#mixedstate

Quantum State Tomography (QST) is vital for identifying a system's unknown quantum state by measuring several identical clones. Since measuring a quantum particle changes or destroys its state, it is difficult to take several measurements on the same quantum system, unlike classical items that may be tested again. Thus, QST needs data from an ensemble of identically prepared quantum states to reconstruct a complete image.

Describe Quantum State Tomography.

QST aims to identify a quantum state's density matrix mathematical description. The density matrix captures all quantitative quantum system attributes in a “mixed state” (a statistical mixing of pure states) or “pure state” (all particles in the ensemble are identical). CT scans use multiple two-dimensional projections, or “slices,” to build a three-dimensional image.

QST also uses several “projections” or measurements to understand a quantum state. Tomographic estimates of the Wigner distribution were proposed in the late 1980s.

See also Quantum Entanglement Battery 2nd Law for Quantum States.

How QST Works

The Measurement Process

In the QST process, the same quantum state is prepared again and measured in various bases using “tomographic” or “projective” methods. Each measurement provides unique status information. Quantum State Tomography (QST) requires measurements in at least three bases for a single qubit (a two-level quantum system, such as photon polarisation). These measurements, often presented on a Poincaré or Bloch sphere, are used to derive state parameters. A single qubit's over-complete set usually has six projective measurements.

Besides polarisation, spatial modes carrying orbital angular momentum (OAM) can encode qubits, and spatial light modulators can describe their states on an OAM Bloch sphere. With more qubits, complexity increases rapidly. QST in multi-particle systems requires contemporaneous projective measurements on every particle. Characterising a two-qubit system requires 36 projective measurements. Tensor products of single-ququbit Pauli matrices and eigenstates are used in these investigations.

Reconstructing Density Matrix

The density matrix must be recreated when measurement data is collected. The “object” (the quantum state) is deduced from its “shadows” (the measurement results) in reverse.

Linear Inversion: Born's formula and observed probability solve a system of linear equations directly in the simplest method, linear inversion. A major downside is its capacity to construct non-physical density matrices with negative probability. Maximum Likelihood Estimation (MLE): This popular method avoids linear inversion by searching for the density matrix in the physically valid space (Hermitian, unit trace, non-negative eigenvalues) that best fits the experimental data. MLE can sometimes produce zero eigenvalues with 100% certainty after finite measurements, which may not always be warranted. Bayesian methods use previous information and “honest” estimations with error ranges to ensure that the reconstructed state is within physical limitations. See also Quantum Photon States: Polarisation, Spin, Entanglement

Issues and Limitations

The biggest challenge for Quantum State Tomography (QST) is system scalability. More qubits require more measurements and computational resources. Systems with more than a few qubits cannot achieve full QST due to the “curse of dimensionality”.

Experimental noise affects measurement results and must be considered. To solve these issues, researchers have developed tomography methods that need fewer measurements or simpler post-processing. Post-processing and local measurements are often used with matrix product states (MPS) for systems with specific correlation structures. Compressed sensing and permutationally invariant quantum tomography reduce measurement costs by assuming state qualities like low rank or symmetry.

Classical Implementations: Instructor and Research Tool

Bright classical light can reproduce and show QST, making it useful for research and education without the issues of single photons.

Based on Klyshko's “time reversal” concept, backprojection with scalar light uses a powerful laser source instead of a quantum detector. Although classical light tracks backward, it faithfully replicates quantum experiments, including entire QSTs. This aids quantum experiment coordination and prediction. QST with Classically Entangled Light vector beams: Quantum State Tomography (QST) using Classically Entangled Light (Vector Beams) uses mathematical similarities between classical and quantum states. Spatially variable polarisation vector beams are “classically entangled” because their spatial and polarisation degrees of freedom are not separable. QST tests can imitate numerous characteristics of quantum entanglement using conventional optical components. Despite accurately representing many quantum events, these classical systems cannot replace quantum experiments for applications like quantum key distribution, which depend on inherent quantum properties.

DIY Lab Implementation and Uses

DIY lab solutions like 3D-printed electromechanical roto-flip stages for automating polarisation optics make QST more accessible. Research and education benefit from faster and more reliable studies.

QST is crucial to modern quantum technologies. This tool is used to debug quantum circuits, characterise entanglement sources, validate quantum algorithms, and compare quantum devices. Concurrence, linear entropy, and reconstructed density matrix fidelity quantify state quality, purity, and entanglement.

Today I figured out I’m having a mixed episode... Simultaneous depression and hypomania.

I think the most common manifestation is when you want to get up and do things and you have a million plans but you can’t get up off the couch and you don’t have the energy to do any of them.  I have felt this way quite often.

For me today was a little more complex.  I was in a very ranty mood and I ranted about at least four topics to my classmates throughout the day on everything from how business casual is the absolute worst and I need to find ways to rebel, to how alcohol is totally fine in moderation and school campaigns shouldn’t teach facts that shame you for consuming even small amount.  (I was ranting loudly and a little disruptively.  One of my professors jokingly came over a couple times during lab and asked my group if I was still “throwing a tantrum.”)

On the alcohol note, there was a sign out for some alcohol abuse campaign the wellness center was putting on.  After reading it the second time and I went up to the volunteers and told them they need to take it down because it was very rape-victim shame-y.  I mean I’m progressive, but not usually outwardly.  I don’t usually make people change what they’re doing because that’s too confrontational.

I was both hyperactive and exhausted.  I’d keep yawning but tapping my feet.  I just wanted to take a nap.

I think the biggest one was that I kept feeling compelled to make jokes, be extra social, and stand up for important (*cough*) things, but then I continuously felt self-conscious, embarrassed, and ashamed every time I did so.   So I’d continuously switch from being very “out there” to wanting to smash my head against the wall.

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