In a quantum computing model, if the probability amplitude for a qubit state is given by \( a + bi \) such that \( |a + bi| = 1 \), determine the values of \( a \) and \( b \) when the state represents a state with a phase difference of \( \frac{\pi}{3} \) from the real axis. Find the sum of all possible values of \( a \).
["Quantum Computing Fundamentals: Understanding Qubit States with Phase and Probability Amplitudes", "In quantum computing, qubits are the fundamental units of quantum information, represented by state vectors in a two-dimensional complex vector space. A general qubit state can be expressed as:", "[\n|\psi\rangle = a + bi\n\quad \ ext{where} \quad a, b \in \mathbb{R}\n]", "This expression uses the condition that the total probability must be normalized, meaning:", "[\n|a + bi|^2 = a^2 + b^2 = 1\n]", "Such a state can be rewritten in polar form using a well-defined probability amplitude that includes both magnitude and phase. Specifically, for a single qubit, we often describe the state as:", "[\n|\psi\rangle = \cos\left(\frac{\ heta}{2}\right) + i\sin\left(\frac{\ heta}{2}\right)\n]", "where ( \ heta ) is the relative phase of the qubit with respect to the real (real-axis) state.", "In this model, the probability amplitudes are split into real and imaginary parts:", "- ( a = \cos\left(\frac{\ heta}{2}\right) )\n- ( b = \sin\left(\frac{\ heta}{2}\right) )", "The phase difference from the real axis corresponds to ( \ heta ). The problem specifies a phase difference of ( \frac{\pi}{3} ) from the real axis. Since the real axis corresponds to ( \ heta = 0 ), the desired phase is:", "[\n\ heta = \frac{\pi}{3}\n]", "Substituting into the expressions for ( a ) and ( b ):", "[\na = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\n]\n[\nb = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\n]", "However, quantum states are defined up to a global phase. This means that adding ( 2\pi ) to ( \ heta ) yields an equivalent state:", "[\n|\psi\rangle = \cos\left(\frac{\ heta + 2\pi}{2}\right) + i\sin\left(\frac{\ heta + 2\pi}{2}\right)\n]", "Thus, the general values for ( a ) and ( b ) satisfying a phase difference of ( \frac{\pi}{3} ) modulo ( 2\pi ) are:", "[\na = \cos\left(\frac{\pi}{3} + \pi k\right), \quad b = \sin\left(\frac{\pi}{3} + \pi k\right), \quad k \in \mathbb{Z}\n]", "But since cosine and sine are periodic with period ( 2\pi ), and ( \frac{\pi}{3} + \pi k ) covers all distinct directions on the unit circle when ( k = 0 ) or ( k = 1 ), we consider the two fundamental solutions:", "- When ( k = 0 ):\n [\n a = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2},\quad b = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\n ]", "- When ( k = 1 ):\n [\n a = \cos\left(\frac{\pi}{3} + \pi\right) = \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2},\quad b = \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}\n ]", "These two pairs represent antipodal points on the Bloch sphere, both describing valid quantum states with ( \ heta = \frac{\pi}{3} ) or ( \ heta = \frac{\pi}{3} + \pi = \frac{4\pi}{3} ), which differ only by a global phase of ( -1 ).", "The question asks for the sum of all possible values of ( a ). From above, the only distinct real values are:", "[\na = \frac{1}{2} \quad \ ext{and} \quad a = -\frac{1}{2}\n]", "Summing these:", "[\n\frac{1}{2} + \left(-\frac{1}{2}\right) = 0\n]", "Conclusion:\nThe sum of all possible real values of ( a ) corresponding to a qubit state with a phase difference of ( \frac{\pi}{3} ) from the real axis—accounting for global phase ambiguity—is:", "[\n\boxed{0}\n]"]
