Basics of Quantum Theory

1. Basics of Quantum Theory

2. 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 AB) iff A is completely contained within B. • Later, we may try to make these definitions more formal & precise. B A

3. 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)

4. 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.

5. 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 functionp:[0,1] stating that the system is in state S with probability p(S). • Quantum mechanics: • Give a complex-valued wavefunction:  ℂ, |(S)|1, implying the system is instate S with probability |(S)|2.

6. States & State Spaces • A possible stateS 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…

7. 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 (st), and that’s all there is to it. • In quantum mechanics (i.e. in reality): • There are pairs of states st 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.

8. 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  ofsystem A can be identified with a unit-length vector in the Hilbert space ℋ. t s 

9. (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

10. Hilbert spaces • A Hilbert spaceℋ 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: xy = (yx)* (* = complex conjugate) xx  0 xx= 0 if and only if x = 0 xy is linear, under scalar multiplication and vector addition within both x and y “Component”picture: y Another notation often used: x xy/|x| “bracket”

11. Review: The Complex Number System • It is the extension of the real number system via closure under exponentiation. • (Complex) conjugate: c* = (a + bi)*  (a bi) • Magnitude or absolute value: |c|2= c*c =a2+b2 +i The “imaginary”unit c b  + a “Real” axis “Imaginary”axis i

12. Review: Complex Exponentiation • Powers of i are complex units: • Note: ei/2 = i ei= 1 e3 i/2 =  i e2 i= e0 = 1 ei +i  1 +1 i

13. 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: s0s1s2si-1si 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: • And the inner product is given by:

14. Dirac’s Ket Notation • Note: The inner product definition is the same as the matrix product of x, as a conjugated row vector, times y, as a normal column vector. • This leads to the definition, for state s, of: • The “bra” s| means the row matrix [c0* c1* …] • The “ket” |s means the column matrix  • The adjoint operator † takes any matrix Mto its conjugate transpose M†MT*, sos| can be defined as |s†, and xy = x†y. “Bracket”

15. Distinguishability of States, again • State vectors s and t are (perfectly) distinguishable or orthogonal (write st) 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, andnon-identical. Not perfectly distinguishable. • We say, “the amplitude of state s, given state t, is s†t”. Note: amplitudes are complex numbers.

16. 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.

17. 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 • 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”.

18. Linear Operators • V,W: Vector spaces. • A linear operator A from V to W is a linear function A:VW. An operator onV 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.

19. 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 eigenvectorv, 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

20. Observables • A Hermitian operator H on the set V is called an observable if there is an orthonormal (all unit-length, 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.

21. Wavefunctions • Given any set Sℋ of system states, • Whether all mutually distinguishable, or not, • a quantum state vector v can be translated to a wavefunction:Sℂ, giving, for each state sS, 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.

22. Time Evolution • Postulate 4: (Closed) systems evolve (change state) over time via unitary transformations. t2 = Ut1t2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

23. 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 • Substitute H = i∂/t and p = i∂/x: • Apply to wavefunction Ψ over position states x: (Where ∂/a≝ ∂/∂a)

24. Multidimensional Form For a system with states given by(x,t) where t is a global time coordinate, and xdescribes 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 (2nd-order, linear, partial) differential equation:

25. 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

26. Gaussian wave packet moving to the right;Array of small sharp potential-energy barriers

27. Initial reflection/refraction of wave packet

28. A little later

29. Aimed a little higher

30. A faster-moving particle

31. 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=AB 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, thenC has state c = ab= ca0cb0|a0b0 + ca0cb1|a0b1 +ca1cb0|a1b0 + ca1cb1|a1b1

32. 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 = ab, • 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

33. 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.

34. Unitary Transformations • A matrix (or linear operator) U is unitary iff its inverse equals its adjoint: U1 = 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: • Corresponds to a change of basis, from one orthonormal basis to another. • Or, a generalized rotation of in Hilbert space

35. 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).

36. 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.

37. 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.

38. Simulating the Schroedinger Wave Equation A Perfectly Reversible Discrete Numerical Simulation Technique

39. Simulating Wave Mechanics • The basic problem situation: • Given: • A (possibly complex) initial wavefunctionin an N-dimensional position basis, and • a (possibly complex and time-varying) potential energy function , • a time t after (or before) t0, • Compute: • Many practical physics applications...

40. 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 . • Project back onto position basis. • Problem: • It may be intractable to find the eigenstates! • We resort to numerical methods...

41. 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 + It1·It+1 • 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. (R=real, I=imag., t=time step index)

42. 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.

43. Centered Difference Equations • Discretize derivatives in a symmetric fashion: • Leads to update rules like:x(t + t) = x(t  t) + 2t ·f(x(t)) • Problem: States at odd- vs. even-numbered time steps not constrainedto stay close to each other! 2t·f x1 g x2 + g x3 + g x4 +

44. Centered Schrödinger Equation • Schrödinger’s equation for 1 particle in 1-D: • Replace time (& also space) derivatives with centered differences. • Centered difference equation has realpart at odd times that depends only onimaginary part at even times, &vice-versa. • Drift not an issue - real & imaginaryparts represent different state components! R1 g  I2 g R3 + g  I4

45. 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: convergencestability. • 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.

46. Phenomena Observed in Model • Perfect reversibility • Wave packet momentum • Conservation of probability mass • Harmonic oscillator • Tunneling/reflection at potential energy barriers • Interference fringes • Diffraction

47. 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