Principles of Quantum Mechanics Pt. 2

Transcription

1 Principles of Quantum Mechanics Pt. 2 PHYS Southern Illinois University February 9, 2017 PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13

2 The Composition of Systems Postulate The Composition of Systems Postulate If A and B are two quantum systems with state spaces H A and H B respectively, the state space of the their combined physical system is the tensor product space H A H B. This is often called a bipartite system. The bipartite product state ψ 1 A ψ 2 B H A H B describes system A in state ψ 1 A and system B in state ψ 2 B. But H A H B is much larger than the set of tensor product states. A bipartite state Ψ AB is entangled if it is not a tensor product state; i.e. Ψ AB ψ 1 A ψ 2 B. A non-entangled state is also called a separable state. In an entangled state, no state vector can be assigned to the individual subsystems. PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13

3 Entanglement How to determine if a state Ψ AB is entangled? For two qubits, expand Ψ AB in the computational basis Ψ AB = a 00 + b 01 + c 10 + d 11, and suppose that Ψ AB = ψ 1 A ψ 2 B = (x x 1 1 ) (y y 1 1 ) = x 0 y x 0 y x 1 y x 1 y Comparing coefficients, Ψ AB is a product state if and only if ad = bc. Proof: PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13

4 Entanglement A more elegant method for deciding entanglement Recall the isomorphism H A H B = L(H A, H B ) given by d A,d B i,j=1 c ij i A j B d A,d B i,j=1 c ij i A j B. For product states: ψ 1 ψ 2 = d A,d B i,j=1 (x i i ) (y j j ) d A,d B i,j=1 (x i i ) (y j j ) = ψ 1 ψ 2 where ψ2 := d B j=1 y j j. The operator ψ 1 ψ2 is rank one! PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13

5 Entanglement To decide entanglement: 1 For a bipartite state Ψ AB, expand in the computational product basis and obtain its corresponding operator T Ψ via the mapping Ψ AB = d A,d B i,j=1 c ij i A j B T Ψ = d A,d B i,j=1 2 Write out T Ψ in matrix form and compute its rank. c ij i A j B. 3 Recall that a matrix has rank r if and only if all of its (r + 1)-minors vanish, and there exists at least one nonvanishing r-minor 4 Ψ AB is entangled if and only if T Ψ is rank one. PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13

6 Entanglement Example Is the state Ψ AB = a 00 + ab 01 + c 02 + ab 10 + b 11 + c b/a 12 c 22 entangled? PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13

7 Unitary Evolution on Composite Systems By the second postulate, a closed bipartite system will evolve unitarily. That is, ψ 0 AB ψ 1 AB = U ψ 0 AB. The operator U is sometimes called a gate. Any tensor product unitary U = U A U B is called a local unitary (LU). Two states Ψ and Ψ are called LU equivalent, denoted by Ψ LU Ψ, if there exists a local unitaries U V such that Ψ = U V Ψ. Example Let Ψ = λ 1 a 1 b 1 + λ 2 a 2 b 2 + λ 3 a 3 b 3 be a Schmidt decomposition of Ψ. Then Ψ LU Ψ = λ λ λ PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13

8 Unitary Evolution on Composite Systems LU operators cannot generate entanglement! (U V ) α β = U α V β = α β U, V, α, β. Any unitary that cannot be written as a tensor product is called a nonlocal unitary. Example The SWAP operator F is nonlocal on H 2 : F α β = β α α, β. Proof: PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13

9 Unitary Evolution on Composite Systems An important class of nonlocal gates on two-qubits is controlled unitaries. One type of controlled unitary has the form U AB = 0 0 I V, where V is some unitary on system B. The unitary V is applied to B iff A is in state 1 : U 00 = 00, U 01 = 01, U 10 = 1 U 0, U 11 = 1 U 1. For this U, we say system A is the source and system B is the target. The operator Û = FUF switches the roles of A and B so that B is the source and A is the target. PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13

10 Unitary Evolution on Composite Systems The controlled-not (CNOT) gate is one of the most useful operations in quantum computing: It has the action U CNOT = 0 0 I σ x. U CNOT 00 = 00, U CNOT 10 = 11 U CNOT 01 = 01, U CNOT 11 = 10. U CNOT does not generate entanglement when acting on the computational product basis. However, it can generate entanglement when acting on superpositions: U CNOT + 0 = 1/2( ). PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13

11 Non-Entangling Gates We have seen that every local unitary cannot generate entanglement. But is the converse true? Does every nonlocal unitary generate entanglement? The answer is no. SWAP (which is nonlocal) cannot generate an entangled state from a product state since F α β = β α. In the homework, you will prove that SWAP and LU are the only types of non-entangling gates. However, SWAP becomes entangling when acting on subsystems of some larger system. For example, if Ψ AA is entangled, then swapping systems A and B in the state Ψ AA 0 B generates Ψ AB 0 A. So systems AA and B are entangled even though they started in a product state. PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13

12 The Bell Basis For two-qubit systems, so far we have primarily been working in the computational product basis { 00, 01, 10, 11 }. However, another highly useful basis for C 2 C 2 is called the Bell basis, and it consists of four orthonormal entangled states: Φ + := Φ 00 = 1/2( ) Ψ + := Φ 01 = 1/2( ) Φ := Φ 10 = 1/2( ) Ψ := Φ 11 = 1/2( ). PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13

13 The Bell Basis Notice that the Bell states are LU equivalent to one another. Φ b1 b 2 = σ b 1 z σ b 2 x I Φ 00 b 1, b 2 {0, 1}. In Φ b1 b 2, b 1 is called the phase bit and b 2 is called that amplitude bit (remember the ordering PA ). Example Express the state Ψ = a 00 + b 01 + c 10 + d 11 in the Bell basis. PHYS Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, / 13