Quantum-Information-Theory Wintersemester 2000/2001

Quantum-Information-Theory Wintersemester 2000/2001 Prof. M. Lewenstein Institute for Theoretical Physics University of Hannover March 27, 2006 Release 1.04 Dear Reader ! This script is based on the lecture by MACIEJ LEWENSTEIN during the winter semester 2000/2001. We, the authors of this script, tried to unify notation and added some additional notes. You may print out this document for your personal use and you may also give copies (either eletronically or on paper) to third persons as long as those copies are not modified by you in any way and you don’t receive any fee except for a reimbursement of copying cost. You may not distribute modified versions of this document without prior written consent of the authors. This version contains the entire lecture and an additional talk given by ANTONIO ACÍN – found in appendix A – which we gratefully acknowledge. We would also like to thank FLORIAN HULPKE and DAGMAR BRUSS for suggestions and corrections. You can find the latest version of this document (fully hyperlinked) always at http://lodda.iqo.uni-hannover.de/download.php?language=en If you find any errors, omissions or ambiguous statements (as well as typing mis- takes of course) we would be glad to hear about them. Please mail your comments to [email protected] or [email protected]. We hope you enjoy reading this script. KAI ECKERT HELGE KREUTZMANN Since March 2004, this script is revised and corrected by RODION NEIGOVZEN whom you can reach at [email protected]. c 2000-2002 KAI ECKERT and HELGE KREUTZMANN 1 Contents 1 Introduction 4 2 Entanglement and Separability 9 2.1 EntanglementofPureStates . 9 2.2 EntanglementandSeparabilityofMixedStates . .. 12 2.3 EntanglementCriteria. 12 3 PPT Entangled States 25 3.1 Definition .............................. 25 3.2 ACriterionofSeparability . 25 2 4 C 3.2.1 Examplein C .................... 26 3.3 EdgeStates .............................⊗ 28 4 Entanglement Witnesses and Positive Maps 31 4.1 EntanglementWitnesses . 31 4.1.1 TechnicalPreface. 31 4.1.2 EntanglementWitness . 32 4.1.3 Examples.......................... 34 4.2 PositiveMaps............................ 37 4.2.1 Introduction......................... 37 4.2.2 Examples.......................... 38 4.2.3 DecomposableMaps . 39 4.2.4 JamiołkowskiIsomorphism . 40 4.2.5 ComparisonofWitnessesandMaps . 43 5 Classification of Separable States, EW and PM 45 5.1 Separability in 2 N CompositeQuantumSystems . 48 × 6 Schmidt Number Witnesses 52 6.1 Introduction............................. 52 6.2 ExampleforaSchmidtNumberWitness . 55 6.3 The3 3Case ........................... 57 × A Generalization of the Schmidt Decomp. for the Three Qubit System 60 A.1 Motivation.............................. 60 A.2 TheBarcelonaApproach . 61 A.3 TheSudberyApproach . 63 A.4 TheInnsbruckapproach . 64 Bibliography 67 2 Motivation This lecture intends to describe the theory from the foundations [1] up to the cur- rent research front (see e.g. reviews from [2–4], and recent publications from [5] and [6]). Its main emphasis is on mathematical describtion of the theory, rather than on possible applications, see [7] for those. Although the intention is to prove all theorems, some previuos mathematical knowledge (as found in e.g. [8], [9], [10] and [11]) is expected. Quantum information theory is strongly related to entanglement theory: 1. Quantum “paradoxes” (EPR, SCHRÖDINGER cat, BELL inequalities). 2. Applications in Quantum Information Processing (QIP) (teleportation, cryp- tography (i.e. for military communications), data compression and quantum computing). 3. Basic and fundamental aspects of quantum mechanics (quantum correla- tions). 4. Connections to important challenges of modern mathematics (i.e. theory of positive maps on C∗ algebras). 3 1 Introduction We consider two or sometimes three (quantum) systems which we label A, B and C. They will also be given names of persons: Alice, Bob and Charlie. Each system has a finite HILBERT space and we arrange the systems such, that the entire HILBERT space can be written as H = HA HB dimHA = M N = dimHB. (1.1) ⊗ ≤ We adopt the notation M ei HA ψA = ∑ ai ei (1.2) {| i} ∈ | i i=1 | i N fi HB ψB = ∑ b j f j . (1.3) {| i} ∈ | i j=1 | i The basis used is arbitrary but fixed. All basis changes will be explicitly noted. Thus any state can be written as ψ = ∑cij ei f j ∑cij e j f j HA HB (1.4) | i ij | i⊗| i ≡ ij | i ∈ ⊗ where we will omit the direct product sign in future equations. The dimension of the combined space is ⊗ dimH = dimHA dimHB = M N. (1.5) · · If Alice and Bob have a system with only two possible eigenstates 0 and 1 (each one is said to have a qubit) we can explicitly write down states in| i the com-| i bined HILBERT space in the following way: 1 1 ψ− = [ 0 1 1 0 ] = ( 01 10 ) (1.6) | i √2 | i| i−| i| i √2 | i−| i Alternatively we can also write the state vectors as vectors of a four dimensional space: 1 0 0 A = 1 A = (1.7) | i 0 | i 1 A A 1 0 0 B = 1 B = (1.8) | i 0 | i 1 B B 1 0 0 1 1 ψ1 = 0 A 0 B = ψ− = (1.9) | i | i | i 0 | i √2 1 − 0 0 4 The translation is done as follows. Write down Alice vector but reserve N components for each of the M components of Alice. Then write Bobs vector into each component of this created vector multiplying Bobs components with the with the corresponding component from Alice. So in the above case for ψ1 first write | i down 0 B multiplied by 1, then below 0 B multiplied by 0. Instead| ofi describing each state by its wave| i function it is usually more convenient to use the density matrix (usually labeled ρ) instead, since this concept is more general and allows to describe mixed state also. Each operator O can be written as ij O = ∑ Okl ei f j ek fl (1.10) ijkl | i⊗| ih |⊗h | where both Alice ei and Bob f j have an orthonormal basis: {| i} {| i} ei e j = δij fk fl = δkl (1.11) h | i h | i This way the combined basis in H = HA HB is also orthogonal and normal. If ⊗ we denote each pair of indices as one index k,k′ 1,...,NM then we can write the operator O as ∈{ } k O = ∑ O ψk ψk . (1.12) k′ | ih ′ | k,k′ Def. 1.1 1. ρ is an operator on H = HA HB ⊗ 2. ρ is hermitian, i.e. ρ = ρ†. ϕ (ρ ψ )=( ϕ ρ) ψ . Written component ⇔h | | i h | | i wise this means ρkk = ρ∗ . ′ k′k Using the spectral theorem we can write ρ = ∑ ρ ψk ψ = ∑λl ϕl ϕl (1.13) kk′ | ih k′ | | ih | k,k′ l where λl are eigenvalues of ρ and ϕl its eigenvectors. | i 3. ρ > 0 ψ : ψ ρ ψ 0. This is equivalent to the statement that all ⇔ ∀| i h | | i ≥ eigenvalues λl 0. ≥ 4. Tr(ρ) = ∑l λl = 1. The last two definitions allow to interpret the density matrix in terms of probabil- ities. Def. 1.2 The Kernel of ρ is defined as K ρ = ψ H : ρ ψ = 0 .1 { } {| i ∈ | i } 1Note that this is a linear equation. 5 We have ρ = ρ† K ρ is a subspace which is spanned by the eigenvectors with zero eigenvalue. ⇒ { } Def. 1.3 Range of ρ : R ρ = ψ H : ϕ : ρ ϕ = ψ . { } {| i ∈ ∃| i | i | i} Since ρ = ρ† R ρ is a linear subspace of H spanned by the eigenvectors of ρ { } with λ > 0, i.e. if ρ φ1 = ψ1 and ρ φ2 = ψ2 then | i | i | i | i ρ (α φ1 + β φ2 ) = α ψ1 + β ψ2 . (1.14) | i | i | i | i Further since ρ = ρ† we have R ρ K ρ . (1.15) { } ⊥ { } Proof: Take ψ1 R ρ and ψ2 K ρ . Then φ ρ φ = ψ1 and | i ∈ { } | i ∈ { } ∃| i | i | i † ψ2 ψ1 = ψ2 ρ φ = ρ ψ2 φ = ρψ2 φ = 0 φ = 0. (1.16) h | i h | | i h | i h | i ·| i We call r ρ := dimR ρ the rank of ρ. Similar we have k ρ := dimK ρ . Since eqn.{ } (1.15) we{ have} r ρ +k ρ = dimH . We can{ also} proof this{ by} explicit construction: { } { } If we call the eigenvectors of ρ φl with l = 1,...,r ρ with eigenvalues λl > 0 we can write ρ as | i { } r ρ r ρ { } { } ρ = ∑ λl φl φl and ψ = ∑ al φl . (1.17) l=1 | ih | | i l=1 | i where ψ is an arbitrary state in R ρ where at least one al = 0. Now we can explicitly| i construct the state χ which{ } will be projected on ψ 6 : | i | i r ρ χ { } al φ ρ χ ψ = ∑ λ l = (1.18) | i l=1 l | i ⇒ | i | i Without proof we make the general remark that if A = A† we have R A K A† and R A† K A . We will not need this property.6 { } ⊥ { } For this{ lecture} ⊥ we{ } adapt the notation dimH O = ∑ O k k′ (1.19) kk′ | ih | k,k′ dimH T O = ∑ O k′ k andthus (1.20) kk′ | ih | k,k′ T O = Ok k. (1.21) kk′ ′ 6 Def. 1.4 Partial transpose: We again have a ρ which acts on HA HB. If we write ⊗ ρ ρij = ∑ kl i A j B k A l B then (1.22) ijkl | i ⊗| i h | ⊗h | ρTA ρij = ∑ kl k A j B i A l B and (1.23) ijkl | i ⊗| i h | ⊗h | ij ρTA ρkj kl = il.

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