ed the commutator definition [A, B] = AB - BA and the fact that the observables are Hermitian. The property (4/0) > 0 (with (4/6) real) is inherent to the definition of inner product. We also have the property (V/X2/1/5) > 0 for any observable X = A, B and state 1/4), see note. Hence, i(y [A, B]/V must be real, = Eq. (03) is of the polynomial form P(2) = with real coefficients a, b, C satisfying a, b, C > 0. Since P(2) > 0, it can only have zero or one root, which corresponds to a discriminant 8 = b2 - 4ac 0 0. We, thus, have See, for instance: C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Paris: Hermann, 1977), Vol. 1, pp. 28-7. Appendix 0 which can be put in the inequality form: Now define the new observables A, B from the previous observables A, B according to BB-BYB-(B) It is easily verified that [A, B]=[A,B]. Substituting the definitions in Eq. (06) into Eq. (05), we obtain It is also easily verified that 2 for any operator X = A, B defined by Eq. (06). Hence, we finally obtain the Heisenberg uncertainty relation: (010) The property > 0 for any observable X and state 1/5/ is shown as follows. Because X is an observable, there exists an orthonormal or eigenstate base {|2i}}, whose state elements satisfy X/2i) = dilhi), where di are the corresponding real eigenvalues. The state /V/S can, thus, be uniquely decomposed into this base with coordinates Mi, according to Hence, we obtain: Appendix P (Chapter 18) Two-qubit teleportation In this appendix, I formally demonstrate the possibility of simultaneously teleporting two qubits from Alice's location to Bob's. The proposed six-qubit quantum circuit, shown in Fig. P1, is an original, symmetrical variant of that described in Gottesman and Chuang (1999). 1 Such an example also represents a test case for the analysis of quantum circuits and Bell measurements, as a full illustration of the concepts and formalism described in Chapter 18, hence, the detailed calculations presented here. In the circuit shown in Fig. P1, the boxes B, B' stand for Bell-state 