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School of Physics something and Astronomy FACULTY OF MATHEMATICAL OTHER AND PHYSICAL SCIENCES Introduction to entanglement Jacob Dunningham Paraty, August 2007 School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Vlatko pic October 2004 1 www.quantuminfo.org School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Vlatko pic October 2004 October 2005 1 9 www.quantuminfo.org School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Vlatko pic October 2004 October 2005 October 2006 1 9 ~ 25 www.quantuminfo.org School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES October 2010 (projected) Overview • Lecture1: Introduction to entanglement: Bell’s theorem and nonlocality Measures of entanglement Entanglement witness Tangled ideas in entanglement Overview • Lecture1: Introduction to entanglement: Bell’s theorem and nonlocality Measures of entanglement Entanglement witness Tangled ideas in entanglement • Lecture 2: Consequences of entanglement: Classical from the quantum Schrodinger cat states Overview • Lecture1: Introduction to entanglement: Bell’s theorem and nonlocality Measures of entanglement Entanglement witness Tangled ideas in entanglement • Lecture 2: Consequences of entanglement: Classical from the quantum Schrodinger cat states • Lecture 3: Uses of entanglement: Superdense coding Quantum state teleportation Precision measurements using entanglement History Both speakers yesterday referred to how Schrödinger coined the term “entanglement” in 1935 (or earlier) History Both speakers yesterday referred to how Schrödinger coined the term “entanglement” in 1935 (or earlier) "When two systems, …… enter into temporary physical interaction due to known forces between them, and …… separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives [the quantum states] have become entangled." Schrödinger (Cambridge Philosophical Society) Entanglement Superpositions: Superposed correlations: Entanglement (pure state) Entanglement Tensor Product: Separable Entangled Separability Separable states (with respect to the subsystems A, B, C, D, …) Separability Separable states (with respect to the subsystems A, B, C, D, …) Everything else is entangled e.g. The EPR ‘Paradox’ 1935: Einstein, Podolsky, Rosen - QM is not complete Either: 1. Measurements have nonlocal effects on distant parts of the system. 2. QM is incomplete - some element of physical reality cannot be accounted for by QM - ‘hidden variables’ An entangled pair of particles is sent to Alice and Bob. The spin in measured in the z, x (or any other) direction. The measurement Alice makes instantaneously affects Bob’s….nonlocality? Hidden variables? Bell’s theorem and nonlocality 1964: John Bell derived an inequality that must be obeyed if the system has local hidden variables determining the outcomes. CHSH: S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2 Bell’s theorem and nonlocality 1964: John Bell derived an inequality that must be obeyed if the system has local hidden variables determining the outcomes. CHSH: S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2 a b a’ b’ Alice’s axes: a and a’ Bob’s axes: b and b’ Bell’s theorem and nonlocality 1964: John Bell derived an inequality that must be obeyed if the system has local hidden variables determining the outcomes. CHSH: S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2 0o (a)’ + + + + - - - - 45o (b)’ + + + - - - - + 90o (a’) + + - - - - + + 135o (b’) + - - - - + + + a b a’ b’ Alice’s axes: a and a’ S = +1 - (-1) +1 -1 = 2 Bob’s axes: b and b’ S = +1 -(+1) +1 +1 = 2 Bell states 00 11 00 11 i( 01 10 ) i( 01 10 ) Bell’s theorem and nonlocality S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2 a b Without local hidden variables, e.g. for Bell states a’ b’ E(a,b) = cos E(a,b’) = cos = - sin E(a’,b) = cos= sin E(a’,b’) = cos S = | 2 cos sin When =45o, we have S = i.e no local hidden variables >2 Measures of entanglement Bipartite pure states: Schmidt decomposition Positive, real coefficients Measures of entanglement Bipartite pure states: Schmidt decomposition Positive, real coefficients Reduced density operators Same coefficients Measure of mixedness Measures of entanglement Bipartite pure states: Schmidt decomposition Positive, real coefficients Reduced density operators Same coefficients Measure of mixedness Unique measure of entanglement (Entropy) Example Consider the Bell state: Example Consider the Bell state: This can be written as: Example Consider the Bell state: This can be written as: Maximally entangled (S is maximised for two qubits) “Monogamy of entanglement” Measures of entanglement Bipartite mixed states: • Average over pure state entanglement that makes up the mixture • Problem: infinitely many decompositions and each leads to a different entanglement • Solution: Must take minimum over all decompositions (e.g. if a decomposition gives zero, it can be created locally and so is not entangled) Measures of entanglement Bipartite mixed states: • Average over pure state entanglement that makes up the mixture • Problem: infinitely many decompositions and each leads to a different entanglement • Solution: Must take minimum over all decompositions (e.g. if a decomposition gives zero, it can be created locally and so is not entangled) Entanglement of formation von Neumann entropy Minimum over all realisations of: Entanglement witnesses An entanglement witness is an observable that distinguishes entangled states from separable ones Entanglement witnesses An entanglement witness is an observable that distinguishes entangled states from separable ones Theorem: For every entangled state , there exists a Hermitian operator, A, such that Tr(A)<0 and Tr(A)>=0 for all separable states, Corollary: A mixed state, , is separable if and only if: Tr(A)>=0 Entanglement witnesses An entanglement witness is an observable that distinguishes entangled states from separable ones Theorem: For every entangled state , there exists a Hermitian operator, A, such that Tr(A)<0 and Tr(A)>=0 for all separable states, Corollary: A mixed state, , is separable if and only if: Tr(A)>=0 Thermodynamic quantities provide convenient (unoptimised) EWs Covalent bonding Covalent bonding relies on entanglement of the electrons e.g. H2 Lowest energy (bound) configuration Overall wave function is antisymmetric so the spin part is: Entangled The energy of the bound state is lower than any separable state - witness Covalent bonding is evidence of entanglement Covalent bonding Covalent bonding relies on entanglement of the electrons e.g. H2 NOTE: It is not at all clear that this entanglement could be used in quantum processing tasks. You will often hear people distinguish “useful” entanglement from other sorts The energy of the bound state is lower than any separable state - witness Covalent bonding is evidence of entanglement Detecting Entanglement • State tomography •Bell’s inequalities •Entanglement witnesses (EW) Detecting Entanglement • State tomography •Bell’s inequalities •Entanglement witnesses (EW) Remarkable features of entanglement • It can give rise to macroscopic effects • It can occur at finite temperature (i.e. the system need not be in the ground state) • We do not need to know the state to detect entanglement • It can occur for a single particle Remarkable features of entanglement • It can give rise to macroscopic effects • It can occur at finite temperature (i.e. the system need not be in the ground state) • We do not need to know the state to detect entanglement • It can occur for a single particle Let’s consider an example that exhibits all these features…. Molecule of the Year Molecule of the Year Overall state: Atoms are not entangled Free quantum fields Use Entanglement Witnesses for free quantum fields e.g. Bosons Free quantum fields Use Entanglement Witnesses for free quantum fields e.g. Bosons “Biblical” operators - more on these later….. Free quantum fields Use Entanglement Witnesses for free quantum fields e.g. Bosons Want to detect entanglement between regions of space Energy • Particle in a box of length L where • In each dimension: Energy • Particle in a box of length L where • In each dimension: • For N separable particles in a d-dimensional box of length L, the minimum energy is: QuickTime™ and a TIFF (U ncompressed) decompressor are needed to see this picture. QuickTime™ and a TIFF (U ncompressed) decompressor are needed to see this picture. Energy as an EW • M spatial regions of length L/M Energy as an EW • M spatial regions of length L/M Thermodynamics Internal energy, temperature, and equation of state Internal energy, temperature, and equation of state Ketterle’s experiments The critical temperature for BEC in an homogeneous trap is: Comparing with the onset of entanglement across the system These differ only by a numerical factor of about 2 ! Entanglement as a phase transition Ketterle’s experiments Typical numbers: This gives: In experiments, the temperature of the BEC is typically: Entanglement in a BEC (even though it can be written as a product state of each particle) Munich experiment A reservoir of entanglement - changes the state of the BEC Ref: I. Bloch et al., Nature 403, 166 (2000) Entanglement & spatial correlations The Munich experiment demonstrates long-range order (LRO) Interference term Phase coherence It is tempting to think that LRO and entanglement are the same Entanglement & spatial correlations The Munich experiment demonstrates long-range order (LRO) Interference term Phase coherence It is tempting to think that LRO and entanglement are the same A GHZ-type state is clearly entangled: BUT They are, however, related Ongoing research Tangled ideas in entanglement 1. Entanglement does not depend on how we divide the system 2. A single particle cannot be ‘entangled’ 3. Nonlocality and entanglement are the same thing Entanglement and subsystems Entanglement depends on what the subsystems are Entanglement and subsystems Entanglement depends on what the subsystems are Entangled Single particle entanglement? “Superposition is the only mystery in quantum mechanics” R. P. Feynman What about entanglement? Single particle entanglement? “Superposition is the only mystery in quantum mechanics” R. P. Feynman What about entanglement? Instead of the superposition of a single particle, we can think of the entanglement of two different variables: Single particle entanglement? “Superposition is the only mystery in quantum mechanics” R. P. Feynman What about entanglement? Instead of the superposition of a single particle, we can think of the entanglement of two different variables: Is this all just semantics? Can we measure any real effect, e.g. violation of Bell’s inequalities? Single particle entanglement? Single photon incident on a 50:50 beam splitter: Single particle entanglement? Single photon incident on a 50:50 beam splitter: Single particle entanglement? Single photon incident on a 50:50 beam splitter: Entangled “Bell state” Entanglement must be due to the single particle state “The term ‘particle’ survives in modern physics but very little of its classical meaning remains. A particle can now best be defined as the conceptual carrier of a set of variates. . . It is also conceived as the occupant of a state defined by the same set of variates... It might seem desirable to distinguish the ‘mathematical fictions’ from ‘actual particles’; but it is difficult to find any logical basis for such a distinction. ‘Discovering’ a particle means observing certain effects which are accepted as proof of its existence.” A. S. Eddington, Fundamental Theory, (Cambridge University Press., Cambridge, 1942) pp. 30-31. “The term ‘particle’ survives in modern physics but very little of its classical meaning remains. A particle can now best be defined as the conceptual carrier of a set of variates. . . It is also conceived as the occupant of a state defined by the same set of variates... It might seem desirable to distinguish the ‘mathematical fictions’ from ‘actual particles’; but it is difficult to find any logical basis for such a distinction. ‘Discovering’ a particle means observing certain effects which are accepted as proof of its existence.” A. S. Eddington, Fundamental Theory, (Cambridge University Press., Cambridge, 1942) pp. 30-31. We need a field theory treatment of entanglement Nonlocality and entanglement Nonlocality implies position distinguishability, which is not necessary for entanglement Confusion arises because Alice and Bob are normally spatially separated Nonlocality and entanglement Nonlocality implies position distinguishability, which is not necessary for entanglement Confusion arises because Alice and Bob are normally spatially separated Example: This state is local, but can be considered to have entanglement 1 PBS 2 Summary • What is entanglement • Bell’s theorem and nonlocality • Measures of entanglement • Entanglement witness in a BEC • Confusing concepts in entanglement