QMCh7_2008

QMCh7_2008 - Chapter 7 Two-qubit problems and an...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
Two-qubit problems and an introduction to quantum communications Chapter 7
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Why studying two qubits? • Demonstration of quantum correlation and applications. e.g. Bell inequality, quantum teleportation, dense coding, quantum cryptography. • Quantum computation can be decomposed into single qubit operations and two-qubit operations. Quantum Computer in Ψ out Ψ = Two-state subsystem 00000000 in ψ = e.g. out X X CX = X = Binary representation of a (N-bit) number
Background image of page 2
Two spin ½ particles as a composite quantum system AB Ψ A a b ψ ⎛⎞ = ⎜⎟ ⎝⎠ B c d = ac ad bc bd = ↑↑ ↑↓ ↓↑ ↓↓ A BA B A B ac ad bc bd =↑ + +↓ + () ( ) A AB B abcd + + Separable state formed by
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
A general state vector for a composite system with two subsystems A B 11 1 2 1 21 2 2 2 12 , ,..., , , ,........ , , MM NN N M ab a b a b a b abab Bases vectors Bases vectors 123 , , ,..... , M bb b b A+B system: Any state vector can by expressed as NM A Bi j i j ij ca b == Ψ= ∑∑ , , , N aa a a i.e., the bases vectors of A and B are: 2 ij c = Joint probability of the system A at and the system B at i a j b
Background image of page 4
Example: Spin states of a two-electron system 1 234 11 10 01 0 0 cc c c Ψ= + + + non-separable states: () 1 11 0 0 2 ± ± 1 10 01 2 ± Φ= ± Show that ,,, + −+− Ψ ΨΦΦ form a orthogonal and complete set of bases vectors. These are called Bell’s bases .
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Separable states AB Ψ ( ) ( ) 12 1 2 aa b b αβ μ ν =+ + 11 21 1 2 2 2 a b ab a b α μβ μα β + + Non-Separable (Entangled) states A BA B ψ Ψ≠ () 1 0 0 2 Ψ= ± e.g.
Background image of page 6
A B Measure the state of A What is the state of B after the measurement ?? ( gaining information of A) AB nm before A B nm cnm Ψ= ∑∑ If the measurement tells that the subsystem A is in the state . What is the probability? A k AB km after A B m ckm γ Only survives A k km A B m kc m = A and B are now separable generally non-separable γ = normalization constant Quantum measurement of a subsystem
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Quantum measurement of a subsystem: Separable systems 11 NM A Bi j i ij ca b == Ψ= ∑∑ j is said to be The state of A+B system ij i j cf g = A i j j f ag b Making a measurement of A does not change the state of B. (No information of B can be gained from A for separable states) : i.e., separable if and only if 1 () M AB jj j collapse state of A gb = Ψ→
Background image of page 8
Quantum measurement of a subsystem: Entangled systems 11 NM A Bi j i ij ca b == Ψ= ∑∑ j is said to be The state of A+B system ij i j c fg A i j j fa g b Ψ≠ Making measurement of A “change” the state of B because some information of B can be gained from the measurement result of A: i.e., inseparable if and only if 1 () M A Bj j j collapse state of A hb = Ψ→ jj hg
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Maximally entangled states ( ) 1 11 0 0 2 A BA B A B Ψ= + A B 1 A 0 A 50% 50% The result of A completely determines the state of B without “touching” B. However, it is not possible if is separable AB Ψ 1 B state of B is 0 B state of B is detector But is such a correlation classical ? Can it be simulated by classical physics?
Background image of page 10
where are eigenvectors of 1
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
No-Cloning theorem 0 Ψ s AB 0 Ψ A 0 Ψ unitary evolution B Given an unknown
Background image of page 12
Image of page 13
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

Page1 / 46

QMCh7_2008 - Chapter 7 Two-qubit problems and an...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online