Violation of Heisenberg's Uncertainty Principle

Violation of Heisenberg’s uncertainty principle

Koji Nagata1 and Tadao Nakamura2 1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea E-mail: ko mi [email protected] 2Department of Information and− Computer− Science, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan E-mail: [email protected] ( Dated: July 28, 2015) Recently, violation of Heisenberg’s uncertainty relation in spin measurements is discussed [J. Erhart et al., Nature Physics 8, 185 (2012)] and [G. Sulyok et al., Phys. Rev. A 88, 022110 (2013)]. We derive the optimal limitation of Heisenberg’s uncertainty principle in a specific two-level system (e.g., electron spin, photon polarizations, and so on). Some physical situation is that we would measure σx and σy, simultaneously. The optimality is certified by the Bloch sphere. We show that a violation of Heisenberg’s uncertainty principle means a violation of the Bloch sphere in the specific case. Thus, the above experiments show a violation of the Bloch sphere when we use 1 as measurement outcome. This conclusion agrees with recent researches [K. Nagata, Int. J. Theor.± Phys. 48, 3532 (2009)] and [K. Nagata et al., Int. J. Theor. Phys. 49, 162 (2010)].

PACS numbers: 03.65.Ta, 03.67.-a

I. INTRODUCTION position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard [18] later that year The quantum theory (cf. [1—5]) gives accurate and at- and by Hermann Weyl [19] in 1928. times-remarkably accurate numerical predictions. Much Recently, Ozawa discusses universally valid reformu- experimental data has fit to quantum predictions for long lation of the Heisenberg uncertainty principle on noise time. and disturbance in measurement [20]. And experimen- As for the foundations of the quantum theory, Leggett- tal demonstration of a universally valid error-disturbance type non-local variables theory [6] is experimentally in- uncertainty relation in spin-measurements is discussed vestigated [7—9]. The experiments report that the quan- [21]. Violation of Heisenberg’s error-disturbance uncer- tum theory does not accept Leggett-type non-local vari- tainty relation in neutron spin measurements is also dis- ables interpretation. cussed [22]. As for the applications of the quantum theory, the im- In this paper, we derive the optimal limitation of plementation of a quantum algorithm to solve Deutsch’s Heisenberg’s uncertainty principle in a specific two-level problem [10] on a nuclear magnetic resonance quantum system (e.g., electron spin, photon polarizations, and so computer was reported firstly [11]. An implementation on). Some physical situation is that we would measure σx of the Deutsch-Jozsa algorithm on an ion-trap quantum and σy, simultaneously. The optimality is certified by the computer is also reported [12]. There are several at- Bloch sphere. We show that a violation of Heisenberg’s tempts to use single-photon two-qubit states for quan- uncertainty principle means a violation of the Bloch tum computing. Oliveira et al. implemented Deutsch’s sphere in the specific case. Especially, the Schrödinger algorithm with polarization and transverse spatial modes uncertainty relation is equivalent to the Bloch sphere re- of the electromagnetic field as qubits [13]. Single-photon lation in the physical situation. Thus, the experiments Bell states are prepared and measured [14]. Also the [21, 22] show a violation of the Bloch sphere when we decoherence-free implementation of Deutsch’s algorithm use 1 as measurement outcome. This conclusion agrees was reported by using such single-photon and by using with± recent researches [23, 24]. two logical qubits [15]. More recently, a one-way based experimental implementation of Deutsch’s algorithm was reported [16]. In quantum mechanics, the uncertainty principle is any II. THE SCHRODINGER¨ UNCERTAINTY of the variety of mathematical inequalities asserting a RELATION fundamental limit to the precision with which certain pairs of physical properties of a particle known as com- plementary variables, such as its position x and momen- In this section, we review the Schrödinger uncer- tum p, can be known simultaneously. For instance, in tainty relation. The derivation shown here incorpo- 1927, Werner Heisenberg stated that the more precisely rates and builds off of those shown in Robertson [25], the position of some particle is determined, the less pre- Schrödinger [26] and standard textbooks such as Grif- cisely its momentum can be known, and vice versa [17]. fiths [27]. The formal inequality relating the standard deviation of For any Hermitian operator Aˆ, based upon the defini- 2 tion of variance, we have and 2 ˆ ˆ ˆ ˆ σA = (A A )Ψ (A A )Ψ . (1) f g + g f = AˆBˆ Aˆ Bˆ + BÂˆ Aˆ Bˆ − | − | | − − We let f = (Aˆ Aˆ )Ψ and thus = A,ˆ Bˆ 2 Aˆ Bˆ , (12) | | − { } − σ2 = f f . (2) ˆ ˆ A | where we have introduced the anticommutator, A, B = ˆ ˆ ˆ ˆ { } Similarly, for any other Hermitian operator Bˆ in the same AB + BA. state We now substitute the above two equations above back into Eq. (8) and get σ2 = (Bˆ Bˆ )Ψ (Bˆ Bˆ )Ψ = g g (3) B − | − | 2 1 ˆ ˆ ˆ ˆ 2 1 ˆ ˆ 2 for g = (Bˆ Bˆ )Ψ . f g = ( A, B A B ) + ( [A, B] ) . | | − | | | 2{ } − 2i The product of the two deviations can thus be ex- (13) pressed as Substituting the above into Eq. (6) we get the σ2 σ2 = f f g g . (4) A B | | Schrödinger uncertainty relation In order to relate the two vectors f and g , we use the | | Cauchy-Schwarz inequality [28] which is defined as 1 2 1 2 σAσB ( A,ˆ Bˆ Aˆ Bˆ ) + ( [A,ˆ Bˆ] ) . ≥ 2{ } − 2i f f g g f g 2, (5) R | | ≥ | | | (14) and thus Eq. (4) can be written as σ2 σ2 f g 2. (6) A B ≥ | | | III. MAIN RESULT Since f g is in general a complex number, we use the | fact that the modulus squared of any complex number z is In this section, we present an example that the 2 defined as z = zz∗, where z∗ is the complex conjugate Schrödinger uncertainty relation is optimal. The opti- | | of z. The modulus squared can also be expressed as mality is certified by the Bloch sphere. In fact, a violation of the Schrödinger uncertainty relation means a z 2 = (Re(z))2 + (Im(z))2 | | violation of the Bloch sphere in the specific case. We z + z z z = ( ∗ )2 + ( − ∗ )2. (7) derive the Schrödinger uncertainty relation by using the 2 2i Bloch sphere relation in the specific case. Let V (X) be 2 2 We let z = f g and z = g f and substitute these into the variance of X, i.e., (X) X . ∗ − the equation above| to get | Statement 1 f g + g f f g g f 2 2 2 1 2 2 2 f g = ( | | ) + ( | − | ) . (8) V (σx)V (σy) ( [σx, σy] ) + σx σy . | | | 2 2i ≥ 2i 2 R The inner product f g is written out explicitly as (15) | 2 2 2 f g = (Aˆ Aˆ )Ψ (Bˆ Bˆ )Ψ , (9) Proof. By using 1 σx σy σz , we have | − | − − − ≥ ˆ ˆ and using the fact that A and B are Hermitian operators, V (σx)V (σy) we find 2 2 = (1 σx )(1 σy ) ˆ ˆ ˆ ˆ − 2 − 2 2 2 f g = Ψ (A A )(B B )Ψ = 1 σx σy + σx σy | | − − − − = Ψ (AˆBˆ Aˆ Bˆ Bˆ Aˆ + Aˆ Bˆ )Ψ σ 2 + σ 2 σ 2 | − − z x y ˆ ˆ ˆ ˆ ≥ = Ψ ABΨ Ψ A B Ψ 1 2 2 2 | − | = ( [σx, σy] ) + σx σy . (16) Ψ Bˆ Aˆ Ψ + Ψ Aˆ Bˆ Ψ 2i − | | = AˆBˆ Aˆ Bˆ Aˆ Bˆ + Aˆ Bˆ Thus, − − = AˆBˆ Aˆ Bˆ . (10) − 1 2 2 2 V (σx)V (σy) ( [σx, σy] ) + σx σy . Similarly it can be shown that g f = BÂˆ Aˆ Bˆ . ≥ 2i | − 2 R For a pair of operators Aˆ and Bˆ, we may define their (17) commutator as [A,ˆ Bˆ] = AˆBˆ BÂˆ. Thus we have − QED We define N as follows: f g g f = AˆBˆ Aˆ Bˆ BÂˆ + Aˆ Bˆ | − | − − 1 2 2 2 = [A,ˆ Bˆ] (11) N := ( [σx, σy] ) + σx σy . (18) 2i 3

We deﬁne S as follows: outcome. This conclusion agrees with recent researches Schrodinger term [23, 24].

1 2 1 2 S := ( σx, σy σx σy ) + ( [σx, σy] ) . )2{ } − N. 2i I IV. CONCLUSIONS (19) We discuss the relation between N and S in the following In conclusions, we have derived the optimal limitation statement. of Heisenberg’s uncertainty principle in a specific two- Statement 2 level system (e.g., electron spin, photon polarizations, and so on). Some physical situation has been that we N = S. (20) would measure σx and σy, simultaneously. The optimality has been certified by the Bloch sphere. We have Proof. We have the following relation: shown that a violation of Heisenberg’s uncertainty principle means a violation of the Bloch sphere in the specific σx, σy = 0. (21) { } case. Thus, the experiments [21, 22] have shown a vio- QED lation of the Bloch sphere when we use 1 as measure- Thus the Schrödinger uncertainty relation is optimal in ment outcome. This conclusion has agreed± with recent the specific case. The optimality is certified by the Bloch researches [23, 24]. sphere. A violation of the Schrödinger uncertainty rela- It is very interesting to study whether Heisenberg’s tion means a violation of the Bloch sphere in the specific uncertainty principle would violate when we would use case. Thus, the experiments [21, 22] show a violation a new measurement theory 1/√2 [24] as measurement of the Bloch sphere when we use 1 as measurement outcome. ± ±

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