^{1}

^{*}

^{2}

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
**σ
_{x}
and
σ
_{y}
, 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)]. **

The quantum theory (cf. [

As for the foundations of the quantum theory, Leggett-type non-local variables theory [

As for the applications of the quantum theory, the implementation of a quantum algorithm to solve Deutsch’s problem [

In quantum mechanics, the uncertainty principle is any of the variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle known as complementary variables, such as its position x and momentum p, can be known simultaneously. For instance, in 1927, Werner Heisenberg stated that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa [

Recently, Ozawa discusses universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement [

In this paper, 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

In this section, we review the Schrödinger uncertainty relation. The derivation shown here incorporates and builds off of those shown in Robertson [

For any Hermitian operator

We let

Similarly, for any other Hermitian operator

for

The product of the two deviations can thus be expressed as

In order to relate the two vectors

and thus Equation (4) can be written as

Since

We let

The inner product

and using the fact that

Similarly it can be shown that

For a pair of operators

Thus we have

and

where we have introduced the anticommutator,

We now substitute the above two equations above back into Equation (8) and get

Substituting the above into Equation (6) we get the Schrödinger uncertainty relation

In this section, we present an example that the Schrödinger uncertainty relation is optimal. The optimality is certified by the Bloch sphere. In fact, a violation of the Schrödinger uncertainty relation means a violation of the Bloch sphere in the specific case. We derive the Schrödinger uncertainty relation by using the Bloch sphere

relation in the specific case. Let

Statement 1

Proof. By using

Thus,

QED

We define N as follows:

We define S as follows:

We discuss the relation between N and S in the following statement.

Statement 2

Proof. We have the following relation:

QED

Thus the Schrödinger uncertainty relation is optimal in the specific case. The optimality is certified by the Bloch sphere. A violation of the Schrödinger uncertainty relation means a violation of the Bloch sphere in the specific case. Thus, the experiments [

In conclusions, we have derived 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 has been that we would measure

It is very interesting to study whether Heisenberg’s uncertainty principle would violate when we would use a new measurement theory

Koji Nagata,Tadao Nakamura, (2015) Violation of Heisenberg’s Uncertainty Principle. Open Access Library Journal,02,1-6. doi: 10.4236/oalib.1101797