(x) may be inherently difficult to calculate. If Bob and Alice were able to exchange qubits, instead of just classical bits, and if Bob agreed to calculate f(x) using a unitary transformation U f, then Alice could achieve her goal in just one correspondence with Bob, using the following algorithm. Alice has an qubit register to store her query in, and a single qubit register which she will give to Bob, to store the answer in. She begins by preparing both her query and answer registers in a superposition state. Bob will evaluate f(x) using quantum parallelism and leave the result in the answer register. Alice then interferes states in the superposition using a Hadamard transformation (a unitary transformation), H =(σ z + σ x )/, on the query register, and finishes by performing a suitable measurement to determine whether f was constant or balanced. Let us follow the quantum states through this algorithm. The input state is ψ 0 = 0 1. () Here the query register describes the state of qubits all prepared in the 0 state. After the Hadamard transformation on the query register and the Hadamard gate on the answer register we have ψ 1 = x {0,1} x [ ] 0 1. (4) The query register is now a superposition of all values, and the answer register is in an evenly weighted superposition of 0 and 1. ext, the function f is

9 Planck constant and the Deutsch-Jozsa algorithm 981 evaluated (by Bob) using U f : x, y x, y f(x), giving ( 1) f(x) x [ ] 0 1. (5) ψ = ± x Here y f(x) is the bitwise XOR (exclusive OR) of y and f(x). Alice now has a set of qubits in which the result of Bob s function evaluation is stored in the amplitude of the qubit superposition state. She now interferes terms in the superposition using a Hadamard transformation on the query register. To determine the result of the Hadamard transformation it helps to first calculate the effect of the Hadamard transformation on a state x. By checking the cases x = 0 and x = 1 separately we see that for a single qubit H x = z ( 1)xz z /. Thus H x 1,...,x z = 1,...,z ( 1) x 1z 1 + +x z z 1,...,z. (6) This can be summarized more succinctly in the very useful equation H z ( 1)x z z x =, (7) where x z is the bitwise inner product of x and z, modulo. Using this equation and (5) we can now evaluate ψ, ψ = ± [ ] ( 1) x z+f(x) z 0 1. (8) z x Alice now observes the query register. ote that the absolute value of the amplitude for the state 0 is x ( 1)f(x) /. Let s look at the two possible cases f constant and f balanced to discern what happens. In the case where f is constant the absolute value of the amplitude for 0 is +1. Because ψ is of unit length it follows that all the other amplitudes must be zero, and an observation will yield (+ 1 )s for all qubits in the query register. Thus, global measurement outcome is (+ 1 ). If f is balanced then the positive and negative contributions to the absolute value of the amplitude for 0 cancel, leaving an amplitude of zero, and a measurement must yield a result other than + 1 (i.e., 1 ) on at least one qubit in the query register. Summarizing, if Alice measures all (+ 1 )s and global measurement outcome is (+ 1 ) the function is constant; otherwise the function is balanced. We notice that the difference between + 1 and 1 is approximately zero when 1. We question if the Deutsch-Jozsa algorithm in the macroscopic scale is possible or not. This question is open problem.

10 98 K. agata and T. akamura. Sphere settings model..1 Original Planck constant Assume that we have a set of spins 1. Each of them is a spin-1/ pure state. Assume that one source of uncorrelated spin-carrying particles emits them in a state which can be described as a multi spin-1/ pure uncorrelated state. Parameterize the settings of the jth observer with a unit vector n j (its direction along which the spin component is measured) with j = 1,...,. We can introduce the Planck correlation function which is the average of the product of the results of measurements E Planck ( n 1, n,..., n )= r( n 1, n,..., n ) avg (9) where r is the result of measurements. The value of r is ±1 (in ( /) unit) which is obtained if the measurement directions are set at n 1, n,..., n if we accept Planck constant. Also we can introduce a quantum correlation function with the system in a multi spin-1/ pure uncorrelated state E QM ( n 1, n,..., n )=Tr[ρ n 1 σ n σ n σ]. (0) denotes the tensor product. denotes the scalar product in R. σ = (σ z,σ x,σ y ) denotes the vector of Pauli operator. ρ denotes a multi spin-1/ pure uncorrelated state ρ = ρ 1 ρ ρ (1) with ρ j = Ψ j Ψ j. Ψ j is a spin-1/ pure state. We can write the observable (unit) vector n j in a spherical coordinate system as follows n j (θ j,φ j ) = sin θ j cos φ j x (1) j + sin θ j sin φ j x () j + cos θ j x () j () where x (1) j = z j, x () j = x j, and x () j = y j are the Cartesian axes. We derive a necessary condition to be satisfied by the quantum correlation function with the system in a multi spin-1/ pure uncorrelated state given in (0). In more detail we derive (E QM,E QM ) which is the value of the scalar product of the quantum correlation function E QM given in (0). We use decomposition (). We introduce the measure dω j = sin θ j dθ j dφ j for the system of the jth observer. We introduce simplified notations as T i1 i...i =Tr[ρ x (i 1) 1 σ x (i ) σ x (i ) σ] ()

11 Planck constant and the Deutsch-Jozsa algorithm 98 and Then we have c j =(c 1 j,c j,c j) = (sin θ j cos φ j, sin θ j sin φ j, cos θ j ). (4) (E QM,E QM )= ( dω 1 dω i 1,...,i =1 T i1...i c i 1 1 c i ) = ( ) 4π Ti 1...i i 1,...,i =1 ( ) 4π (5) where we use the orthogonality relation dω j c α j cβ j = 4πδ α,β. The value of i 1,...,i =1 T i 1...i is bounded as i 1,...,i =1 T i 1...i 1. We have j=1 i j =1 (Tr[ρ j x (i j) j σ]) 1. (6) It is important that this inequality is saturated iff ρ is a multi spin-1/ pure uncorrelated state. The reason of the inequality (5) is due to the Bloch sphere (Tr[ρ j x (i j) j σ]) 1. (7) i j =1 The inequality (7) is saturated iff ρ j = Ψ j Ψ j and Ψ j is a spin-1/ pure state. The inequality (5) is saturated iff the inequality (7) is saturated for every j. Thus we have the maximal possible value of the scalar product as a quantum proposition concerning our physical world (E QM,E QM ) max = ( ) 4π (8) when the system is in a multi spin-1/ pure uncorrelated state. On the other hand a correlation function satisfies Planck constant if it can be written as E Planck ( n 1, n,..., n ) = lim r( n 1, n,..., n,l) m (9) where l denotes a label and r denotes the result of measurements of the dichotomic observables parameterized by directions of n 1, n,..., n.

12 984 K. agata and T. akamura Assume the quantum correlation function with the system in a multi spin- 1/ pure uncorrelated state given in (0) admits Planck constant. We have the following proposition concerning Planck constant E QM ( n 1, n,..., n ) = lim r( n 1, n,..., n,l). (40) m In what follows we show that we cannot assign the truth value 1 for the proposition (40) concerning Planck constant. Assume the proposition (40) is true. We have same quantum correlation function by changing the label l into l and by changing the label m into m as follows l E QM ( n 1, n,..., n ) = lim =1 r( n 1, n,..., n,l ). (41) m m The value of the right-hand-side of (40) is equal to the value of the righthand-side of (41) because we only change the labels. We abbreviate r( n 1, n,..., n,l)tor(l). We abbreviate r( n 1, n,..., n,l ) to r(l ). We have We use the following (E QM,E QM ) ( = dω 1 dω ( = dω 1 dω ( dω 1 dω ( = dω 1 dω lim r(l) m lim lim lim m m m lim m lim l =1 m lim m lim m ) l =1 r(l ) m m r(l)r(l ) ) l =1 r(l)r(l ) m ) m l =1 m ) =(4π). (4) r( n 1, n,..., n,l) r( n 1, n,..., n,l ) =1. (4) The inequality (4) is saturated since we have {l r( n 1, n,..., n,l)=1 l } = {l r( n 1, n,..., n,l )=1 l } and {l r( n 1, n,..., n,l)= 1 l } = {l r( n 1, n,..., n,l )= 1 l }.(44)

13 Planck constant and the Deutsch-Jozsa algorithm 985 Hence we have the following proposition concerning Planck constant (E QM,E QM ) max =(4π). (45) We cannot assign the truth value 1 for two propositions (8) (concerning our physical world) and (45) (concerning Planck constant) simultaneously when the system is in a multi spin-1/ pure uncorrelated state. Each of them is a spin-1/ pure state. We are in the contradiction when the system is in a multi spin-1/ pure uncorrelated state. Thus we cannot accept the validity of the proposition (40) (concerning Planck constant) if we assign the truth value 1 for the proposition (8) (concerning our physical world). In other words Planck constant does not reveal our physical world... Reformulation of Planck constant We formulate Planck constant. We have the maximal possible value of the scalar product as a quantum proposition concerning our physical world ( ) 4π (E QM,E QM ) max = (46) when the system is in a multi spin-1/ pure uncorrelated state. On the other hand we have the following proposition concerning Planck constant (E QM,E QM ) max =(4π). (47) We cannot assign the truth value 1 for two propositions (46) (concerning our physical world) and (47) (concerning Planck constant) simultaneously when the system is in a multi spin-1/ pure uncorrelated state. We are in the contradiction when the system is in a multi spin-1/ pure uncorrelated state. We formulate Planck constant as follows. Formulated Planck constant: h. (48) The value of r is ±( 1 ) (in ( /) unit) which is obtained if the measurement directions are set at n 1, n,..., n if we accept formulated Planck constant. The proposition (47) (concerning Planck constant) becomes the following new proposition concerning formulated Planck constant when we accept formulated Planck constant (E QM,E QM ) max = ( ) 4π. (49)

14 986 K. agata and T. akamura We can assign the truth value 1 for both two propositions (46) (concerning our physical world) and (49) (concerning formulated Planck constant) simultaneously when the system is in a multi spin-1/ pure uncorrelated state. We are not in the contradiction when the system is in a multi spin-1/ pure uncorrelated state. We solve the contradiction presented in the previous section by formulating Planck constant... ew type of the Deutsch-Jozsa algorithm The earliest quantum algorithm, the Deutsch-Jozsa algorithm, is representative to show that quantum computation is faster than classical counterpart with a magnitude that grows exponentially with the number of qubits. Let us follow the argumentation presented in [8]. The application, known as Deutsch s problem, may be described as the following game. Alice, in Amsterdam, selects a number x from 0 to 1, and mails it in a letter to Bob, in Boston. Bob calculates the value of some function f : {0,..., 1} {0, 1} and replies with the result, which is either 0 or 1. ow, Bob has promised to use a function f which is of one of two kinds; either the value of f(x) is constant for all values of x, or else the value of f(x) is balanced, that is, equal to 1 for exactly half of all the possible x, and 0 for the other half. Alice s goal is to determine with certainty whether Bob has chosen a constant or a balanced function, corresponding with him as little as possible. How fast can she succeed? In the classical case, Alice may only send Bob one value of x in each letter. At worst, Alice will need to query Bob at least / + 1 times, since she may receive / 0s before finally getting a 1, telling her that Bob s function is balanced. The best deterministic classical algorithm she can use therefore requires / + 1 queries. ote that in each letter, Alice sends Bob bits of information. Furthermore, in this example, physical distance is being used to artificially elevate the cost of calculating f(x), but this is not needed in the general problem, where f(x) may be inherently difficult to calculate. If Bob and Alice were able to exchange qubits, instead of just classical bits, and if Bob agreed to calculate f(x) using a unitary transformation U f, then Alice could achieve her goal in just one correspondence with Bob, using the following algorithm. Alice has an qubit register to store her query in, and a single qubit register which she will give to Bob, to store the answer in. She begins by preparing both her query and answer registers in a superposition state. Bob will evaluate f(x) using quantum parallelism and leave the result in the answer register.

15 Planck constant and the Deutsch-Jozsa algorithm 987 Alice then interferes states in the superposition using a Hadamard transformation (a unitary transformation), H =(σ z + σ x )/, on the query register, and finishes by performing a suitable measurement to determine whether f was constant or balanced. Let us follow the quantum states through this algorithm. The input state is ψ 0 = 0 1. (50) Here the query register describes the state of qubits all prepared in the 0 state. After the Hadamard transformation on the query register and the Hadamard gate on the answer register we have ψ 1 = x {0,1} x [ ] 0 1. (51) The query register is now a superposition of all values, and the answer register is in an evenly weighted superposition of 0 and 1. ext, the function f is evaluated (by Bob) using U f : x, y x, y f(x), giving ( 1) f(x) x [ ] 0 1. (5) ψ = ± x Here y f(x) is the bitwise XOR (exclusive OR) of y and f(x). Alice now has a set of qubits in which the result of Bob s function evaluation is stored in the amplitude of the qubit superposition state. She now interferes terms in the superposition using a Hadamard transformation on the query register. To determine the result of the Hadamard transformation it helps to first calculate the effect of the Hadamard transformation on a state x. By checking the cases x = 0 and x = 1 separately we see that for a single qubit H x = z ( 1)xz z /. Thus H x 1,...,x z = 1,...,z ( 1) x 1z 1 + +x z z 1,...,z. (5) This can be summarized more succinctly in the very useful equation H z ( 1)x z z x =, (54) where x z is the bitwise inner product of x and z, modulo. equation and (5) we can now evaluate ψ, ψ = ± z x ( 1) x z+f(x) z Using this [ 0 1 ]. (55)

16 988 K. agata and T. akamura Alice now observes the query register. ote that the absolute value of the amplitude for the state 0 is x ( 1)f(x) /. Let s look at the two possible cases f constant and f balanced to discern what happens. In the case where f is constant the absolute value of the amplitude for 0 is +1. Because ψ is of unit length it follows that all the other amplitudes must be zero, and an observation will yield (+ 1 )s for all qubits in the query register. Thus, global measurement outcome is (+ 1 ). If f is balanced then the positive and negative contributions to the absolute value of the amplitude for 0 cancel, leaving an amplitude of zero, and a measurement must yield a result other than + 1 (i.e., 1 ) on at least one qubit in the query register. Summarizing, if Alice measures all (+ 1 )s and global measurement outcome is (+ 1 ) the function is constant; otherwise the function is balanced. We notice that the difference between + 1 and 1 is approximately zero when 1. We question if the Deutsch-Jozsa algorithm in the macroscopic scale is possible or not. This question is open problem. Conclusions In conclusion we have investigated whether Planck constant meets our physical world. We have derived a proposition concerning a quantum expectation value under the assumption of the existence of the orientation of reference frames in spin-1/ systems. The quantum predictions within the formalism of Planck constant have violated the proposition with a magnitude that grows exponentially with the number of particles. Planck constant cannot have depictured our physical world with a violation factor that grows exponentially with the number of particles. Planck constant cannot have met our physical world. We have formulated Planck constant. Our formulation has been equivalent to changing Planck constant (h) to new constant (h/ Dimension). References [1] K. agata, Int J Theor Phys: Volume 48, Issue 1 (009), Page 5: DOI: /s z. [] Planck constant - Wikipedia, the free encyclopedia. [] J. von eumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, ew Jersey, 1955).

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18 990 K. agata and T. akamura [] K. agata and T. akamura, Adv. Studies Theor. Phys. 4, 197 (010). [] K. agata and T. akamura, arxiv: [4] K. agata and T. akamura, Advances and Applications in Statistical Sciences, Volume, Issue 1, (010), Page 195. [5] K. agata and T. akamura, Adv. Studies Theor. Phys. (010) Does singleton set meet Zermelo-Fraenkel set theory with the axiom of choice? Received: September, 010