Transcription
Efficient storage and transmission of information, or in other words, compression of information, is one of the fundamental tasks in classical information theory. The father of this field was Claude Shannon, whose works addressed the concepts of information itself, the information content of the message, and the minimum requirements to represent such information.
In the last two decades, quantum computing knowledge has dramatically increased, and a new paradigm in information theory based on the principles of quantum theory has emerged. In this video, we will reveal some basic concepts of Shannon's classical information theory and explain the basic idea behind quantum information theory.
A classical message is formed by a set of letters which belong to an alphabet. These letters can be represented as short sequences of symbols, such as binary symbols. This is called source coding. Shannon's compression theorem states that the Shannon entropy is the fundamental limit for the compression rate of the message. That is, if one compresses at a rate above the Shannon entropy, then it is possible to recover the compressed data perfectly in the asymptotic limit, and otherwise, it is not possible to do so. Shannon's entropy represents the average amount of information per message. As an example, the message "hi Alice" cannot be compressed, but if we had a letter with a much higher probability, as in the second message, then we can reduce its length without reducing the transmitted information.
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Now, let us enter into the quantum world. Therefore, the letters will correspond to quantum states. If a one-qubit system is assumed, its letter will be defined by the following expression, where alpha and beta are complex coefficients. Then, we can have an alphabet of N quantum letters, such as this. As we are dealing with quantum states, such letters can be written in the form of a density matrix. Knowing that an analogous expression to classical Shannon entropy could be built with a quantum message, firstly, an average letter state can be created as the weighted sum of the density matrices of the message. From here, it can be defined the von Neumann entropy, which represents a measure of the amount of the statistical uncertainty in a quantum state, where the lambdas are the eigenvalues of the density matrix. This new magnitude, which can be interpreted as the quantum Shannon's entropy, is key to establish the limits of quantum data compression, which will be explained next.
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Without this theory, we can now go to the specific features of quantum data compression. In classical theory, we would use the Shannon entropy as an asymptotic limit to compress the data. In quantum theory, there exists the quantum noiseless coding theorem, which states that by coding the quantum message in blocks of K letters, K times the von Neumann entropy cubed is enough to encode each block in the asymptotic limit, K tending to infinity. The big difference between classical compression and quantum compression is the fact that classically, if all the letters in a message are equally likely, then no compression is possible according to Shannon's theory. However, if we are dealing with a quantum message, it can be compressed even if the letters are equally likely, as long as they are not orthonormal. This is due to the fact that von Neumann entropy only takes its largest value when the letter states are orthonormal, independently of their likelihood. This phenomenon supposes a big leap in the data compression rate of a message, so it opens up to filter exploitation in quantum computing.
In order to better understand the concept of quantum compression and the applied protocol to perform it, a two-qubit system compression will be described. The basis states of a two-qubit ensemble are |00⟩, |01⟩, |10⟩, |11⟩. Let us consider a two-level system with alpha the same for the two letters, and both alpha and beta real numbers. Then, the two-letter state will be... The usual way to proceed is to take the most important component of the basis with the biggest coefficients of the system and make a new basis. We have the ones with less work. This is better known as a measurement in the typical subspace. Let us consider in this case that β₁ and β₂ are much smaller than α, so that the three-dimensional subspace created by the three largest eigenvalues is spanned by |00⟩, |01⟩, and |10⟩. The rules to compress are: the elements in |00⟩, |01⟩, and |10⟩ will be left the same, and the elements in the state |11⟩ will be compressed as |00⟩. We have to translate every state into another state. We will have to apply reversible processes to be able to decode it later. To decode this system, we will just let the system as it is. Notice that we have lost a bit of information because the states |11⟩ cannot be retrieved, but during this process, we have transformed a two-qubit system into a 1.58-qubit system. The laws of information will have to be kept in check using the fidelity of the compression-decompression system.
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Where PA and PÂ are the probabilities of the two-qubit system to lie in the typical subspace and its complementary, respectively. Putting numbers in this example, if we choose α² to be 0.9, then β² is 0.1, and the fidelity is 0.99. If we compute the von Neumann entropy of the compressed state, it gives 0.79 bits per letter state. Notice that here we have assumed that the two letters were equally likely, and still, a compression with a very high fidelity is achieved. Since the two letter states are not orthonormal, fulfilling the Schumacher quantum noiseless theorem, this indeed is a remark that must be done on the compressibility of quantum systems.
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Another method for compressing a system is the quantum sure whale transport. This method can compress a quantum system of 2 to the N dimensions to a system of just N+1 dimensions. This means it performs a logarithmic compression. To perform this transform, the states have to be part of an ensemble of pure identical qubits. It is really useful to obtain the parameters α and β from the key without coming in a process with great precision. This process must be performed multiple times to get a statistic with a mean and a standard deviation. This way, we can make an estimation of the values of α and β. The quantum sure whale transform reduces the amount of qubits needed to make this provision as accurate as 1:1. It uses the fact that an N-qubit system has, as I mentioned, 2 to the N, but the relevant information is stored in just N+1, given the symmetries of the system. The other dimensions are just permutations of the qubits, which is irrelevant in the case of identical qubits. The earlier relevant information is the one from the angular momentum of the multi-qubit system. Continuing with the example, the method to implement this transform is this circuit. Then, if we enter three identical qubits of this form, the final state will be... We can see that the third qubit will be redundant and won't bring any information. Therefore, the information is encoded in the first two qubits. Hence, the compressed system contains the same information as the initial system.
In conclusion, quantum data compression can bring in the future many applications that exceed the limits of classical compression by using quantum properties such as superposition and entanglement. All this potential can be unlocked in the near future, but more developments will have to be made in the theoretical and, mostly, in the experimental setups to be able to make it a reality.
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