MP 472 Quantum Information and Computation

Transcription

1 MP 472 Quantum Information and Computation Outline Open quantum systems The density operator ensemble of quantum states general properties the reduced density operator Quantum noise (decoherence) Quantum error correction Fault-tolerant quantum computation

2 Density operator (review) ρ = Σ i p i ψ i ><ψ i Characterization of density operators: An operator ρ is the density operator associated to some ensemble {p i, ψ i >} iff it satisfies the conditions: (1) (Trace condition) tr(ρ) = 1 (2) (Positivity) ρ is positive operator Proof: (1) tr(ρ) = Σ i p i tr( ψ i ><ψ i ) = Σ i p i =1 (2) Suppose φ> is an arbitrary vector in the Hilbert space <φ ρ φ> = Σ i p i <φ ψ i ><ψ i φ> = Σ i p i <φ ψ i > 2 P 0 Conversely, suppose ρ is any operator satisfying (1) and (2). Since ρ is positive it must have a spectral decomposition ρ = Σ j λ i j><j where the vectors j> are orthogonal and λ i 2 R are nonnegative eigenvalues of ρ. From the trace condition Σ j λ i = 1. Therefore a system in state j> with probability λ i will have the density operator ρ.

3 Schmidt decomposition Theorem (Schmidt s decomposition): Suppose ψ> is a pure state of a bipartite composite system, AB. Then there exist orthonormal states i A > for system A, and i B > for system B s.t. ψ> = λ i i A > i B > Σi where λ i are non-negative real numbers satisfying known as Schmidt coefficients. Σ λ i2 = 1 i Proof: Let j> and k> be any fixed orthonormal bases for A and B (the relevant state spaces are here of the same dimension). Then ψ> can be written ψ> = Σ jk a jk j> k> for some matrix a of complex numbers a jk. By the singular value decomposition a=udv, where d is a nonnegative diagonal matrix, and u and v are unitary matrices, ψ> = Σ ijk u ji d ii v ik j> k> Defining i A > = Σ j u ji j>, and i B > = Σ k v ik k>, and l i = d ii, this gives ψ> = Σ i λ i i A > i B > i A > form orthonormal set due to unitarity of u and orthonormality of j>, and similarly i B > form an orthonormal set.

4 Schmidt decomposition: examples The number of non-zero values of the Schmidt coefficients λ i is called the Schmidt number. If it equals to 1 then the state is the product state. A) ψ> = (2-1/2 )( 00> + 01>) 1) construct matrices a and a + a: 2) diagonalize a + a: a = (2-1/2 ) a + a= (2-1 ) characteristic equation det(a + a αi) = 0 α 1 = λ 12 = 1 and α 2 = λ 2 2 = 0 There is only one nonzero Schmidt coefficient and thus ψ> is a product state. B) ψ> = (2-1/2 )( 00> + 11>) 1) construct a and a + a: 2) diagonalize a + a: a = (2-1/2 ) a + a= (2-1 ) characteristic equation det(a + a αi) = 0 α 1 = λ 12 = 1/2 and α 2 = λ 2 2 = 1/2 There are two nonzero Schmidt coefficients and thus ψ> is an entangled state. Homework: What is the Schmidt number for the state (3-1/2 )( 00>+ 01>+ 11>)?

5 Reduced density operator Suppose we have physical systems A and B, whose state is described by a density matrix ρ AB. The reduced density operator for system A is ρ A = tr B (ρ AB ) where tr B is an operator map known as partial trace over system B. It is defined as ρa = tr B ( a 1 ><a 2 5 b 1 ><b 2 ) = a 1 ><a 2 tr( b 1 ><b 2 ) where a 1 and a 2 are any two vectors in A and b 1 and b 2 are any two vectors in B. tr( b 1 ><b 2 ) is the usual trace, so tr( b 1 ><b 2 ) = <b 2 b 1 > (via completeness relation!) Physical interpretation: The reduced density matrix ρ A above provides correct measurement statistics for measurements on system A. Completeness relation: Let { i>} be orthonormal basis for the vector space V, so an arbitrary vector can be written v> = Σ i v i i> for some set of v i 2 C. Note that v i = <i v>, thus:(σ i i><i ) v> = Σ i i><i v> = Σ i v i i> = v>. Since this is true for all v>, it follows that Σ i i><i = I. tr( b 1 ><b 2 ) = Σ i <i b 1 ><b 2 i> = Σ i <b 2 i><i b 1 > = <b 2 (Σ i i><i ) b 1 > = <b 2 b 1 >

6 Reduced density operator: independent subsystems A state of the composite system AB consisting of independent subsystems is a product state which is described by the density operator ρ AB = ρ 5 σ Example: Ψ> = (2-1/2 )( 00> + 10>) = [(2-1/2 )( 0> + 1>)] 0> = ψ A > 5 ψ B > ρ AB = Ψ><Ψ = ψ A ><ψ A 5 ψ B ><ψ B = ρ 5 σ (Remark: in general ρ and σ can describe mixed states!) The reduced density operators for the composite system in a product state is calculated as follows: ρ A = tr B (ρ AB ) = tr B (ρ 5 σ) = ρ tr(σ) = ρ ρ B = tr A (ρ AB ) = tr A (ρ 5 σ) = tr(ρ) σ = σ

7 Reduced density operator: entangled subsystems Example: Two-qubit system in the Bell state β 00 > = (2-1/2 )( 00> + 11>): ρ = (2-1/2 ) 2 ( 00> + 11>)(<00 + <11 ) = (1/2)( 00>< >< >< ><11 ) Partial trace over the second qubit is ρ 1 = tr 2 [(1/2)( 00>< >< >< ><11 )] = = (1/2)( 0><0 <0 0> + 0><1 <1 0> + 1><0 <0 1> + 1><1 <1 1>) = = (1/2)( 0><0 + 1><1 ) = I/2 Is this single qubit mixed state? tr((i/2) 2 ) = ½ < 1 The state of the joint system of two qubits is a pure state, i.e. it is known exactly, however, the first qubit is in mixed state, i.e. a state about which we do not have maximal knowledge. characteristics of quantum entanglement Homework: ρ 1 = (1/2)I is a state of one qubit - what is a Bloch vector of this state?

8 Reduced density operators and the Schmidt decomposition Suppose ψ> is a pure state of a bipartite composite system, AB. The Schmidt decomposition is given as ψ> = Σ λ i i A > i B > i where the Schmidt coefficients λ i are real and satisfy Σ i λ i2 = 1. The density matrix of the system is ρ = ψ><ψ = λ i2 i A ><i A 5 i B ><i B Σi and the reduced density matrices are ρ A = λ Σi i2 i A ><i A ρ B = λ Σi i2 i B ><i B Note that the eigenvalues of ρ A and ρ B are identical and are equal to λ i2. (recall that i A > and i B > form orthonormal sets).