量子信息中的纠缠问题

Entanglement in quantum information theory Xiaoguang Wang(王晓光 ) Zhejiang Institute of Modern Physics, Zhejiang University Outline 1. Introduction to entanglement. 2. Entanglement measures. 3. Application of entanglement theory to magnetic systems. Entanglement Product state in a composite system: )10|01(| 2 1 |   21 ||  A pure state is entangled if it cannot be written in the above form. For instance, The entanglement results from the superposition principle and the tensor product structure. An example Entangled or not )11|10|01|00(| 2 1 | )11|10|01|00(| 2 1 | 2 1   The second state is entangled, while the first one is not. )|1||0(| 2 1 | )1|0(| 2 1 |,||| 2 1   Entanglement                    212121 2211 2 221121 212121 21 1 2 1 ))(()())(( 2 1 1 2 1 2 1 1 2 1 1 2 1     zz zz yyxxzz i H  From this one can prove that this is a swap operator. Bell states )11|00(| 2 1 | )11|00(| 2 1 | )10|01(| 2 1 | )10|01(| 2 1 |         Singlet states and triplet states         || || || || 12 12 12 12 S S S S Entanglement measure and Entropy An arbitrary bipartite pure state is given by |)(| )log( ),(1 21 121 2 1    Tr TrE TrE von L        jiA m i n j ij ||| 1 1 The linear entropy or Von Neumann entropy can be used to measure the entanglement in the above pure state Pure State and Linear Entropy The linear entropy and the reduced density matrix can be written as .)(, ),(1 ** 21 AAAAAA AAAATrE T       .2/1)(1 )2/1,2/1( 11|00|2/1| 4    AtrE diagA Example, Bell states (maximally entangled states): Schmidt decomposition From SVD, an arbitrary pure state can be decomposed as '' 1 ' |||    kk r k k  ?10|01| 2 1 ).(log ,1 1 2 1 2        k r k kVon r k kL E E   Linear entropy and Von Neumann entropy: Mixed state 1, '''   i iii i i pp  A mixed state: If a mixed state is non-separable, then the state is entangled. Separable state 1|,|   i iii i i pp  Quantification of entanglement Entanglement of formation )()( BVonAVon EE   |)(|min)( }{| ii i iEpE i     Quantification of entanglement For two qubits, the entanglement of formation can be obtained from the concurrence , ).1(log)1(log)( , 2 11 22 2 xxxxxh C hE            The concurrence itself is a good measure of two-qubit entanglement. Quantification of entanglement Calculations of the concurrence Spin flipped density matrix ).(*)(~ yyyy   The concurrence is },,0max{ 4321  C where k are the square roots of the four real eigenvalues of ~ in decreasing order. Example 1: Two-qubit pure states   .1 10|01|2/1|   C .11|10|01|00||  dcba    2 1 )(|,||| i iy iC  is the complex conjugation operator. Bell states:  .||2 bcadC  is anti-unitary time-reversal operator. An arbitrary two-qubit state Example 2: the Werner state || 4 )1( 4   p I pw )1(2/1|| 12S  12 4 24 )1( S pIp w    }2/)13(,0max{  pC 0,3/1  Cp Example 2: the Werner state                                                         4 13 ,0max2 4 )1( ,0max2 24 )1( 2 ,0max2 4 )1( 4 )1( 2 24 )1( 24 )1( 24 )1( 12 4 p p p ppp C p pp pp pp S pIp w Example 3: Three-qubit states   .0 111|000|2/1|   C GHZ   .3/2 100|010|001|3/1|   C W Two classes of three-qubit states: Thermal entanglement Ze kTHT / / The system state at finite temperature is given by the Gibb’s density operator Z is the partition function. T is the temperature. k is the Boltzmann’s constant. As the density matrix represents a thermal state, the entanglement in this state is called thermal entanglement. The thermal entanglement is in general determined by both eigenvalues and eigenstates. Application to magnetic systems: A simple example   12211 2 1 SH    ]1,,1[ /])sinh()[cosh( / 12 12 x H T S ZS Ze         Thus, the thermal state is just the Werner state with p given by  ]sinh2)cosh(4/[)sinh(2 /)sinh(2     Zp The Werner state is naturally realized in the two-qubit isotropic Heisenberg model. In addition, the ground state is a singlet state, which is maximally entangled. Application to magnetic systems: A simple example                                                          ee ee ee eeee Z C Z ZS Ze HT 3,0max 3 1 3,0max 22 1 )cosh(2sinh4,0max sinh2cosh4 1 )cosh(sinh2,0max 2 )sinh()cosh( )cosh()sinh( )sinh()cosh( )sinh()cosh( 1 /])sinh()[cosh( / 12 The concurrence          3 3 ,0max 2 2   e e C The threshold value of T 3ln/2thT When the temperature is larger than the threshold value, the entanglement disappears. Partial transpose Definition of partial transpose  knAmlmnAkl T |||| 1 There is the following identity:      BABA T T         1 A conclusion ''' ii i ip   unit trace, Hermitian. So it can be considered as a legitimate density matrix A separable state ''*' 1 ii i i T p   '* i If is separable, all eigenvalues of 1 T are non-negative. In other words, if the partially transposed density matrix has negative eigenvalues, the state is entangled. For 2 by 2 and 2 by 3 systems, we can further claim that if the partially transposed density matrix has non- negative eigenvalues, the state is not entangled. Trace Norm, negativity, and logarithmic negativity AAtrA 1||||Trace Norm: Logarithmic negativity: Negativity: ||||log)( 22 T E   An arbitrary pure state   .2/1 10|01|2/1|   N Bell A pure state with Schmidt coefficients   .1 22|11|00|3/1|   N