Quantum entanglement and Quantum state Tomography Zoltán Scherübl Nanophysics Seminar – Lecture BUTE

What is Quantum Tomography? Measuring a QM system: 1 measurement (eg. x): 1 physical parameter -> random 1 type of measurement on many copy of same system: |ψ (x)| 2 density function To reconstruct the ψ(x) wavefunction: more type of measurements needed Quorum: Complet set of measurable quantities, (operator basis in the Hilbert-space) Continuus variables: Wigner function (eg. light polarization) Discrete variables: density matrix (eg. qubit) Fidelity: probability of correctly identifying the states Quantum Process Tomography: Using different inputs (complete basis) -> QST on the output QM Black Box Input Output

Wigner function Classical system: Density function in phase-scape W(x,p) non negativ, normalized Marginal distribution: QM system: Heisenberg’s uncertainty principle: x and p cannot be measured at the same time -> neither the phase-space probability density But: X or P can be measured, so the marginal distributions too And always exists a quasi-probability density function (Wigner- function), which: Normalized It’s marginal distributions are exists (as above) But indefinit (not necessarily non negativ)

Entanglement, mixed state, density matrix QM system: Entire system: wavefunction Multiple subsystems: Entangled: Not entangled: Subsystem: density matrix Entangled entire system: mixed state There is not one, but more wavefunction: with p i probability Not entangled entire system: pure state Exist a wavefunction:

Properties of density matrix Hermitian Positive semidefinit real, ≥0 eigenvalues spectral representation: Normalized: Pure state Mixed state In two level syetem (CSB): Expectation values: If ρ 1 pure: If mixed:

Full system: Wavefunction: Density matrix: It cannot be written as a product -> Entangled state Subsystem: first spin: Up with ½, down with ½ probability Not the same as: An example: Spin singlet pair

Entanglement measures There’s no operator such as = the degree of entanglement But exists some quanitity, that can tell if the Qstate is entangled or not: General 2-qubit wavefunction: Not entangled state can be writen as a product: Then: Statemanet: if c 1 c 4 -c 2 c 3 ≠0, then the Qstate is entangled It can be generalized for bigger systems

Entanglement measures II - Von Neumann entropy As other entropies, it measures the lack of our knowlendge of the Qstate Where p i -s are the eigenvalues of ρ Pure state: S=0, because: ρ=| ψ ><ψ| (p 1 =1, p i≠1 =0) Subsystem: Maximal entropy: p i (1) =1/M, S 1 =log 2 M diagonal reduced density matrix -> maximally entangled state

Decoherence I Two interacting subsystem (system and environment): Together a closed system, well defined energy and phase The energy and phase of subsystem are timedependent/undefined do to the interaction Relaxiation: with energy transfer In Q system always followed by decoherence Decoherence/dephasing: without energy transfer Fluctuation of an external parameter (eg flux, magnetic field) (assumption: Guassian distribution) -> the phase of the system fluctuates in time -> time average -> decay in coherence -> loss of phase information The time average can be seen as ensamble average (eg. Slightly different N qubit, or spatial fluctuation of the parameter) Losing ability to interfere In density matrix picture: rapid vanishing of the off-diagonal elements -> just the classical occupation probabilities remain. The off diagonal elements are also called „coherence”

Decoherence II Let’s take N qubit, coupled to the same bath Each qubit gets a phase from the bath: The phase has a Gaussian distribution so it’s needed to average out to the phase: So the density matrix:

Decoherence III Time evolution: Unitary: a closed system always have unitary time evolution The state is always pure, so A subsystem: Consider a time evolution for the subsystem (not unitary): Where L[ρ 1 (t)] is the so called Lindblad decoherence term Mostly L[ρ 1 (t)]=-γρ ij, so it describes an exponential relaxation

Spin measurement: Projective operator measurement: But the output of the measurement can be 1 or 0 -> need to measure multiple times Other coefficients: Recipe: prepare the same state, measure σ x, σ y, σ z many times (3 type of measurement) -> calculate ρ ij -> you have ρ Basic idea of QST (1 QUBIT) Where, andis the Bloch-vector because Pure state Mixed state

QST is multiqubit system 4 N -1 real parameter N Qubit measurement: M qubit measurement: some σ ji =1 Notation: If N-qubit measurements are possible -> one qubit operations are enough Multiqubit measurement is not (hardly) realizable is solid-state systems If only single qubit measurements are possible -> one two qubit operation is required Theorem: Every M-qubit operation can be decomposed to the product of single qubit operations and one two operation.

One-qubit measurement Without loss of generality: ε lα, J lm αβ are positive real numbers Optimal case: every parameter is switchable σ z : σ y : σ x : Notation: In charge qubit system ε ly -s are always zero. In most real systems ε lz -s are not switchable By setting special J lm αβ we can get Heisenberg, XXZ, XY etc. Models Charge qubit: Fully controllable parameters

One qubit measurement – charge qubit r z : r y : Set Ф x =0 rotation around x axis POM r x : Set Ф x =Ф 0 /2,n g =0 R z (t=ħπ/8E C )=R z (-π/2) Set Ф x =0, n g =1/2 R x (t=ħπ/2E J (0))=R x (-π/2) Set Ф x =Ф 0 /2,n g =0 R z (t=3ħπ/8E C )=R z (-3π/2) POM

Two qubit measurement Basic two-qubit operation: time evolution: Assumption: N=2, ε 1α =ε 2α =ε α, J lm αβ =J lm α δ αβ Eg.: r zx in XY model (J mn x =J mn y, J mn z =0) Charge qubit: The interaction is switchable by the flux Ф i

Multi-qubit measurement Theorem: with one two qubit and all single qubit operation, every m-qubit operation can be performed For an m-qubit measurement at least m-1 2-qubit operation needed Eg.: r zzx in the XY model: Not necessary to do exactly these measurements, its enough to do at least 4 N -1 linearly independent measurment, so you have at least 4 N -1 equation for 4 N -1 variable. If there are more equation, than variable, solve by RMS method.

Rehearsal: Josephson junction, phase qubits Washboard potential

Measurement of entangled phase qubits I Anharmonic potential: different level spacings f 10 =5.1 GHz, ~30% tunability with bias current 1-qubit operations: rotation around z: current pulse on bias line rotation around x/y: microwave pulse the phase of the pulse defines the rotation axis the duration defines the rotation angle Measurement: strong current pulse: |1> tunnels out Two coupled qubit: At resonance: oscillation with S/h=10MHZ freq between |01> and i|10> 2-qubit operation Avioded crossing

Measurement of entangled phase qubits II |00> -> |01> Not eigenstate t free =25 ns: entangeled state: But: pulselength: 10, 4 ns -> not negligible: -> t free =16 ns 90 z rotation: eigenstate No oscillation (destruction of coherence?) -> 180 z pulse T 1 =130 ns T 2 *=80 ns

Measurement of entangled phase qubits III Single qubit fidelities: F 0 =0.95, F 1 =0.85 Fidelity for |ψ1> F=0.75 After correction with single qubit fidelities: F=0.87 Estimated maximal fidelity: F=0.89 Cause of fidelity loss: single qubit decoherence

References Y. V. Nazarov: Quantum Transport: Introduction to Nanoscience, Cambridge University Press, Yu-xi Liu et al. Europhys. Lett. 67 (6), pp (2004) Yu-xi Liu et al. PRB, 72, (2005) M. Steffen et al. Science, 313, 1423 (2006)