Quantum State tomography and MLE Roy Philip George K A, Ravikant, V. Narayanan, and Subhashish Banerjee

Research Excellence

Quantum bits or qubits play a vital role in Quantum computation and to be able to manufacture qubits at will is a prerequisite for any quantum lab. Qubits differ from normal bits in the fact that instead of being limited to either 1 or 0, it exists in superposition and can be entangled. Our short term goal was to manufacture and characterize two qubit Bell states using spontaneous parametric downconversion (SPDC) which would act as the groundwork for future quantum experiments. Bell states are four maximally entangled 2 qubit states, all of which have a maximal value of . These are 2 qubit states which are in an equal superposition that takes one of the following four forms

To confirm that we are indeed producing the Bell state or any other entangled state we choose, some means of characterizing and also representing the produced state needs to be adopted and quantum state tomography is just that.

The concept of the quantum state is key to understanding quantum tomography. Any state, Quantum or Classical, is simply the set of all the things that we know about a particular system. A state can contain any number of ‘facts’ about the system that may be of use. These facts can be used to describe the current configuration or situation of the system under examination. For example, to describe the state of a ball we can specify its position with respect to some origin, its momentum or velocity if it is in motion, its size, colour, brand or any other property or attribute that may help to set our ball apart from everything else. This is the classical example of a state.

A quantum state is analogous to the previous example but there are some very stark differences. Until measured, a quantum system is assumed to not be in any single state. In other words, a quantum system is thought of as existing in all possible states simultaneously. This property is referred to as superposition. This means that Quantum mechanics deals in probabilities instead of actual values.

Measurement returns a single value but disturbs the quantum system in a way that it is no longer in superposition. In other words a measurement will give a classical state from a quantum state by taking away the superposition.

It is therefore essential to discuss the methods of state representation. There are two types of states: Pure states and mixed states. If the state in question can be described by a state vector it is considered pure otherwise it is a mixed state. A density matrix can be used to describe both mixed and pure states. There are certain rules that a density matrix must adhere to in order to remain a physical and realistic representation of the state of quantum system. An important property worth noting is that the trace of the square of the density matrix must not exceed one. Most quantum systems exist in mixed states and as a result the concept of the density matrix is essential to understand what state has been produced at the end of an experiment. So now the focus is on how to extract information about a state from an experiment and form its density matrix. But there is a problem here; once a quantum state is measured, it loses its superposition and collapses to a classical state. This means that if we attempt to find out what the state of a quantum system is, it will cease to be a quantum system. This would imply that it is impossible to use any quantum resource if measuring it will destroy the property that made it so appealing in the first place. So why not simply create the same state n- number of times? The no-cloning theorem forbids specifically this. According to this postulate of quantum mechanics, it is impossible to create an exactly identical copy of an arbitrary unknown quantum state. Things might seem hopeless at this point but there is a way out and it is called Quantum state Tomography. To get around the no cloning theorem, it is possible to manufacture a large number of identically prepared states and conduct a series of projective measurements on them and from the results of these measurements arrive at a reasonably accurate density matrix. The process of reconstructing the density matrix of an arbitrary unknown state through above mentioned process is referred to as quantum state tomography. For photon polarization based two qubit state characterization, projective measurements are essential for the reconstruction.

Projective measurements in experiments related to SPDC utilize the Stoke’s parameters to selectively project the unknown state onto a set of known states of polarization, one by one. Here the measurement parameter is the photon count that is registered by the detector after the projective apparatus is set to transmit only one particular state. To project the unknown state onto a polarization state of the experimenter’s choosing, three optical elements are placed in front of each detector in the following order: a quarter wave plate, a half wave plate and a linear polarizer or a polarizing beam splitter. The angles of the fast axes of both wave plates can be set arbitrarily which means that the unknown state can be projected onto any desired state by setting the relative angles of the fast axes of the wave plates.

Figure 1: The kind of setup used for tomography of the state obtained from SPDC The figure shows the kind of setup used for tomography of the state obtained from SPDC. However this is the case for 2 qubit tomography.

The qubit is the basic unit of quantum information and computing. It is analogous to the classical bit which can either be a zero or a one and is the basis of current computational technology. The qubit differs in the fact that it is more of a superposition of the two states |0> and |1>. With superposition, we can encode an exponential amount of information that can scale a solution better than classical computing. Such qubits are represented by an ideal two-state quantum system.

In the specific case of polarization qubit systems, the measurable parameter will be the coincidence counts .The coincidence counter registers a count only when both detectors click within a very small time window. Light from the source will fall on the detector past the Stoke’s parameter arrangement only when there is a component of the projector state in the unknown state. The unknown state will have to be projected onto 4n different known states, where n is the number of qubits in the unknown state.

The goal of tomography is reconstruct the density matrix of the unknown quantum state from the set of coincidence counts, . For this we will need to introduce a family of matrices which are in some way connected to the projector states. These are termed the

matrices. These matrices are required to possess certain properties. They must form an orthonormal basis and more importantly, they should be able to express any matrix in terms of a product between itself and the matrix. Mathematically this will look like where is any matrix of the same dimensions as . Conveniently, these matrices can be derived from the Pauli spin matrices. The second property allows for the density matrix to be expressed in terms of the matrices. Cutting to the end, it can be shown that the density matrix can be obtained from the coincidence counts through an expression that involves a family of matrices derived from the matrices through some rigorous mathematical manipulation as

A sample set of coincidence count values lifted from literature can be used to verify the competence of the Mathematica code written. For the coincidence data set

We were able to the correct density matrix.

This meant that when we eventually acquire coincidence data from our own experiment, we could obtain the tomographic density matrix at the click of a button with the full knowledge that it was accurate.

However, the results obtained through tomographic measurement techniques often have certain problems. Most of the time, the results obtained violate basic but very important principles like positivity. The density matrices obtained using this technique are often unphysical. The culprits here are the statistical fluctuations of the coincidence counts as well as experimental inaccuracy. What this means is that the counts collected from an experiment may not be as accurate as we imagined. There are several factors that can cause a coincidence count irregularity such as spurious counts from the detector, stray background light and other factors we have no control over. However this does not mean that quantum tomography is useless. The Technique of Maximum Likelihood Estimation can be employed in this scenario to iron out the kinks. MLE is a constrained optimization technique where the space of all allowed, physical density matrices is searched using the optimization algorithm for the one with the highest probability to result in the measured counts.

MLE is a three step process:

1. Generate a formula for an explicitly physical density matrix. This matrix should inherently possess all the necessary properties of a proper density matrix such as being Hermetian, normalized and positive semi-definite. This matrix should be a function of 16 real variables { t1, t2, t3….. t16}, called the t-parameters and the matrix itself shall be represented by . 2. Introduce a likelihood function which quantifies how good the density matrix is in relation to the experimental data obtained. 3. Use some standard optimization technique to find the optimal set of t-parameter values for which the likelihood function is maximized.

Once the optimal t-parameter values are obtained, these values can be substituted into the expression for and the resulting matrix will be the best estimate for the density matrix derived from the experimental data. This matrix can be represented by . We performed MLE on the density matrix obtained earlier and ended up with a perfectly physical density matrix

To be able to characterize the state being generated in any quantum optics experiment is paramount and at the moment we are fully equipped to tackle 2 qubit characterization and have the knowhow to extend these techniques to characterize states with any number of qubits.

References

[1] H. Singh, Arvind, and K. Dorai, “Constructing valid density matrices on an NMR quantum information processor via maximum likelihood estimation,” Physics Letters A, vol. 380, no. 38, pp. 3051–3056, Sep. 2016, doi: 10.1016/j.physleta.2016.07.046.

[2] Z. Hradil, “Quantum-state estimation,” Phys. Rev. A, vol. 55, no. 3, pp. R1561–R1564, Mar. 1997, doi: 10.1103/PhysRevA.55.R1561.

[3] A. C. Keith, C. H. Baldwin, S. Glancy, and E. Knill, “Joint quantum-state and measurement tomography with incomplete measurements,” Phys. Rev. A, vol. 98, no. 4, p. 042318, Oct. 2018, doi: 10.1103/PhysRevA.98.042318.

[4] J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, “Photonic State Tomography,” in Advances In Atomic, Molecular, and Optical Physics, vol. 52, Elsevier, 2005, pp. 105–159 [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/S1049250X05520032.

[5] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A, vol. 64, no. 5, p. 052312, Oct. 2001, doi: 10.1103/PhysRevA.64.052312.

Roy Philip George K A

Project Student, Department of Physics

Ravikant

Project Student, Department of Physics

V. Narayanan

Assistant Professor, Department of Physics

Email: [email protected]

Subhashish Banerjee

Associate Professor, Department of Physics

Email; [email protected]