Quantum Computation and Information Chap 8 Quantum Noise and Quantum Operations

1. Quantum Computation and Information Chap 8 Quantum Noise and Quantum Operations Dr. Charles Tappert The information presented here, although greatly condensed, comes almost entirely from the course textbook: Quantum Computation and Quantum Information by Nielsen & Chuang

2. Introduction • So far we dealt with closed quantum systems • But there are no perfectly closed systems • Real systems have unwanted interactions with the outside world that show up as noise • Noise is a central topic in this 3rd book part • The mathematical formalism of quantum operations is a key tool in describing the dynamics of open quantum systems

3. 8.1 Classical Noise and Markov Processes • Imagine a bit stored on a hard disk drive of an ordinary classical computer • The bit starts out in state 0 or 1 but in time stray magnetic fields can cause the bit to flip

4. 8.1 Classical Noise and Markov Processes • Suppose p0 and p1 are the initial probabilities of the bit being in states 0 and 1, respectively • Let q0 and q1 be the corresponding probabilities after noise has occurred • Let X be the initial state of the bit and Y the final state, then by the law of total probability

5. 8.1 Classical Noise and Markov Processes • Writing these equations out explicitly we have

6. 8.1 Classical Noise and Markov Processes • Now take a more complex example of noise • Our circuit consists of a single input bit X • To which are applied two consecutive NOT gates resulting in a stochastic process known as a Markov process • For a single stage process, the output probabilities are related to the input probabilities by • where E is an evolution matrix of transition probabilities

7. 8.1 Classical Noise and Markov Processes • and must be valid probability distributions • This is equivalent to two conditions on E • E is non-negative (positivity requirement) • Columns of E sum to 1 (completeness requirement) • Summary • There is a linear relationship between input and output probabilities described by a transition matrix • Classical noise processes are described by Markov processes provided the noises are independent

8. 8.2 Quantum OperationsBackground 2.4 Density Operator • We have formulated quantum mechanics using the language of state vectors • An alternative formulation is using the density operator or density matrix • Mathematically equivalent to using state vectors • But provides a more convenient language for some of the encountered scenarios

9. 8.2 Quantum Operations2.4.1 Ensembles of Quantum States • The density operator can describe systems whose state is not completely known • Suppose a quantum system is in one of a number of states with respective probs pi • An ensemble of pure states is called • The density operator is • All the postulates of quantum mechanics can be reformulated in the density operator language

10. 8.2 Quantum Operations2.4.2 General Properties of the Density Operator • This allows us to reformulate the postulates of quantum mechanics using the density operator

11. 8.2 Quantum Operations2.4.2 General Properties of the Density Operator • Postulate 1: state vector => ensemble

12. 8.2 Quantum Operations2.4.2 General Properties of the Density Operator • Postulate 2: evolution =>

13. 8.2 Quantum Operations2.4.2 General Properties of the Density Operator • Postulate 3: measurement =>

14. 8.2 Quantum Operations2.4.2 General Properties of the Density Operator • Postulate 4: composite physical system =>

15. 8.2 Quantum Operations2.4.2 General Properties of the Density Operator • Pure and mixed states • Given a density operator • with equality if and only if is a pure state

16. 8.2 Quantum Operations2.4.3 The Reduced Density Operator • The reduced density operator can describe subsystems of composite systems • Given systems A and B, the reduced density operator for system A is • partial trace over B • are any two vectors in the state space of A • are any two vectors in the state space of B • The reduced density operator is indispensable in the analysis of composite systems

17. 8.2 Quantum Operations2.4.3 The Reduced Density Operator • Example calculation: Bell state • Density operator • Trace out the second qubit from the reduced density operator of the first qubit

18. 8.2 Quantum Operations2.4.3 The Reduced Density Operator • Note that this state is a mixed state since • But the joint system of two qubits is a pure state known exactly • This strange property is a hallmark of quantum entanglement

19. 8.2 Quantum Operations2.4.3 The Reduced Density Operator Detailed calculations expand qubits factor out system 1, trace(outer product)=inner product