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The Essence of Quantum Computing Part 2 of 3 Part Series

A.5.4 Trace of a matrix

Another important matrix function is the trace of a matrix, tr(A), defined as the sum of the diagonal elements of A (which need not be a diagonal matrix),

The trace is easily seen to be cyclic, that is, tr(AB) = tr(BA), and linear, i.e., tr(A + B) = tr(A) + tr(B), tr(zA) = z tr(A), where A and B are arbitrary matrices, and z is a complex number. It also follows from the cyclic property that the trace of a matrix is invariant under the unitary similarity transformation, A→UA† , since

We define the trace of an operator A to be the trace of any matrix representation of A. The invariance of the trace under unitary similarity transformations ensures that the trace of an operator is well defined. It also means that if we choose a unitary transformation that diagonalizes A, then tr(?) = Σ? λi,i.e., the trace of A is also the sum of its eigenvalues, λi.

An extremely important formula (presented here without proof) for evaluating tr(A|Ψ〉〈Ψ|) , where |Ψ〉 is a unit vector and A is an arbitrary operator, is

This result is extremely useful in evaluating the trace of an operator. It also plays a crucial role in deciding the probability with which a quantum system will collapse to a particular state when that system is measured. If A is the identity operator, then  tr(|Ψ〉〈Ψ| = 〈Ψ|Ψ〉).

A.5.5 Commutator and anti-commutator

The commutator between two operators A and B is defined as

[A, B] ≡ AB – BA.

If [A, B] = 0, that is, AB = BA, then we say A commutes with B. Similarly, the anti-commutator of two operators is defined by

{A, B} ≡ AB + BA.

A is said to anti-commute with B if {A, B} = 0. It so happens that many important properties of pairs of operators can be deduced from their commutator and anti-commutator. One such is the simultaneous diagonalization theorem, which states that if A and B are Hermitian operators, then [A, B] = 0 if and only if there exists an orthonormal basis such that both A and B are diagonal with respect to that basis. Thus, A and B are simultaneously diagonalizable in this case. It turns out that non-zero commutators play an essential role in the famous Heisenberg’s uncertainty principle in quantum mechanics. Indeed, non-commuting operators lie at the heart of quantum mechanics.

A.5.6 Tensor product

The tensor product of |v〉 and |w〉, represented by |v〉⊗ |w〉 (alternatively, by |v〉 |w〉 or |v , w〉 or |vw〉), produces a matrix. The tensor product puts vector spaces together to form larger vector spaces. This simple method is crucial for constructing the Hilbert space for multi-qubit systems. Let A be a m×n matrix, and B a p×q matrix. Then the way vector spaces are put together is as follows:

Here each element AijB represents a p×q submatrix whose entries are the entries of B multiplied by Aij. The fully expanded matrix form of A⊗B is a larger mp×nq matrix.

Example 1.

Finally, we have the useful notation |v⊗k , which stands for |v〉 tensored with itself k times. For example,

By definition,

We shall shortly see that an analogous notation is used for operators (as they also have a matrix representation) on tensor product spaces.

By definition, the tensor product satisfies the following basic properties:

One may also show that

where the superscripts *, T, and †, respectively, denote complex conjugate, transpose, and Hermitian conjugate42 (or adjoint) operations on the entity they operate on.

Finally, we come to the nature of linear operators that act on the space |V〉⊗|W〉. By convention (and this is important), a matrix representation of a linear operator will imply that the representation is with respect to orthonormal input and output bases. Further, if the input and output spaces for a linear operator are the same, then the input and output bases are assumed to be the same, unless noted otherwise.

Suppose |v〉 and |w〉 are vectors in V and W, and A and B are linear operators on V and W, respectively. Then we define a linear operator A⊗B on V⊗W by the equation

The definition of A⊗B is then extended to all elements of V⊗W in the natural way to ensure linearity of A⊗B, that is,

One can show thatA⊗B so defined is a well-defined linear operator on V⊗W. This notion of the tensor product of two operators extends in the obvious way to the case where A: V→V’ and B: W→W’ map between different vector spaces. Indeed, an arbitrary linear operator C mapping V⊗W to V’⊗W’ can be represented as a linear combination of tensor products of operators mapping V to V’ and W to W’,

where by definition

The inner products on the spaces V and W can be used to define a natural inner product on V⊗W.
Define

42 A† ≡ (AT)* ≡ (A*)T is the Hermitian conjugate or adjoint of matrix A. An operator A whose adjoint is also A is known as a Hermitian or self-adjoint operator.

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