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Characterization of Two-qubit Operators R. Sankaranarayanan Department of Physics National Institute of Technology Tiruchirappalli – 620015 (www.nitt.edu) [email protected] 1 OUTLINE • Introduction Model for computation Some perspectives Quantum States Quantum Algorithms Single qubit and Two-qubit operators • Characterization Tools Geometry Local Invariants Entangling Power Operator Entanglement • Conclusion 2 INTRODUCTION 3 Model for Computation Fundamental unit: 0 and 1 – bit (classical) Single Input gate: NOT (reversible) Two Input gates: OR, AND, XOR, NOR, NAND (irreversible) General Logic Gate f : 0, 1 0, 1 - sequence of n input bits converted to another sequence of m output bits n NAND m Universal gate 4 Some Perspectives Logical gates are intrinsically irreversible (given the output of the gate, input is not uniquely determined) eg. Output of NAND gate is 1 for the inputs 00,01,10. - information input to the gate is lost irretrievably when the gate operates (information is erased) Moore’s law: computer power will double for constant cost roughly once very two years. (true for last 4 decades!) - quantum effects will begin to interfere in the functioning of electronic devices as they are made smaller and smaller! 5 Quantum States P1. Associated to any ‘isolated’ physical system is a complex vector space with inner product (Hilbert Space) known as state space. The system is completely described by a state vector in the state space. Simplest quantum mechanical system has two-dimensional state space. a 0 b 1 ; a 2 b 2 1 - Two-level system P2. Evolution of ‘closed’ quantum system is described by a unitary transformation. U ; U U I ; U 1 (reversible) If the state space is n-dimensional, U is n n unitary matrix. 6 P2’. Time evolution of the state of a closed quantum system is described by the Schrodinger equation d i Hˆ dt ; Ĥ – Hamiltonian operator P3. Quantum measurements are described by a collection of measurement operators, M i i i ; i 1,2,n - n dimensional Hilbert space For two-level system, M 0 0 0 ; M1 1 1 M 0 M 0 a such that 2 ; M1 M1 b 2 7 State space of two-qubit (composite) system H H1 H 2 Four dimensional Hilbert space spanned by the linearly independent vectors: 0 0 , 0 1 , 1 0 , 1 1 General state a 00 b 01 c 10 d 11 with the normalization condition 2 2 2 2 1 or a b c d 1 . State space of n-qubit (composite) system H H1 H 2 H 3 H n - 2 n Hilbert space 8 State of a two-qubit (composite) system a 00 b 01 c 10 d 11 H H1 H 2 Let 1 0 1 H1 and 2 0 1 H 2 . If the state 1 2 - product (or) untangled state condition: ad bc 0 If the state 1 2 - entangled state condition: ad bc 0 9 Product (unentangled) states (i) 1 00 10 1 0 1 0 2 2 (ii) 1 00 i 01 10 i 11 2 1 0 1 1 0 i 1 2 2 Entangled states (i) 1 00 i 01 10 i 11 2 (ii) 1 00 11 2 ; ad bc ; ad bc i 2 1 2 10 For state of a two-qubit system: a 00 b 01 c 10 d 11 Concurrence, C ( ) 2 ad bc Range: 0 C ( ) 1 For product state: C ( ) 0 For maximally entangled state: C ( ) 1 eg. Bell states Superposition and Entanglement - New resource for computation 11 Superdense coding 2 bits Problem: A B 00,01,10,11 1 00 11 2 1. each qubit of a two-qubit state shared by A & B 2. A applies I , X , Z , Y gates for 00,01,10,11 respectively 3. A sends her qubit to B 4. B applies CNOT followed by H gate to the 1st qubit - two-qubit state 00 , 01 , 10 , 11 5. B makes measurement to retrieve 00,01,10,11 Transfer of one qubit = transferring two bits C.H. Bennett and S.J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992) 12 Quantum search Problem: Search an item in a random collection of N items. eg. Search a name corresponds to a phone number in telephone directory Best case – 1 search, Worst case – N search On average: 1 1 2 N ~ N N 2 search Grover’s quantum logic – average no. of search ~ N - quadratic improvement in random search L.K. Grover, Phys. Rev. Lett. 79, 325 (1997) 13 Prime factorization Problem: factoring n-bit integer into prime factors N p q ~ 2 n ; where p,q – primes eg. 137703491 7919 17389 best classical algorithm – number field sieve no. of operations – Oexp n1/ 3 log 2 / 3 n eg. 130 digit number – super computer (1012 flops) - 42 days to factorize 400 digit number – 1010 years to factorize ! Quantum logic – no. of operations - On 2 log n log log n - factorization is exponentially faster 400 digit number – 3 years to factorize ! 14 P.W. Shor, SIAM J. Comp. 26 (5), 1484 (1997) Single qubit gates Acts on single qubit system – SU(2) Group Pauli-X Pauli-Y Pauli-Z X Y Z b 0 1 a a 1 0 b b 0 i a i a i 0 b a 1 0 a b 0 1 b 0 1 0 i1 0 1 0 1 i 0 1 1 0 15 Hadamard H S 1 a b 1 1 1 a 2 a b 2 1 1 b 0 1 1 2 1 2 0 0 Phase 1 1 a 1 0 a ib 0 i b 0 0 1 i1 Pi/8 T a 1 0 i i e 4 b 0 e 4 0 1 a b 0 i e4 1 16 Two-qubit gates Acts on two-qubit system – SU(4) Group SU (4) SU (2) SU (2) - Local gates – do not produce entanglement If ( A B) 1 2 A 1 B 2 1' 2' SU (4) SU (2) SU (2) - Nonlocal gates – produce entanglement U 1 2 1' 2' Local gates Nonlocal gates Perfect Entanglers (Bell state for some input product state) 17 CNOT SWAP 00 00 ; 01 01 00 00 ; 01 10 10 11 ; 11 10 10 01 ; 11 11 U CN 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 U SW AP 0 0 0 0 1 0 0 1 0 0 0 0 0 1 18 CHARACTERIZING TOOLS 19 Geometry An arbitrary two-qubit gate can be written as - Pauli matrices, Geometrical structure of 3-Torus with period - real numbers : . 20 Local Invariants Two operators are called locally equivalent if they differ only by local operations: Bell basis: Transformation of from standard basis to the basis: Local gates are real orthogonal matrices in the Bell basis where 21 Theorem: The complete set of local invariants of a two-qubit gate with is given by the set of eigenvalues of the matrix . The spectrum of the matrix is completely described by a complex number and a real number . Local invariant of two-qubits are defined as: Y. Makhlin, Quantum Info. Process 1, 243 (2003) 22 Eigenvalues of : Local invariants of a two-qubit gate: Properties: 1. For every 2. For every 3. For every , and , and , , Local invariants are unaffected by local operations. Zhang J. et. al. Phys. Rev. A 67, 042313 (2003) 23 Symmetry in the base L = CNOT A3 = SWAP Tetrahedron (Weyl Chamber): Perfect entanglers in the Polyhedron: (a) and (b) Zhang J. et. al., Phys. Rev. A 67, 042313 (2003) 24 Local invariants of Weyl Chamber edges O = Local gate L = CNOT A3 = SWAP α OA3 = SWAP A1 A3= SWAP -α 26 Local invariants of Polyhedron edges Entangling Power Average entanglement generated by the action of a two-qubit gate on all product states : where is the linear entropy of the state. Properties: 1. For every , 2. For every , 3. For every , Entangling power is unaffected by local operations. Zanardi P. et. al, Phys. Rev. A 62, 030301 (2000) 28 Entangling power of Weyl Chamber edges Entangling power of Polyhedron edges Entangling power and local invariants where . Since If , we have , . . This is the case for the edges QP, MN and PN. Perfect entanglers are such that, (c) and (d) Local invariants classify two-qubit gates as Perfect entanglers and Non-perfect entanglers. 31 32 Operator Entanglement Nonlocal part of U can also be written as where Defining are complex functions. as Schmidt coefficients, we have . Measures of operator entanglement: (i) Schmidt strength, (ii) Linear Entropy, Nielson M.A. et. al., Phys. Rev. A 67, 052301 (2003) 33 Properties: 1. For every , , 2. For every , , 3. For every , , Schmidt strength and Linear entropy are unaffected by local operations. 34 Schmidt coefficients of Weyl Chamber edges Schmidt coefficients of Polyhedron edges Schmidt coefficients of Polyhedron edges (continued) Linear Entropy and Schmidt Strength are not equivalent measures 38 Linear Entropy and Local Invariants Linear entropy of an operator U can also be written as We can show that , the maximum value only for the gates on the edge A2A3 . Further, we have 39 O = Local gate L = CNOT A3 = SWAP α OA3 = SWAP A1 A3= SWAP-α A2 A3 – maximally entangled gates 40 Linear entropy of Weyl Chamber edges Linear entropy of Polyhedron edges Local invariants of Polyhedron edges Conclusion • • • • All the geometrical edges of two-qubit gates are one-parametric. SWAP α and its inverse form two edges of the Weyl chamber. Relation between entangling power and local invariant. Local invariants classify gates as perfect and nonperfect entanglers. • Two measures of operator entanglement, (i) linear entropy and (ii) Schmidt strength are not equivalent. • Linear entropy of an operator in terms of geometrical points and local invariants. 44 References • Balakrishnan, S. and R. Sankaranarayanan, Entangling characterization of (SWAP)1/m and controlled unitary gates, Phys. Rev. A 78, 052305 (2008). • Balakrishnan, S. and R. Sankaranarayanan, Characterizing the geometrical edges of nonlocal two-qubit gates, Phys. Rev. A 79, 052339 (2009). • Balakrishnan, S. and R. Sankaranarayanan, Entangling power and local invariants of two-qubit gates, Phys. Rev. A 82, 034301 (2010). • Balakrishnan, S. and R. Sankaranarayanan, Operator-Schmidt decomposition and the geometrical edges of two-qubit gates, Quantum Inf. Process. 10(4), 449 – 461 (2011). • Balakrishnan, S. and R. Sankaranarayanan, Measures of operator entanglement of two-qubit gates, Phys. Rev. A 83, 062320 (2011). Thank you for your attention! 45