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Mathematical Introduction to Quantum Information Processing

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Mathematical Introduction to Quantum Information Processing (growing lecture notes, SS2019) Michael M. Wolf June 22, 2019 Contents 1 Mathematical framework 5 1.1 Hilbert spaces . .5 1.2 Bounded Operators . .8 Ideals of operators . 10 Convergence of operators . 11 Functional calculus . 12 1.3 Probabilistic structure of Quantum Theory . 14 Preparation . 15 Measurements . 17 Probabilities . 18 Observables and expectation values . 20 1.4 Convexity . 22 Convex sets and extreme points . 22 Mixtures of states . 23 Majorization . 24 Convex functionals . 26 Entropy . 28 1.5 Composite systems and tensor products . 29 Direct sums . 29 Tensor products . 29 Partial trace . 34 Composite and reduced systems . 35 Entropic quantities . 37 1.6 Quantum channels and operations . 38 Schrödinger & Heisenberg picture . 38 Kraus representation and environment . 42 Choi-matrices . 45 Instruments . 47 Commuting dilations . 48 1.7 Unbounded operators and spectral measures . 51 2 Basic trade-offs 53 2.1 Uncertainty relations . 53 Variance-based preparation uncertainty relations . 54 Joint measurability . 55 2 CONTENTS 3 2.2 Information-disturbance . 56 No information without disturbance . 56 2.3 Time-energy . 58 Mandelstam-Tamm inequalities . 58 Evolution to orthogonal states . 59 These are (incomplete but hopefully growing) lecture notes of a course taught in summer 2019 at the department of mathematics at the Technical University of Munich. 4 CONTENTS Chapter 1 Mathematical framework 1.1 Hilbert spaces This section will briefly summarize relevant concepts and properties of Hilbert spaces. A complex Hilbert space is a vector space over the complex numbers, equipped with an inner product h·; ·i : H × H ! C and an induced norm k k := h ; i1=2 w.r.t. which it is complete.1 Hence, every Hilbert space is in particular a Banach space. We will use the physicists convention that the inner prod- uct is linear in the second and conjugate-linear in its first argument so that h ; c'i = ch ; 'i = hc ; 'i, 8c 2 C. The most import inequality for the inner product is the Cauchy-Schwarz inequality, which immediately follows2 from the identity 1 2 k k2 k'k2 − jh ; 'ij2 = k'k2 − h'; i' ≥ 0: k'k2 This also shows that equality holds iff ' and are linearly dependent. A characteristic property of any norm that is induced by an inner product is that it satisfies the parallelogram law k + 'k2 + k − 'k2 = 2 k k2 + k'k2 : (1.1) In fact, whenever a norm satisfies Eq.(1.1) for all ; ', then we can reconstruct a corresponding inner product via the polarization identity, which in the case of a complex space reads 3 1 X k k 2 h ; 'i = i ' + i : (1.2) 4 k=0 1That is, every Cauchy sequence converges. 2Note that the derivation of Cauchy-Schwarz does not use that h ; i = 0 ) = 0. It only requires that h ; i ≥ 0. 5 6 CHAPTER 1. MATHEMATICAL FRAMEWORK A central concept that is enabled by an inner product is orthogonality: ; ' are called orthogonal if h ; 'i = 0. In that case k + 'k = k − 'k so that the parallelogram law becomes the Pythagoras identity k + 'k2 = k k2 + k'k2. For any subset S ⊆ H the orthogonal complement S? is defined as the subset of H whose elements are orthogonal to every element in S. S? is then necessarily a closed linear subspace. Every closed linear subspace H1 ⊆ H, in turn, gives rise to a unique decomposition of any element 2 H as = 1 + 2, where ? 1 2 H1 and 2 2 H2 = H1 . In this way, the Hilbert space decomposes into an orthogonal direct sum H = H1 ⊕H2. The i’s can equivalently be characterized as those elements in Hi closest to . Uniqueness of the i’s enables the definition of two orthogonal projections Pi : H!Hi; 7! i, which are linear idempotent maps related via P2 = 1 − P1, where 1 denotes the identity map on H. If we think the idea of orthogonal decompositions of a Hilbert space further, we are led to the concept of an orthonormal basis. An orthonormal basis is a set feig ⊆ H whose linear span is dense in H and whose elements satisfy hei; eji = δij. Its cardinality defines the dimension of the Hilbert space. Separability of H means that there is a countable orthonormal basis. In that case, for P every 2 H we have = ihei; iei (converging in norm) and the Parseval 2 P 2 identity k k = i jhei; ij holds. An orthonormal set of vectors can always be extended to an orthonormal basis. Another property that Hilbert spaces share with their Euclidean ancestors is expressed by the Riesz representation theorem: it states that every continuous linear map from H into C is of the form 7! h'; i for some ' 2 H, and vice versa. In other words, there is a conjugate linear bijection between H and its topological dual space H0 (i.e. the space of all continuous linear functionals). The possible identification of H and H0 motivates the so-called Dirac-notation that writes j i for elements of H and h'j for elements of H0. These symbols are then called ket and bra, respectively and the inner product in this nota- tion reads h'j i (forming a “bra(c)ket”). When we would restrict ourselves to Euclidean spaces, kets and bras would be nothing but column vectors and row vectors, respectively. Dirac notation also enables the introduction of a ket-bra j ih'j : H!H that defines a map jφi 7! j ih'jφi. Using ket-bras, a necessary and sufficient condition for a set of orthonormal vectors to form a basis of a separable Hilbert space is given by X jekihekj = 1: (1.3) k To write expressions of this form even more compactly, the elements of a fixed orthonormal basis are often simply specified by their label so that one writes jki instead of jeki. So far, this has been abstract Hilbert space theory. Before we proceed, some concrete examples of Hilbert spaces: Example 1.1. Cn becomes a Hilbert space when equipped with the standard Pn inner product h ; 'i = i=1 i'i. 1.1. HILBERT SPACES 7 N CN P 2 Example 1.2. The sequence space l2( ) := 2 j k j kj < 1 be- comes a Hilbert space when equipped with the standard inner product h ; 'i = P k k'k. The standard orthonormal basis in this case is given by sequences ek; k 2 N such that the l’th element in ek equals δlk. R R C R 2 Example 1.3. The function space L2( ) := f : ! R j (x)j dx < R 2 1 = ∼ where ∼ ' , R j (x) − '(x)j dx = 0 becomes a separable Hilbert R space with h ; 'i = R (x)'(x)dx. Example 1.4. The space Cn×m of complex n × m matrices becomes a Hilbert space with hA; Bi = tr [A∗B]. Two Hilbert spaces H1 and H2 are called isomorphic if there is a bijection U : H1 !H2 that preserves all inner products. U, which is called a Hilbert space isomorphism, is then necessarily linear and it turns out that Hilbert spaces are isomorphic iff they have the same dimension. Hence, all separable Hilbert n spaces are isomorphic to either C or l2(N), in particular, L2(R) ' l2(N). Sometimes one has to deal with inner product spaces that are not complete. In these cases the following theorem comes in handy and allows to ‘upgrade’ every such space to a Hilbert space: Theorem 1.1 (Completion theorem). For every inner product space X there is a Hilbert space H and a linear map V : X!H that preserves all inner products3 so that V (X ) is dense in H and equal to H if X is complete. The space H is then called the completion of X . It is unique in the sense that if (V 0; H0) give rise to another completion, then there is a Hilbert space isomorphism U : H!H0 s.t. V 0 = U ◦ V . As in the more general case of metric spaces, the completion is constructed by considering equivalence classes of Cauchy-sequences in X . Usually, this con- struction is, however, hardly used beyond the proof of this theorem, and it is sound to regard H as a superspace of X that has been constructed from X by adding all the elements that were missing for completeness. We finally state a property that distinguishes Hilbert spaces from almost all other normed spaces and has various applications in the form of a dimension reduction argument: Lemma 1.2 (Johnson-Lindenstrauss). There is a universal constant c 2 R such that for any 2 (0; 1], Hilbert space H, n 2 N, 1; : : : ; n 2 H there is a linear map L : H!Hd that is a multiple of an orthogonal projection onto a d-dimensional subspace Hd with c d ≤ log n; 2 so that for all i; j 2 f1; : : : ; ng : 2 2 2 (1 − ) k i − jk ≤ kL i − L jk ≤ (1 + ) k i − jk : (1.4) 3In other words, V is an isometry; see next section for the definition. 8 CHAPTER 1. MATHEMATICAL FRAMEWORK This is often stated and used for real Hilbert spaces, but equally valid for complex ones. From now on, we will tacitly assume that all Hilbert spaces H; H1; H2; etc. are complex and separable. Exercise 1.1. Show that the closed unit ball of any Hilbert space is strictly convex. Exercise 1.2. Show that any linear map U : H1 !H2 that preserves norms also preserves inner products. Exercise 1.3.
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