Note that while we cannot speak of an observable having a value for a particular state, we can speak of the probability of its having any specified value for the state (i.e., the probability of it being obtained when one makes a measurement) and we can define an average value for the state. It is permissible for a quantum state to be simultaneously an eigenstate of two observables; the chances for such an existence are most favorable if the two observables commute and rather exceptional when they do not. When the observables commute, there exist so many simultaneous eigenstates that they form a complete set.89 The case when two observables commute is special in the sense that the observations are non-interfering or compatible in such a way that one can give a meaning to the two observations being made simultaneously and discuss the probability of their being obtained. Indeed, one may view the two observations as a single observation of a more complicated type, the result of which is two real numbers rather than one. A natural generalization of this is that any two or more commuting observable-operators may be treated as a single observable-operator whose action produces two or more numbers. The states for which this measurement is certain to lead to one particular result are the simultaneous eigenstates.
After measurement, the system remains in its collapsed state unless altered. Till altered, any repetition of the measurement will only reproduce the first measurement. This is the deep connection between the set of eigenvalues of an observable-operator and the possible results of measurement associated with that observable-operator. One may, of course, alter the post-measurement state of the system by measuring a different observable or measuring in a different basis or by applying a quantum (unitary) operation.
7.3 Global and relative phase factor
Global phase factor
Consider the state eiϒiϒ|Ψ〉 is equal to |Ψ〉, up to a global phase factor eiϒ. What is interesting is that the measurement statistics predicted for these two states are identical. To see this, suppose Mm is a measurement operator associated to some quantum measurement, then the respective probabilities that an outcome corresponding to index m will occur are
Thus, from a measurement point of view the two states are identical and global phase factors are irrelevant to the observed properties of physical systems. This symmetry, of all things, implies the conservation of electric charge in the Universe!90
Relative phase factor
Consider the states a|0〉 + b|1〉 and c|0〉 + d|1〉. We say that two complex amplitudes, such as, a and c differ by a relative phase factor if there is a real θ (relative phase shift) such that a = eiθc. We likewise say that complex amplitudes b and d differ by a relative phase factor if there is a real ∅ such that b = ei∅d. Note that θ and ∅ depend on the basis being used to represent the states.
Example. In the two states (1/√2) (|0〉 + |1〉) and (1/√2) (|0〉 – |1〉), the amplitudes of |0〉 are the same in the two states, hence they have a relative phase factor of 1, while the amplitudes of |1〉 differ by a relative phase factor of -1.
The difference between the relative phase factors and global phase factors is that for relative phase the phase factors may vary from amplitude to amplitude, which makes the relative phase a basis-dependent concept unlike the global phase. As a result, states, which differ only by relative phases in some basis, give rise to physically observable differences in measurement statistics, and it is no longer possible to regard these states as being physically equivalent.
7.4 Unitary operators
A matrix or an operator U is unitary if U† U = I. An operator is unitary if and only if each of its matrix representations is unitary. U is invertible, normal, and has a spectral decomposition (i.e., it can be diagonalized), and preserves inner products between vectors. Each eigenvalue of a unitary matrix has modulus 1 (i.e., it has the form eiθ for some real θ). The tensor product of two unitary operators is unitary. Unitarity simply means the equality of two lengths. Any unitary transformation is equivalent to a rotation, and an L2 length that remains invariant to a rotation. A L2 length has the form of a sum of squares.
Rather surprisingly, the unitarity constraint is the only constraint required of quantum gates. Thus, any unitary matrix specifies a valid quantum gate! Further, it can be shown that any arbitrary 2×2 unitary matrix may be decomposed as
Quantum computing uses unitary operators to change the state of discrete state vectors when that change is being made under postulate 2. This change can be reversed by the corresponding inverse operator (which always exists) if it is applied before any measurement is made on the quantum system. Reversible operators (gates) only move states around; they have the same number of inputs and outputs and have a one-to-one mapping between input vectors and output vectors. Every reversible gate can be described by a permutation. Thus, a reversible operator’s input and output are uniquely retrievable from each other. As no information is lost, energy is conserved. Finally, since the evolution of a quantum system is governed by unitary operators, non-unitary operations (such as irreversible operations) are not admissible when operating under postulate 2. Thus, quantum gates do not allow (1) feedback loops from one part of the system to another, (2) fan-in operations (that is, bitwise OR of inputs), and (3) fan-out operations (because it would violate the no-cloning theorem).
7.5 Important qubit transformations
We now look at some qubit transformations (gates) which are central to quantum computing. The reader should commit these transformations to memory and become familiar with their use. The 1-qubit transformations (gates), namely the Pauli gates, and the Hadamard gate, and the 2-qubit Cnot gate are particularly important. They are all unitary matrices. In matrix form, by convention, the basis states are lexicographically ordered, e.g., 3-qubit gates are ordered according to |000〉, |001〉, |010〉, |011〉, |100〉, |101〉, |110〉, and |111〉.
In quantum computing, Pauli gates (also called σ-matrices), named after Wolfgang Pauli who won the 1945 Nobel Prize in physics for his “decisive contribution through his discovery of a new law of Nature, the exclusion principle or Pauli principle”, play a central role in the manipulation of single qubits. They are represented by 2×2 matrices as shown in Table 7.1. Except for I, all have trace zero. Some authors do not include I in the set of Pauli matrices (also called σ-matrices). We will include I. Any 2×2 matrix M can be expressed as the linear sum M =α I +β X + γY + δZ, where α,β ,γ ,δ are complex constants. Thus, there are infinitely many 2×2 unitary matrices, and therefore infinitely many single qubit gates, but each is expressible as linear combinations of the matrices, I, X, Y, and Z. This contrasts with classical information theory, where only two logic gates are possible for a single bit, namely the identity and the logical NOT operation.
 See, e.g., Peskin & Schroeder (1995), Section 7.4 (“The Ward-Takahashi identity”).
 A half-silvered mirror effects this transformation on a photon when it encounters the mirror.