1 Qubits, time and the equations of physics Salomon S. Mizrahi Departamento de Física, CCET, Universidade Federal de São Carlos Time and Matter October 04 – 08, 2010, Budva - Montenegro

2 V FEYNFEST-2011 FEYNMAN FESTIVAL – BRAZIL MAY 02-06, XII ICSSUR-2011 INTERNATIONAL CONFERENCE ON SQUEEZED STATES AND UNCERTAINTY RELATIONS MAY 02-06, 2011 WWW,ICSSUR2011.UFSCAR.BR

3 TIME: At 2060 The decay of the earth According to Newton

4 Quantum Mechanics Quantum computation, quantum criptography, search algorithms, plus Information and Communication Theories Quantum Information theory P. Benioff, R. Feynman, D. Deutsch, P. Schor Grover New vision QM! Would it be a kind of information theory?

5 What simple concepts of information theory can tell about the very nature of QM? Essentially, what can we learn about the more emblematic equations of QM: the Schrödinger and Dirac equations? Dirac Eq. is Lorentz covariant, spatial coordinate plus spin (intrinsic dof), S-O interaction Pauli-Schrödinger Eq. spatial coordinate plus spin (intrinsic dof) Schrödinger Eq. spatial coordinate

6 Digest Using the concept of sequence of actions I will try to show that this is a plausible perspective, I will use very simple formal tools. A single qubit is sufficient for nonrelativistic dynamics. Relativistic dynamics needs two qubits. The dynamics of the spatial degree of freedom is enslaved by the dynamical evolution of the IDOF.

7 1.Bits, qubits in Hilbert space, 2.Action and a discrete sequence of actions. 3.Uniformity of Time shows up 4.The reversible dynamical equation for a qubit. 5.Information is physical, introducing the qubit carrier, a massive particle freely moving, the Pauli-Schrödinger 6.The Dirac equation is represented by two qubits, 7.Summary and conclusions Dynamical equations for qubits and their carrier

8 Bits and maps

9 Bits, Hilbert space, action, map The formal tools to be used: [I,X] = 0

10 Single action operation, map U(  ) is unitary

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12 we construct an operator composed by n sequential events

13 THE LABEL OF THE KET IS THE LINEAR CLASSICAL MAP

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15 The sequence of events is reversible and norm conserving

16 Now we go from a single bit to a qubit We assume the coefficients real and on a circle of radius 1

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18 If one requires a sequence of actions to be reversible

19 one needs Thus necessarily Parametrizing as So, the i enters the theory due to the requirement of reversibility and normalization of the vectors

20 The n sequential action operators become And we call

21 Now, applying the composition law then necessarily, = n uniformity and linearity follow

22 Computing the difference between consecutive actions, we get the continuous limit and

23 As an evolved qubit state is given by it obeys the first order diferential equation for the evolution of a qubit

24 The generator of the motion can be generalized An arbitrary initial state is A qubit needs a carrier We recognize as a mean value

25 Instead of trying to guess what should be we write the kinetic energy of the free particle The solution (in coordinate rep.) is the superposition

26 If one sets T(p) = μ, a constant,the variable q becomes irrelevant Where, the carrier position becomes correlated with qubit state. The probabilities are

27 The relation Finaly leads to the Pauli-Schrödinger equation For an arbitrary generator for a particle under a generic field

28 In the absence of the field that probes the qubit, or spin, We have a decoupling

29 For the electron Dirac found the sound generator All the 4X4 matrices involved in Dirac´s theory of the electron can be written as two- qubit operato rs

30 the matrices have structure of qubits Dirac hamiltonian is

31 In the nonrelativistic case Z 1 is absent.

32 Going back the the usual representation, one gets the entangled state The solution to Dirac equation is a superposition of the nonrelativistic component plus a relativistic complement, known as (for λ=1) large and small components. However, they are entangled to an additional qubit that controls the balance between both components

33 Summary 1. One bit and an action U 2. An action depending on exclusive {0,1} parameters is reversible, 3. A sequence actions is reversible, keeping track of the history of the qubit state. 4. An inverse action with real parameters on a circle of radius 1 does not conserve the norm neither the reversibility of the vectors. 5. Reversibility is restored only if one extends the parameters to the field of complex numbers 6. Time emerges as a uniform parameter that tracks the sequence of actions and we derive a dynamical equation for the qubit. 7. A qubit needs a carrier, a particle of mass m, its presence in the qubit dynamical equation enters with its kinetic energy, leading to the Schrödinger equation 8. Dirac equation is properly characterized by two qubits.

34 Conclusion It seems plausible to see QM as a particular Information Theory where the spin is the fundamental qubit and the massive particle is its carrier whose dynamical evolution is enslaved by the spin dynamics. Both degrees of freedom use the same clock (a single parameter t describes their evolution) Thank you!

35 Irreversible open system The master equation has the solution Whose solution is At there is a fix point

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37 The generator of the motion can be generalized Whose eigenvalues and eigenvectors are For a generic state

38 choosing The probabilities for each qubit component are And is the mean energy:

39 For an initial superposition

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42 A qubit needs a carrier or, information is physical R. Landauer, Information is Physical, Physics Today, 44, (1991). Doing the generalization

43 Instead of trying to guess what should be we write the kinetic energy of the free particle