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Basics of Quantum Theory Systems and Subsystems • Intuitively speaking, a physical system consists of a region of spacetime & all the entities (e.g. particles & fields) contained within it. – The universe (over all time) is a physical system – Transistors, computers, people: also phys. systs. • One physical system A is a subsystem of another system B (write AB) iff A is A completely contained within B. • Later, we may try to make these definitions more formal & precise. B Closed vs. Open Systems • A subsystem is closed to the extent that no particles, information, energy, or entropy (terms to be defined) enter or leave the system. – The universe is (presumably) a closed system. – Subsystems of the universe may be almost closed • Often in physics we consider statements about closed systems. – These statements may often be perfectly true only in a perfectly closed system. – However, they will often also be approximately true in any nearly closed system (in a well-defined way) Concrete vs. Abstract Systems • Usually, when reasoning about or interacting with a system, an entity (e.g. a physicist) has in mind a description of the system. • A description that contains every property of the system is an exact or concrete description. – That system (to the entity) is a concrete system. • Other descriptions are abstract descriptions. – The system (as considered by that entity) is an abstract system, to some degree. • We nearly always deal with abstract systems! – Based on the descriptions that are available to us. System Descriptions • Classical physics: – A system could be completely described by giving a single state S out of the set of all possible states. • Statistical mechanics: – Instead, give a probability distribution function p:[0,1] stating that the system is where in state S with probability p(S). p ( S ) 1 • Quantum mechanics: SΣ – Give a complex-valued wavefunction : ℂ, where |(S)|1, implying the system is in 2 Ψ( S ) 1 state S with probability |(S)|2. SΣ States & State Spaces • A possible state S of an abstract system A (described by a description D) is any concrete system C that is consistent with D. – I.e., it is possible that the system in question could be completely described by the description of C. • The state space of A is the set of all possible states of A. • So far, the concepts we’ve discussed can be applied to either classical or quantum physics – Now, let’s get to the uniquely quantum stuff… Distinguishability of States • Classical & quantum mechanics differ crucially regarding the distinguishability of states. • In classical mechanics, there is no issue: – Any two states s,t are either the same (s=t), or different (st), and that’s all there is to it. • In quantum mechanics (i.e. in reality): – There are pairs of states st that are mathematically distinct, but not 100% physically distinguishable. – Such states cannot be reliably distinguished by any number of measurements, no matter how precise. • But you can know the real state (with high probability), if you prepared the system to be in a certain state. State Vectors & Hilbert Space • Let S be any maximal set of distinguishable possible states s, t, … of an abstract system A. – I.e., no possible state that is not in S is perfectly distinguishable from all members of S. • Identify the elements of S with unit-length, mutually-orthogonal (basis) vectors in an abstract complex vector space ℋ. – The system’s “Hilbert space” • Postulate 1: Each possible state of system A can be identified with a unitlength vector in the Hilbert space ℋ. t s (Abstract) Vector Spaces • A concept from abstract linear algebra. • A vector space, in the abstract, is any set of objects that can be combined like vectors, i.e.: – You can add them • Addition is associative & commutative • Identity law holds for addition to zero vector 0 – You can multiply them by scalars (incl. 1) • Associative, commutative, and distributive laws hold • Note: There is no inherent basis (set of axes) – The vectors themselves are the fundamental objects, rather than being just lists of coordinates Hilbert spaces • A Hilbert space ℋ is a vector space in which the scalars are complex numbers, with an inner product (dot product) operation : ℋ×ℋ C – See Hirvensalo p. 107 for defn. of inner product: xy = (yx)* (* = complex conjugate) xx 0 xx = 0 if and only if x = 0 “Component” xy is linear, under scalar multiplication picture: and vector addition within both x and y y x xy/|x| Another notation often used: x y x y “bracket” Review: The Complex Number System • It is the extension of the real number system via closure under exponentiation. c a bi (c C, a,b R) The “imaginary” Re [c ] a i -1 unit Im [c] b • (Complex) conjugate: c* = (a + bi)* (a bi) • Magnitude or absolute value: c c*c (a bi )(a bi ) a 2 b2 |c|2 = c*c = a2+b2 +i b c a + “Real” axis “Imaginary” i axis Review: Complex Exponentiation • Powers of i are complex units: e cos i sin θi • Note: ei/2 = i ei = 1 e3 i /2 = i e2 i = e0 = 1 ei +i 1 +1 i Vector Representation of States • Let S={s0, s1, …} be any maximal set of mutually distinguishable states, indexed by i. • A basis vector vi identified with the ith such state can be represented as a list of numbers: s0 s1 s2 si-1 si si+1 vi = (0, 0, 0, …, 0, 1, 0, … ) • Arbitrary vectors v in the Hilbert space ℋ can then be defined by linear combinations of the vi: v ci vi (c0 , c1 ,) * i x y x yi • And the inner product is given by: i i Dirac’s Ket Notation • Note: The inner product x y x* yi definition is the same as the “Bracket” i y1 matrix product of x, as a * * x1 x2 y2 conjugated row vector, times y, as a normal column vector. • This leads to the definition, for state s, of: i – The “bra” s| means the row matrix [c0* c1* …] – The “ket” |s means the column matrix c1 † • The adjoint operator takes any matrix M to its conjugate transpose M† MT*, so s| can be defined as |s†, and xy = x†y. c 2 Distinguishability of States, again • State vectors s and t are (perfectly) distinguishable or orthogonal (write st) iff s†t = 0. (Their inner product is zero.) • State vectors s and t are perfectly indistinguishable or identical (write s=t) iff s†t = 1. (Their inner product is one.) • Otherwise, s and t are both non-orthogonal, and non-identical. Not perfectly distinguishable. • We say, “the amplitude of state s, given state t, is s†t”. Note: amplitudes are complex numbers. Probability and Measurement • A yes/no measurement is an interaction designed to determine whether a given system is in a certain state s. • The amplitude of state s, given the actual state t of the system determines the probability of getting a “yes” from the measurement. • Postulate 2: For a system prepared in state t, any measurement that asks “is it in state s?” will say “yes” with probability P(s|t) = |s†t|2 – After the measurement, the state is changed, in a way we will define later. A Simple Example • Suppose abstract system S has a set of only 4 distinguishable possible states, which we’ll call s0, s1, s2, and s3, with corresponding ket vectors |s0, |s1, |s2, and |s0. • Another possible state is then the unit vector 1 i s0 s3 2 2 1 2 0 0 i 2 • Which is equal to the column matrix: • If measured to see if it is in state s0, we have a 50% chance of getting a “yes”. Linear Operators • V,W: Vector spaces. • A linear operator A from V to W is a linear function A:VW. An operator on V is an operator from V to itself. • Given bases for V and W, we can represent linear operators as matrices. • An Hermitian operator H on V is a linear operator that is self-adjoint (H=H†). – Its diagonal elements are real. Eigenvalues & Eigenvectors • v is called an eigenvector of linear operator A iff A just multiplies v by a scalar a, i.e. Av=av – “eigen” (German) means “characteristic” • a, the eigenvalue corresponding to eigenvector v, is just the scalar that A multiplies v by • a is degenerate if it is shared by 2 eigenvectors that are not scalar multiples of each other • Any Hermitian operator has all real-valued eigenvectors, which form an orthogonal set Observables • A Hermitian operator H on the set V is called an observable if there is an orthonormal (all unitlength, and mutually orthogonal) subset of its eigenvectors that forms a basis of V. • Postulate 3: Every measurable physical property of a system can be described by a corresponding observable H. Measurement outcomes correspond to eigenvalues of H. • The measurement can also be thought of as a yes-no test that compares the state with each of the observable’s normalized eigenvectors. Wavefunctions • Given any set Sℋ of system states, – Whether all mutually distinguishable, or not, • a quantum state vector v can be translated to a wavefunction :Sℂ, giving, for each state sS, the amplitude (s) of that state. – When s is some other state vector, and the “actual” state is v, then (s) is just s†v. – Whenever S includes a basis set, determines v. • is called a “wavefunction” because its dynamics takes the form of a wave equation when S ranges over a space of positional states. Time Evolution • Postulate 4: (Closed) systems evolve (change state) over time via unitary transformations. t2 = Ut1t2 t1 • Note that since U is linear, a small-factor change in the amplitude of a particular state at t1 leads to a correspondingly small change in the amplitude of the corresponding state at t2! – Chaotic sensitivity to initial conditions requires an ensemble of initial states that are different enough to be distinguishable (in the sense we defined) • Indistinguishable initial states never beget distinguishable outcomes true chaotic/analog computing doesn’t exist Schrödinger's Wave Equation • Start w. classical Hamiltonian energy equation: H = K + P (K = kinetic, P = potential) • Express K in terms of momentum: K = ½mv2 = p2/2m (Where • Substitute H = i∂/t and p = i∂/x: ∂/a ≝ ∂/∂a) 2 2 i i P ( x, t ) 2 t 2m x • Apply to wavefunction Ψ over position states x: ( x, t ) 2 2 ( x, t ) i i P ( x, t ) ( x, t ) 2 t 2m x Multidimensional Form For a system with states given by (x,t) where t is a global time coordinate, and x describes N/3 particles (p0,…,pN/3−1) with masses (m0,…,mN/3−1) in a 3-D Euclidean space, where each pi is located at coordinates (x3i, x3i+1, x3i+2), and where particles interact with potential energy function P(x,t), the wavefunction (x,t) obeys the following (2ndorder, linear, partial) differential equation: N 1 1 2 ( x, t ) P( x, t ) i ( x, t ) 2 2 j 0 m j / 3 x j t Features of the wave equation • Particles’ momentum state p is encoded by their wavelength , as per p=h/ • The energy of a state is given by the frequency f of rotation of the wavefunction in the complex plane: E=hf. • By simulating this simple equation, one can observe basic quantum phenomena, such as: – Interference fringes – Tunneling of wave packets through potential energy barriers • Demo of SCH simulator Gaussian wave packet moving to the right; Array of small sharp potential-energy barriers Initial reflection/refraction of wave packet A little later Aimed a little higher A faster-moving particle Compound Systems • Let C=AB be a system composed of two separate subsystems A,B each with vector spaces A, B with bases |ai, |bj. • The state space of C is a vector space C=AB given by the tensor product of spaces A and B, with basis states labeled as |aibj. • E.g., if A has state a=ca0|a0 + ca1 |a1, while B has state b=cb0|b0 + cb1 |b1, then C has state c = ab= ca0cb0|a0b0 + ca0cb1|a0b1 + ca1cb0|a1b0 + ca1cb1|a1b1 Entanglement • If the state of compound system C can be expressed as a tensor product of states of two independent subsystems A and B, c = ab, • then, we say that A and B are not entangled, and they have individual states. – E.g. |00+|01+|10+|11=(|0+|1)(|0+|1) • Otherwise, A and B are entangled (basically correlated); their states are not independent. – E.g. |00+|11 Size of Compound State Spaces • Note that a system composed of many separate subsystems has a very large state space. • Say it is composed of N subsystems, each with k basis states: – The compound system has kN basis states! – There are states of the compound system having nonzero amplitude in all these kN basis states! – In such states, all the distinguishable basis states are (simultaneously) possible outcomes (each with some corresponding probability) – Illustrates the “many worlds” nature of quantum mechanics. Unitary Transformations • A matrix (or linear operator) U is unitary iff its inverse equals its adjoint: U1 = U† • Some properties of unitary transformations: – – – – Invertible, bijective, one-to-one. The set of row vectors is orthonormal. Ditto for the set of column vectors. Preserves vector length: |U| = | | • Therefore also preserves total probability over all states: 2 2 ( si ) – Corresponds to a change of basis, from one orthonormal basis to another. – Or, a generalized rotation of in Hilbert space i After a Measurement? • After a system or subsystem is measured from outside, its state appears to collapse to exactly match the measured outcome – the amplitudes of all states perfectly distinguishable from states consistent w. that outcome drop to zero – states consistent with measured outcome can be considered “renormalized” so their probs. sum to 1 • This “collapse” seems nonunitary (& nonlocal) – However, this behavior is now explicable as the expected consensus phenomenon that would be experienced even by entities within a closed, perfectly unitarily-evolving world (Everett, Zurek). Pointer States • For a given system interacting with a given environment, – The system-environment interactions can be considered measurements of a certain observable of the system by the environment, and vice-versa. • For each observable there are certain basis states that are characteristic of that observable. – The eigenstates of the observable • A pointer state of a system is an eigenstate of the system-environment interaction observable. – The pointer states are the inherently stable states. Key Points to Remember: • An abstractly-specified system may have many possible states; only some are distinguishable. • A quantum state/vector/wavefunction assigns a complex-valued amplitude (si) to each distinguishable state si (out of some basis set) • The probability of state si is |(si)|2, the square of (si)’s length in the complex plane. • States evolve over time via unitary (invertible, length-preserving) transformations. Simulating the Schroedinger Wave Equation A Perfectly Reversible Discrete Numerical Simulation Technique Simulating Wave Mechanics • The basic problem situation: – Given: • A (possibly complex) initial wavefunction 0 ( x, t0 ) in an N-dimensional position basis, and • a (possibly complex and time-varying) potential energy function V ( x , t ), • a time t after (or before) t0, – Compute: • ( x, t ) • Many practical physics applications... The Problem with the Problem • An efficient technique (when possible): – – – – – Convert V to the corresponding Hamiltonian H. Find the energy eigenstates of H. Project onto eigenstate basis. Multiply each component by e iH ( t t0 ). Project back onto position basis. • Problem: – It may be intractable to find the eigenstates! • We resort to numerical methods... History of Reversible Schrödinger Sim. See http://www.cise.ufl.edu/~mpf/sch • Technique discovered by Ed Fredkin and student William Barton at MIT in 1975. • Subsequently proved by Feynman to exactly conserve a certain probability measure: Pt = Rt2 + It1·It+1 (R=real, I=imag., t=time step index) • 1-D simulations in C/Xlib written by Frank at MIT in 1996. Good behavior observed. • 1 & 2-D simulations in Java, and proof of stability by Motter at UF in 2000. • User-friendly Java GUI by Holz at UF, 2002. Difference Equations • Consider any system with state x that evolves according to a diff. eq. that is 1st-order in time: x = f(x) • Discretize time to finite scale t, and use a difference equation instead: x(t + t) = x(t) + t ·f(x(t)) • Problem: Behavior not always numerically stable. – Errors can accumulate and grow exponentially. Centered Difference Equations • Discretize derivatives in a symmetric fashion: d x x(t t ) x(t t ) dt 2t • Leads to update rules like: x(t + t) = x(t t) + 2t ·f(x(t)) • Problem: States at odd- vs. evennumbered time steps not constrained to stay close to each other! x1 + x3 + 2t·f g g g x2 + x4 Centered Schrödinger Equation • Schrödinger’s equation for 1 particle in 1-D: 2 d2 d i ( x, t ) V ( x, t ) ( x, t ) 2 dt 2m dx • Replace time (& also space) derivatives with centered differences. (t t ) (t t ) i g ( (t )) • Centered difference equation has real R1 part at odd times that depends only on g I2 imaginary part at even times, & g R3 + vice-versa. – Drift not an issue - real & imaginary parts represent different state components! g I4 Proof of Stability • Technique is proved perfectly numerically stable & convergent assuming V is 0 and x2/t > /m (an angular velocity) • Elements of proof: – Lax-Richmyer equivalence: convergencestability. – Analyze amplitudes of Fourier-transformed basis • Sufficient due to Parseval’s relation – Use theorem (cf. Strikwerda) equating stability to certain conditions on the roots of an amplification polynomial (g,), which are satisfied by our rule. • Empirically, technique looks perfectly stable even for more complex potential energy funcs. Phenomena Observed in Model • • • • • • • Perfect reversibility Wave packet momentum Conservation of probability mass Harmonic oscillator Tunneling/reflection at potential energy barriers Interference fringes Diffraction Interesting Features of this Model • Can be implemented perfectly reversibly, with zero asymptotic spacetime overhead – Every last bit is accounted for! • As a result, algorithm can run adiabatically, with power dissipation approaching zero – Modulo leakage & frictional losses • Can map it to a unitary quantum algorithm – Direct mapping: • Classical reversible ops only, no quantum speedup – Indirect (implicit) mapping: • Simulate p particles on kd lattice sites using pd lg k qubits • Time per update step is order pd lg k instead of kpd