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L2. The geometry of quantum states Oscar Rosas-Ortiz Departamento de Fı́sica Cinvestav México (CADI 2012, ITESM Edo Méx.) L2. The geometry of quantum states – p. 1 Introduction L2. The geometry of quantum states – p. 2 Introduction L2. The geometry of quantum states – p. 3 Pure states For a pure state the ket |Φi = X k ck |φk i provides a complete description of the quantum system. Simplest case: |Φi = a|0i + b|1i, 2 2 |a| + |b| = 1, |0i = 1 0 , |1i = 0 1 . L2. The geometry of quantum states – p. 4 Simplest measuring devices define pure states • Simplest measuring device: A filter checking for |ψi. Πψ = |ψihψ|. • If |ϕi is the initial vector then Πψ |ϕi = Γ|ψi, with Γ := hψ|ϕi. The result of testing |ψi on |ϕi is: ⋆ Affirmative if |ϕi ∝ |ψi ⋆ Negative if |ϕi⊥|ψi ⋆ Uncertain if |ϕi = α|ψi + |ηi, with |ηi⊥|ψi (⇒ α = Γ). |Γ|2 = |hψ|ϕi|2 ⇔ “transition probability” P (ψ) = |Γ|2 = hϕ|ψihψ|ϕi = hϕ|Πψ |ϕi ≡ hΠψ iϕ ≡ hΠϕ iψ L2. The geometry of quantum states – p. 5 Simplest measuring devices define pure states A “projective measurement” is described by an observable A, with spectral decomposition (αk ∈ R) X A= αk Παk ⇒ A|αk i = αk |αk i. k Upon measuring the arbitrary state |αi, the probability P (αk ) of getting result αk is given by ! X P (αk ) = hΠαk iα ⇒ hAiα = αk P (αk ) k Given the outcome αk occurred, the state of the quantum system immediately after the measurement is Παk |αi p . P (αk ) L2. The geometry of quantum states – p. 6 Theory of filters L2. The geometry of quantum states – p. 7 Stern-Gerlach (preparation of pure states) Assume that Z has the eigenvalues +1 and −1 with eigenvectors |0i and |1i. The spectral decomposition of Z is given by 1 X 1 0 . Z= zk Πk = |0ih0| − |1ih1| = 0 −1 k=0 The measurement of Z on the state |0i + |1i |ψi = √ 2 gives +1 with probability −1 with probability 1 2 1 2 L2. The geometry of quantum states – p. 8 Mixed state: insufficient information • Ignorance about which of the mutually exclusive pure states |ψk i has been prepared. X • We can only ascribe probabilities Pk ≥ 0, Pk = 1, to each |ψk i. k • At t = t0 the set of kets |ψ1 i, |ψ2 i, . . ., with statistical weights P1 , P2 , . . ., represent the quantum state. • The average value hAi = X k hAik Pk , X Pk ≥ 0, Pk = 1 k can be managed as follows hAi = X Pk hψk |AI|ψk i = k X = hψℓ | ℓ X Pk hψk |A|ψℓ ihψℓ |ψk i = k,ℓ " X k Pk Π k ! # A |ψℓ i ≡ Tr X Pk hψℓ |Πk A|ψℓ i k,ℓ " X k Pk Π k ! # A L2. The geometry of quantum states – p. 9 Measurements on statistical mixtures Then hAi = TrρA, with X ρ := Pk Πk , Πk ≡ ρk := |ψk ihψk |, k • For A = I we get hIi = Trρ = X Pk TrΠk = k X k • For any |ψi hρiψ = hψ| Pk X k Pk Πk ! Pk ≥ 0, ! X hψi |ρk |ψi i i |ψi = X k X Pk = 1. X Pk = 1. k = k Pk |hψ|ψk i|2 ≥ 0. L2. The geometry of quantum states – p. 10 The density operator • The density operator ρ is Hermitean (H), positive definite (+) and of trace equal to unity (1). • Conversely, any (H+1)-e ρ must have a spectral decomposition X X ρe = λk Πk , λk ≥ 0, λk = 1. k k • Pure state = statistical mixture having |ψi as its sole element. ρ = Πψ = |ψihψ| ρ2 = ρ ⇒ • Mixed state = statistical mixture X X 2 ρ= Pk Πk ⇒ ρ = Pk2 Πk k k ⇒ ⇒ Trρ2 = Trρ = 1. 2 Trρ = X k P 2 ≤ Trρ L2. The geometry of quantum states – p. 11 The density operator Trρ2 = 1 + 2(|ρ12 |2 − ρ11 ρ22 ) ρA = 1 0 1 , Trρ2A = 1, ρB = 1 2 1 0 0 1 1 , Trρ2B = 1, ρC = 2 1 1 , Trρ2C = 7 9 1 • Since ρ|ψk i = Pk |ψk i, for the von Neumann entropy X S(ρ) = −Trρ log2 ρ ≡ − Pk log2 Pk , 0 log2 0 ≡ 0 k ⋆ Pure state: S(ρ) = −1 · log2 (1) = 0 ⋆ Maximally mixed: Pk = 1 N, so that S(ρ) = log2 (N ) L2. The geometry of quantum states – p. 12 Bloch-Poincaré sphere I +α ~ · ~σ ρ= , 2 ||~ α|| ≤ 1 ⋆ ρ is pure iff ||~ α|| = 1 ⋆ The maximally mixed state ρ = I 2 is at origin. L2. The geometry of quantum states – p. 13 Convexity • A set S is convex if for each pair of points x and y in S, the line segment joining x and y is in S αx + βy ∈ S, α ≥ 0, β ≥ 0, α+β =1 • Affine combination y= N X k=1 λk x k , N X k=1 λk = 1, λk ∈ R L2. The geometry of quantum states – p. 14 Convex Set • A convex combination is an affine combination such that λk ≥ 0: ρ := X k Pk Πk , X k Pk = 1, Pk ≥ 0 • THEO. A set S is convex iff every convex combination of points of S lies in S. • The convex hull of a set S is the intersection of all the convex sets which contain S. • THEO. For any set S, the convex hull of S consists precisely of all convex combinations of elements of S. L2. The geometry of quantum states – p. 15 Convex polytope and simplex • THEO. If S is a nonempty subset of Rn then every x in convS can be expressed as a convex combination of n + 1 of fewer points of S. • The convex hull of a finite set of points is called a polytope. If S = {x1 , . . . , xk+1 } and DimS = k, then convS is a k-dimensional simplex. The points xk are called vertices. ⋆ 0-simplex S 0 = a single point {x1 } ⋆ 1-simplex S 1 = a line segment [x1 , x2 ] ⋆ 2-simplex S 2 = a triangle (joining x3 ∈ / affS 1 ) ⋆ 3-simplex S 3 = a tetrahedron (joining x4 ∈ / affS 2 ) ⋆ 4-simplex S 4 = a pentatope (joining x5 ∈ / affS 3 ) L2. The geometry of quantum states – p. 16 Convex polytope and simplex L2. The geometry of quantum states – p. 17 Entanglement Entropy of entanglement for a 3D cross section of the 6D manifold of pure states of two qubits. The hotter the color the more entangled the state. (Bengtsson and Zyczkowski) √ + 2|ϕ i = |00i + |11i, √ + 2|ψ i = |01i + |10i L2. The geometry of quantum states – p. 18 Discussion 1. How close are two quantum states? 2. The trace distance 1 D(ρ1 , ρ2 ) = Tr|ρ1 − ρ2 |, 2 √ |A| = A† A is convex. 3. For instance, given two Bloch vectors I +α ~ k · ~σ , ρk = 2 k = 1, 2 one notice that the distance between two qubit states is equal to one half the ordinary Euclidean distance between them on the Bloch sphere! ||~ α1 − α ~ 2 || D(ρ1 , ρ2 ) = 2 L2. The geometry of quantum states – p. 19 Discussion 4. The unbiased basis description is intrinsically convex 5. Geometric description of quantum phenomena is free of interpretation 6. Study of entanglement, quantum tomography, and quantum decoherence among others. L2. The geometry of quantum states – p. 20 Gracias!! L2. The geometry of quantum states – p. 21