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Theory of Quantum Entanglement

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Theory of Quantum Entanglement Theory of quantum entanglement Mark M. Wilde Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana, USA [email protected] On Sabbatical with Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305 Mitteldeutsche Physik Combo, University of Leipzig, Leipzig, Germany, September 17, 2020 Mark M. Wilde (LSU) 1 / 103 Motivation Quantum entanglement is a resource for quantum information processing (basic protocols like teleportation and super-dense coding) Has applications in quantum computation, quantum communication, quantum networks, quantum sensing, quantum key distribution, etc. Important to understand entanglement in a quantitative sense Adopt axiomatic and operational approaches This tutorial focuses on entanglement in non-relativistic quantum mechanics and finite-dimensional quantum systems Mark M. Wilde (LSU) 2 / 103 Quantum states... (Image courtesy of https://en.wikipedia.org/wiki/Quantum_state) Mark M. Wilde (LSU) 3 / 103 Quantum states The state of a quantum system is given by a square matrix called the density matrix, usually denoted by ρ, σ, τ, !, etc. (also called density operator) It should be positive semi-definite and have trace equal to one. That is, all of its eigenvalues should be non-negative and sum up to one. We write these conditions symbolically as ρ 0 and Tr ρ = 1. Can abbreviate more simply as ρ ( ), to be read≥ as \ρ isf ing the set of density matrices." 2 D H The dimension of the matrix indicates the number of distinguishable states of the quantum system. For example, a physical qubit is a quantum system with dimension two. A classical bit, which has two distinguishable states, can be embedded into a qubit. Mark M. Wilde (LSU) 4 / 103 Interpretation of density operator The density operator, in addition to a description of an experimental procedure, is all that one requires to predict the (probabilistic) outcomes of a given experiment performed on a quantum system. It is a generalization of (and subsumes) a probability distribution, which describes the state of a classical system. All probability distributions can be embedded into a quantum state by placing the entries along the diagonal of the density operator. Mark M. Wilde (LSU) 5 / 103 Mixtures of quantum states A probabilistic mixture of two quantum states is also a quantum state. That is, for σ0; σ1 ( ) and p [0; 1], we have 2 D H 2 pσ0 + (1 p)σ1 ( ): − 2 D H The set of density operators is thus convex. Mark M. Wilde (LSU) 6 / 103 Mixed states and pure states A density operator can have dimension 2 and can be written as ≥ X ρ = ρi;j i j ; j ih j i;j i;j where i ei is the standard basis and ρ are the matrix elements. fj i ≡ g Since every density operator is positive semi-definite and has trace equal to one, it has a spectral decomposition as X ρ = pX (x) φx φx ; x j ih j where pX (x) are the non-negative eigenvalues, summing to one, f g and φx is a set of orthonormal eigenvectors. fj ig A density operator ρ is pure if there exists a unit vector such that ρ = (rank = 1) and otherwise it is mixed (rank >j 1).i j ih j Mark M. Wilde (LSU) 7 / 103 Multiple quantum systems... IBM 65-qubit universal quantum computer (released September 2020) Mark M. Wilde (LSU) 8 / 103 Composite quantum systems If the state of Alice's system is ρ and the state of Bob's system is σ and they have never interacted in the past, then the state of the joint Alice-Bob system is ρA σB : ⊗ We use the system labels to say who has what. Mark M. Wilde (LSU) 9 / 103 Composite quantum systems (ctd.) More generally, a generic state ρAB of a bipartite system AB acts on a tensor-product Hilbert space AB A B . H ≡ H ⊗ H If i A i is an orthonormal basis for A and j B j is an fj i g H fj i g orthonormal basis for B , then i A j B i;j is an orthonormal H fj i ⊗ j i g basis for AB . H Generic state ρAB can be written as X i;k;j;l ρAB = ρ i k A j l B j ih j ⊗ j ih j i;j;k;l where ρi;k;j;l are matrix elements Mark M. Wilde (LSU) 10 / 103 Quantum entanglement... Depiction of quantum entanglement taken from http://thelifeofpsi.com/2013/10/28/bertlmanns-socks/ Mark M. Wilde (LSU) 11 / 103 Separable states and entangled states x x If Alice and Bob prepare states ρA and σB based on a random variable X with distribution pX , then the state of their systems is X x x pX (x)ρA σB : x ⊗ Such states are called separable states [Wer89] and can be prepared using local operations and classical communication (LOCC). No need for a quantum interaction between A and B to prepare these states. By spectral decomposition, every separable state can be written as X z z z z pZ (z) A φ φ B ; z j ih j ⊗ j ih j z z where, for each z, A and φ B are unit vectors. j i j i Entangled states are states that cannot be written in the above form. Mark M. Wilde (LSU) 12 / 103 Example of entangled state A prominent example of an entangled state is the ebit (eee bit): · Φ Φ AB ; j ih j 1 where Φ AB p ( 00 AB + 11 AB ). j i ≡ 2 j i j i In matrix form, this is 2 1 0 0 1 3 1 6 0 0 0 0 7 Φ Φ AB = 6 7 : j ih j 2 4 0 0 0 0 5 1 0 0 1 To see that this is entangled, consider that for every A and φ B j i j i 2 1 Φ AB A φ B jh j j i ⊗ j i j ≤ 2 impossible to write Φ Φ AB as a separable state. ) j ih j Mark M. Wilde (LSU) 13 / 103 Tool: Schmidt decomposition Schmidt decomposition theorem Given a two-party unit vector AB A B ; we can express it as j i 2 H ⊗ H d−1 X AB ppi i i B ; where j i ≡ j iA j i i=0 P probabilities pi are real, strictly positive, and normalized i pi = 1. i A and i B are orthonormal bases for systems A and B. fj i g fj i g ppi i2f0;:::;d−1g is the vector of Schmidt coefficients. Schmidt rank d of AB is equal to the number of Schmidt j i coefficients pi in its Schmidt decomposition and satisfies d min dim( A); dim( B ) : ≤ f H H g Pure state AB is entangled iff d 2. j ih j ≥ Mark M. Wilde (LSU) 14 / 103 Tool: Partial trace The trace of a matrix X can be realized as X Tr[X ] = i X i ; h j j i i where i i is an orthonormal basis. fj ig Partial trace of a matrix YAB acting on A B can be realized as H ⊗ H X TrA[YAB ] = ( i A IB )YAB ( i A IB ); h j ⊗ j i ⊗ i where i A i is an orthonormal basis for A and IB is the identity fj i g H matrix acting on B . H Both trace and partial trace are linear operations. Mark M. Wilde (LSU) 15 / 103 Interpretation of partial trace Suppose Alice and Bob possess quantum systems in the state ρAB . We calculate the density matrix for Alice's system using partial trace: ρA TrB [ρAB ]: ≡ We can then use ρA to predict the outcome of any experiment performed on Alice's system alone. Partial trace generalizes marginalizing a probability distribution: " # X TrY pX ;Y (x; y) x x X y y Y x;y j ih j ⊗ j ih j X = pX ;Y (x; y) x x X Tr[ y y Y ] x;y j ih j j ih j " # X X X = pX ;Y (x; y) x x X = pX (x) x x X ; x y j ih j x j ih j P where pX (x) pX ;Y (x; y). ≡ y Mark M. Wilde (LSU) 16 / 103 Purification of quantum noise... Artistic rendering of the notion of purification (Image courtesy of seaskylab at FreeDigitalPhotos.net) Mark M. Wilde (LSU) 17 / 103 Tool: Purification of quantum states A purification of a state ρS on system S is a pure quantum state RS on systems R and S, such that j ih j ρS = TrR [ RS ]: j ih j P p Simple construction: take RS = x p(x) x R x S if ρS has P j i j i ⊗ j i spectral decomposition p(x) x x S . x j ih j Two different states RS and φ φ RS purify ρS iff they are j ih j j ih j related by a unitary UR acting on the reference system. Necessity: y y TrR [(UR IS ) RS (U IS )] = TrR [(U UR IS ) RS ] ⊗ j ih j R ⊗ R ⊗ j ih j = TrR [ RS ] j ih j = ρS : To prove sufficiency, use Schmidt decomposition. Mark M. Wilde (LSU) 18 / 103 Uses and interpretations of purification The concept of purification is one of the most often used tools in quantum information theory. This concept does not exist in classical information theory and represents a radical departure (i.e., in classical information theory it is not possible to have a definite state of two systems such that the reduced systems are individually indefinite).
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