Editing Quantum indeterminacy (section)

=== Example ===

[[Image:PauliSpinStateSpace.png|frame|right|[[Bloch sphere]] showing eigenvectors for Pauli Spin matrices. The Bloch sphere is a two-dimensional surface the points of which correspond to the state space of a spin 1/2 particle. At the state ''ψ'' the values of ''σ''<sub>1</sub> are +1 whereas the values of ''σ''<sub>2</sub> and ''σ''<sub>3</sub> take the values +1, −1 with probability 1/2.]]

In this example, we consider a single [[Spin-1/2|spin 1/2]] [[Elementary particle|particle]] (such as an electron) in which we only consider the spin degree of freedom. The corresponding Hilbert space is the two-dimensional complex Hilbert space '''C'''<sup>2</sup>, with each quantum state corresponding to a unit vector in '''C'''<sup>2</sup> (unique up to phase). In this case, the state space can be geometrically represented as the surface of a sphere, as shown in the figure on the right.

The [[Pauli matrix|Pauli spin matrices]]
<math display="block">
\sigma_1 = 
\begin{pmatrix}
0&1\\
1&0
\end{pmatrix},
\quad
\sigma_2 = 
\begin{pmatrix}
0&-i\\
i&0
\end{pmatrix},
\quad
\sigma_3 = 
\begin{pmatrix}
1&0\\
0&-1
\end{pmatrix}
</math>
are [[self-adjoint]] and correspond to spin-measurements along the 3 coordinate axes.

The Pauli matrices all have the eigenvalues +1, &minus;1.
* For ''σ''<sub>1</sub>, these eigenvalues correspond to the eigenvectors <math display="block"> \frac{1}{\sqrt{2}} (1,1), \frac{1}{\sqrt{2}} (1,-1) </math>
* For ''σ''<sub>3</sub>, they correspond to the eigenvectors <math display="block"> (1, 0), (0,1) </math>

Thus in the state
<math display="block"> \psi = \frac{1}{\sqrt{2}} (1,1), </math>
''σ''<sub>1</sub> has the determinate value +1, while measurement of ''σ''<sub>3</sub> can produce either +1, &minus;1 each with probability 1/2.  In fact, there is no state in which measurement of both ''σ''<sub>1</sub> and ''σ''<sub>3</sub> have determinate values.

There are various questions that can be asked about the above indeterminacy assertion.
# Can the apparent indeterminacy be construed as in fact deterministic, but dependent upon quantities not modeled in the current theory, which would therefore be incomplete? More precisely, are there ''hidden variables'' that could account for the statistical indeterminacy in a completely classical way?
# Can the indeterminacy be understood as a disturbance of the system being measured?
Von Neumann formulated the question 1) and provided an argument why the answer had to be no, ''if'' one accepted the formalism he was proposing. However, according to Bell, von Neumann's formal proof did not justify his informal conclusion.<ref>J.S. Bell, ''Speakable and Unspeakable in Quantum Mechanics'', Cambridge University Press, 2004, pg. 5.</ref>  A definitive but partial negative answer to 1) has been established by experiment: because [[Bell's inequalities]] are violated, any such hidden variable(s) cannot be ''local'' (see [[Bell test experiments]]).

The answer to 2) depends on how disturbance is understood, particularly since measurement entails disturbance (however note that this is the [[Observer effect (physics)|observer effect]], which is distinct from the uncertainty principle). Still, in the most natural interpretation the answer is also no. To see this, consider two sequences of measurements: (A) that measures exclusively ''σ''<sub>1</sub> and (B) that measures only ''σ''<sub>3</sub> of a spin system in the state ''ψ''. The measurement outcomes of (A) are all +1, while the statistical distribution of the measurements (B) is still divided between +1, &minus;1 with equal probability.