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The Wonderful Journey of Quantum Measurement: How do we obtain quantum information from measurements?
In the world of quantum physics, quantum measurement opens a door to an incredibly mysterious realm. Reconstruction of quantum states - This process is called quantum state tomography, and it allows us to obtain quantum information in seemingly unobservable situations. This approach allows us to understand what quantum states are and how to piece together a complete picture of these states from multiple measurements.
The core of quantum state imaging is to infer relevant information about the same quantum state by repeating multiple measurements.
The process of quantum state imaging starts with the same quantum state prepared by a device or system. The states used can be not only pure quantum states, but also mixed states. To be able to uniquely identify quantum states, these measurements must be imaging complete. This means that the operators of the measurement need to form an operator basis in the Hilbert space of the system in order to provide full information about the state.
The complexity of the quantum measurement process is that any measurement behavior will change the quantum state. Therefore, the purpose of quantum state imaging is to determine the state of the system before making measurements. This is very different from traditional physical measurements. In quantum mechanics, measurement focuses on the two core properties of position and momentum. According to Heisenberg's uncertainty principle, these two cannot be obtained simultaneously, so we must extract information from a probability distribution.
Using multiple measurements and frequency statistics, we can infer the probability of a particle's state.
An important application of quantum state imaging is quantum computing and quantum information theory. In this context, we can imagine a scenario: a person named Bob prepares many identical particles or fields and gives them to Alice for measurement in the same quantum state. If Alice is not sure about the state described by Bob, she may choose to perform quantum state imaging in order to make her own classification of these states.
Methods of quantum state imaging
Linear inversion
The simplest form of quantum state imaging can be derived using Born's rule. Often, however, it is not known at the outset whether a quantum state is pure, but may be a mixed state. This in turn creates the need for many different types of measurements to fully reconstruct the density matrix in a finite-dimensional Hilbert space. This means that the technology developed must have a certain degree of flexibility to ensure that we can obtain accurate quantum information.
One example comes from state imaging of a single quantum bit (qubit). In this example, the density matrix of qubits can be represented by Bloch vectors and Pauli vectors. Using this method allows us to efficiently reconstruct quantum states, and due to the diversity of measurements and the comprehensive nature of the data, this process can often provide us with insights from different perspectives.
Measurement with continuous variables
In infinite-dimensional Hilbert spaces, for example in the case of measuring continuous variables such as position, the method is more complex. Optical isotropic measurements are a prominent example of this. Through balanced co-signal measurements, we can derive the Wigner function and density matrix to depict the quantum state of light.
Through measurements in different rotation directions, we can establish a new probability density distribution for quantum systems.
The key to this process is whether we can accurately measure the probability distribution in the direction θ and complete the inverse Radon transformation, thereby deriving the concentration function of the quantum state. Such technology has been widely used in medical imaging, and has gradually shown its potential in quantum physics research.
Ultimately, quantum measurement is not only a means in science and technology, it is also a window for us to understand the quantum world. Every measurement captures the moment of quantum execution and provides an opportunity for further understanding. When exploring this unknown time domain, we can’t help but wonder: Through the measurement of quantum states, can we gain a deeper understanding of the truth of space and time?
