Quantum tomography


Quantum tomography or quantum state tomography is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states. The source of these states may be any device or system which prepares quantum states either consistently into quantum pure states or otherwise into general mixed states. To be able to uniquely identify the state, the measurements must be tomographically complete. That is, the measured operators must form an operator basis on the Hilbert space of the system, providing all the information about the state. Such a set of observations is sometimes called a quorum. The term tomography was first used in the quantum physics literature in a 1993 paper introducing experimental optical homodyne tomography.
In quantum process tomography on the other hand, known quantum states are used to probe a quantum process to find out how the process can be described. Similarly, quantum measurement tomography works to find out what measurement is being performed. Whereas, randomized benchmarking scalably obtains a figure of merit of the overlap between the error prone physical quantum process and its ideal counterpart.
The general principle behind quantum state tomography is that by repeatedly performing many different measurements on quantum systems described by identical density matrices, frequency counts can be used to infer probabilities, and these probabilities are combined with Born's rule to determine a density matrix which fits the best with the observations.
This can be easily understood by making a classical analogy. Consider a harmonic oscillator. The position and momentum of the oscillator at any given point can be measured and therefore the motion can be completely described by the phase space. This is shown in figure 1. By performing this measurement for a large number of identical oscillators we get a probability distribution in the phase space. This distribution can be normalized and the distribution must be non-negative. So we have retrieved a function which gives a description of the chance of finding the particle at a given point with a given momentum.
For quantum mechanical particles the same can be done. The only difference is that the Heisenberg's uncertainty principle mustn't be violated, meaning that we cannot measure the particle's momentum and position at the same time. The particle's momentum and its position are called quadratures in quantum related states. By measuring one of the quadratures of a large number of identical quantum states will give us a probability density corresponding to that particular quadrature. This is called the marginal distribution, or . In the following text we will see that this probability density is needed to characterize the particle's quantum state, which is the whole point of quantum tomography.

What quantum state tomography is used for

Quantum tomography is applied on a source of systems, to determine the quantum state of the output of that source. Unlike a measurement on a single system, which determines the system's current state after the measurement, quantum tomography works to determine the state prior to the measurements.
Quantum tomography can be used for characterizing optical signals, including measuring the signal gain and loss of optical devices, as well as in quantum computing and quantum information theory to reliably determine the actual states of the qubits. One can imagine a situation in which a person Bob prepares many identical objects in the same quantum states and then gives them to Alice to measure. Not confident with Bob's description of the state, Alice may wish to do quantum tomography to classify the state herself.

Methods of quantum state tomography

Linear inversion

Using Born's rule, one can derive the simplest form of quantum tomography. Generally, being in a pure state is not known in advance, and a state may be mixed. In this case, many different types of measurements will have to be performed, many times each. To fully reconstruct the density matrix for a mixed state in a finite-dimensional Hilbert space, the following technique may be used.
Born's rule states, where is a particular measurement outcome projector and is the density matrix of the system.
Given a histogram of observations for each measurement, one has an approximation
to for each.
Given linear operators and, define the inner product
where is representation of the operator as a column vector and a row vector such that is the inner product in of the two.
Define the matrix as
Here Ei is some fixed list of individual measurements, and A does all the measurements at once.
Then applying this to yields the probabilities:
Linear inversion corresponds to inverting this system using the observed relative frequencies to derive .
This system is not going to be square in general, as for each measurement being made there will generally be multiple measurement outcome projectors. For example, in a 2-D Hilbert space with 3 measurements, each measurement has 2 outcomes, each of which has a projector Ei, for 6 projectors, whereas the real dimension of the space of density matrices is /2=4, leaving to be 6 x 4. To solve the system, multiply on the left by :
Now solving for yields the pseudoinverse:
This works in general only if the measurement list Ei is tomographically complete. Otherwise, the matrix will not be invertible.

Continuous variables and quantum homodyne tomography

In infinite dimensional Hilbert spaces, e.g. in measurements of continuous variables such as position, the methodology is somewhat more complex. One notable example is in the tomography of light, known as optical homodyne tomography. Using balanced homodyne measurements, one can derive the Wigner function and a density matrix for the state of the light.
One approach involves measurements along different rotated directions in phase space. For each direction, one can find a probability distribution for the probability density of measurements in the direction of phase space yielding the value. Using an inverse Radon transformation on leads to the Wigner function,, which can be converted by an inverse Fourier transform into the density matrix for the state in any basis. A similar technique is often used in medical tomography.

Example: single-qubit state tomography

The density matrix of a single qubit can be expressed in terms of its Bloch vector and the Pauli vector :
The single-qubit state tomography can be performed by means of single-qubit Pauli measurements:
  1. First, create a list of three quantum circuits, with the first one measuring the qubit in the computational basis, the second one performing a Hadamard gate before measurement, and the third one performing the appropriate phase shift gate followed by a Hadamard gate before measurement ;
  2. Then, run these circuits, and the counts in the measurement results of the first circuit produces, the second circuit, and the third circuit ;
  3. Finally, if, then a measured Bloch vector is produced as, and the measured density matrix is ; If, it'll be necessary to renormalize the measured Bloch vector as before using it to calculate the measured density matrix.
This algorithm is the foundation for qubit tomography and is used in some quantum programming routines, like that of Qiskit.

Example: homodyne tomography.

Electromagnetic field amplitudes can be measured with high efficiency using photodetectors together with temporal mode selectivity. Balanced homodyne tomography is a reliable technique of reconstructing quantum states in the optical domain. This technique combines the advantages of the high efficiencies of photodiodes in measuring the intensity or photon number of light, together with measuring the quantum features of light by a clever set-up called the homodyne tomography detector.
Quantum homodyne tomography is understood by the following example.
A laser is directed onto a 50-50% beamsplitter, splitting the laser beam into two beams. One is used as a local oscillator and the other is used to generate photons with a particular quantum state. The generation of quantum states can be realized, e.g. by directing the laser beam through a frequency doubling crystal and then onto a parametric down-conversion crystal. This crystal generates two photons in a certain quantum state. One of the photons is used as a trigger signal used to trigger the readout event of the homodyne tomography detector. The other photon is directed into the homodyne tomography detector, in order to reconstruct its quantum state. Since the trigger and signal photons are entangled, it is important to realize that the optical mode of the signal state is created nonlocal only when the trigger photon impinges the photodetector and is actually measured. More simply said, it is only when the trigger photon is measured, that the signal photon can be measured by the homodyne detector.
Now consider the homodyne tomography detector as depicted in figure 4. The signal photon interferes with the local oscillator, when they are directed onto a 50-50% beamsplitter. Since the two beams originate from the same so called master laser, they have the same fixed phase relation. The local oscillator must be intense, compared to the signal so it provides a precise phase reference. The local oscillator is so intense, that we can treat it classically and neglect the quantum fluctuations.
The signal field is spatially and temporally controlled by the local oscillator, which has a controlled shape. Where the local oscillator is zero, the signal is rejected. Therefore, we have temporal-spatial mode selectivity of the signal.
The beamsplitter redirects the two beams to two photodetectors. The photodetectors generate an electric current proportional to the photon number. The two detector currents are subtracted and the resulting current is proportional to the electric field operator in the signal mode, depended on relative optical phase of signal and local oscillator.
Since the electric field amplitude of the local oscillator is much higher than that of the signal the intensity or fluctuations in the signal field can be seen. The homodyne tomography system functions as an amplifier. The system can be seen as an interferometer with such a high intensity reference beam that unbalancing the interference by a single photon in the signal is measurable. This amplification is well above the photodetectors noise floor.
The measurement is reproduced a large number of times. Then the phase difference between the signal and local oscillator is changed in order to 'scan' a different angle in the phase space. This can be seen from figure 4. The measurement is repeated again a large number of times and a marginal distribution is retrieved from the current difference. The marginal distribution can be transformed into the density matrix and/or the Wigner function. Since the density matrix and the Wigner function give information about the quantum state of the photon, we have reconstructed the quantum state of the photon.
The advantage of this balanced detection method is that this arrangement is insensitive to fluctuations in the intensity of the laser.
The quantum computations for retrieving the quadrature component from the current difference are performed as follows.
The photon number operator for the beams striking the photodetectors after the beamsplitter is given by:
where i is 1 and 2, for respectively beam one and two.
The mode operators of the field emerging the beamsplitters are given by:
The denotes the annihilation operator of the signal and alpha the complex amplitude of the local oscillator.
The number of photon difference is eventually proportional to the quadrature and given by:
Rewriting this with the relation:
Results in the following relation:
where we see clear relation between the photon number difference and the quadrature component. By keeping track of the sum current, one can recover information about the local oscillator's intensity, since this is usually an unknown quantity, but an important quantity for calculating the quadrature component.