Stabilizer States as a Basis for Density Matrices

We show that the space of density matrices for n-qubit states, considered as a (2)-dimensional real vector space, has a basis consisting of density matrices of stabilizer states. We describe an application of this result to automated verification of quantum protocols. 1 Definitions and Results We are working with the stabilizer formalism [5], in which certain quantum states on sets of qubits are represented by the intersection of their stabilizer groups with the group generated by the Pauli operators. The stabilizer formalism is defined, explained and illustrated in a substantial literature; good introductions are given by Aaronson and Gottesman [1] and Nielsen and Chuang [7, Sec. 10.5]. In this paper we only need to use the following facts about stabilizer states. 1. The standard basis states are stabilizer states. 2. The set of stabilizer states is closed under application of Hadamard (H), Pauli (X, Y, Z), controlled not (CNot), and phase (P = ( 1 0 0 i ) ) gates. 3. The set of stabilizer states is closed under tensor product. Notation 1 Write the standard basis for n-qubit states as {|x〉 | 0 6 x < 2}, considering numbers to stand for their binary representations. We omit normalization factors when writing quantum states. Definition 1 Let GHZn = |0〉 + |2 − 1〉 and iGHZn = |0〉+ i|2 − 1〉, as n-qubit states. Lemma 1 For all n, GHZn and iGHZn are stabilizer states. Proof: By induction on n. For the base case (n = 1), we have that |0〉+ |1〉 and |0〉+ i|1〉 are stabilizer states, by applying H and then P to |0〉. For the inductive case, GHZn and iGHZn are obtained from GHZn−1 ⊗ |0〉 and iGHZn−1 ⊗ |0〉, respectively, by applying CNot to the two rightmost qubits. Lemma 2 If 0 6 x, y < 2 and x 6= y then |x〉+ |y〉 and |x〉+ i|y〉 are stabilizer states. Proof: By induction on n. For the base case (n = 1), the closure properties imply that |0〉+ |1〉, |0〉+ i|1〉 and |1〉+ i|0〉 = |0〉 − i|1〉 are stabilizer states. For the inductive case, consider the binary representations of x and y. If there is a bit position in which x and y have the same value b, then |x〉+ |y〉 is the tensor product of |b〉 with an (n − 1)-qubit state of the form |x′〉 + |y′〉, where x′ 6= y′. By the induction hypothesis, |x′〉+ |y′〉 is a stabilizer state, and the conclusion follows from the closure properties. Similarly for |x〉+ i|y〉. Otherwise, the binary representations of x and y are complementary bit patterns. In this case, |x〉 + |y〉 can be obtained from GHZn by applying X to certain qubits. The conclusion follows from Lemma 1 and the closure properties. The same argument applies to |x〉 + i|y〉, using iGHZn. Theorem 1 The space of density matrices for n-qubit states, considered as a (2)-dimensional real vector space, has a basis consisting of density matrices of nqubit stabilizer states. Proof: This is the space of Hermitian matrices and its obvious basis is the union of {|x〉〈x| | 0 6 x < 2} (1) {|x〉〈y|+ |y〉〈x| | 0 6 x < y < 2} (2) {−i|x〉〈y|+ i|y〉〈x| | 0 6 x < y < 2}. (3) Now consider the union of {|x〉〈x| | 0 6 x < 2} (4) {(|x〉+ |y〉)(〈x|+ 〈y|) | 0 6 x < y < 2} (5) {(|x〉+ i|y〉)(〈x| − i〈y|) | 0 6 x < y < 2}. (6) This is also a set of (2) states, and it spans the space because we can obtain states of forms (2) and (3) by subtracting states of form (4) from those of forms (5) and (6). Therefore it is a basis, and by Lemma 2 it consists of stabilizer states.