basis) cannot both be made perfect by the use of any "pre-agreement" stored in some hidden variablesbut that quantum mechanics predicts perfect correlations. In a more refined formulation known as the Bell-CHSH inequality, it is shown that a certain correlation measure cannot exceed the value 2 if one assumes that physics respects the constraints of local "hidden-variable" theory (a sort of common-sense formulation of how information is conveyed), but certain systems permitted in quantum mechanics can attain values as high as . Thus, quantum theory violates the Bell inequality and the idea of local "hidden variables." Bell basis Four specific two-qubit states with the maximal value of are designated as "Bell states". They are known as the four maximally entangled two-qubit Bell states and they form a maximally entangled basis, known as the Bell basis, of the four-dimensional Hilbert space for two qubits: (1) (2) (3) (4) Creating Bell states Although there are many possible ways to create entangled Bell states through quantum circuits, the simplest takes a computational basis as the input, and contains a Hadamard gate and a CNOT gate (see picture). As an example, the pictured quantum circuit takes the two qubit input and transforms it to the first Bell state Explicitly, the Hadamard gate transforms into a superposition of . This will then act as a control input to the CNOT gate, which only inverts the target (the second qubit) when the control (the first qubit) is 1. Thus, the CNOT gate transforms the second qubit as follows . For the four basic two-qubit inputs, , the circuit outputs the four Bell states (listed above). More general, the circuit transforms the input in accordance with the equation where is the negation of . Properties of Bell states The result of a measurement of a single qubit in a Bell state is indeterminate, but upon measuring the first qubit in the z-basis, the result of measuring the second qubit is guaranteed to yield the same value (for the Bell states) or the opposite value (for the Bell states). This implies that the measurement outcomes are correlated. John Bell was the first to prove that the measurement correlations in the Bell State are stronger than could ever exist between classical systems. This hints that quantum mechanics allows information processing beyond what is possible with classical mechanics. In addition, the Bell states form an orthonormal basis and can therefore