The Concept of Entropy: Quantum Communication Advanced Quantum Mechanics: Final Report
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The Concept of Entropy: Quantum Communication Advanced Quantum Mechanics: Final Report Medha Goyal PHYS 243 at the University of Chicago 1 Introduction: What is Information Theory? Information theory is the study of how to quantify, store, and communicate information [1]. It is concerned with ideas of transmitting data compactly, efficiently, and without error. In the modern world, where we are all reliant on the internet for information, where increasing amounts of data has to be stored without issue, and where we want to know we can transmit information reliably, the basic ideas of information theory should resonate with us deeply. The founding of information theory as a discipline is largely attributed to Claude Shan- non’s seminal paper from 1948: "A Mathematical Theory of Communication" [2], in which he proposed several ideas we take for granted today. The ideas he put forth in his paper have made a significant impact in fields as diverse as statistics, computer science, electri- cal engineering, cryptography, and even linguistics! The conception of the internet is often traced back to this paper. Among the ideas he put forth are a conception of error-free com- munication, of maximizing the "information content" per message sent, and even of naming a unit of information a "bit". While a lot of these ideas may be familiar to us, there is one concept Shannon introduced that has seeped into the language of the lay person, but which is thrown around carelessly and often misused, so much so that it has been used taken completely out of context to describe the depiction of chaos in art by academics in the field of art history! That concept is ’entropy’. Physicists should find this word familiar, as in thermodynamics it relates to the progression of time through irreversible processes[3], and in statistical mechanics it describes a statistical uncertainty in the state of a physical symptoms based on the number of states it can be in and their relative probabilities. While neither of these two definitions are exactly how entropy is defined in the context of information theory, the second of these two definitions comes very close. This paper starts with the basic ideas of discrete and quantum data transmission, and touch on very simple ideas of quantum error correction and redundancy, all of which to build up to an exploration of entropy in both classical and quantum contexts. 2 Long Distance Communication When Shouting Just Won’t Work 2.1 The Process of Data Transmission The process of data transmission starts with a sender, who picks a message mi from a set of messages M, and transmits a signal corresponding to this message through a communica- tion channel [4]. The original message is considered discrete, but since physical signals are continuous in the real world, the message to be transmitted must first be converted into an 2 Medha Goyal analogue representation. The process of turning the message into a vector of real numbers is called encoding and the process of then choosing the corresponding analogue wave-form is called modulation. The message reaches a receiver, who then decides on what the message must have been by observing the output of the channel and finding the message amidst (a) deterministic distortions and (b) random noise. The process of deciding upon the received message, by minimizing the probability of an error, is called detection. Data Encoding and Modulation: Each message m is turned into a symbol by a vector encoder, represented by a real vector x. Each possible message maps to a vector x with a different value. A modulator then converts each x into a continuous waveform xi(t). The channel then distorts the continuous waveform into yi(t). Data Detection: A demodulator converts yi(t) into a vector y (analogous to x). The vector is then decoded to get the message ˆmi. Hence the probability of error is defined as Pe ≡ P (mi 6= ˆmi). Code: We use vector xk to denote a vector sent at time t = k. A code C is a set of one or more indexed sequences, or codewords xk which are formed by concatenating symbols from the output of the encoder. Each codeword in the code has a one to one mapping with encoder-input messages. 2.2 Interlude: When Things Go Wrong in Quantum Data Transmission Errors due to noisy transmission is not limited to classical information transmission1.A budding subfield of quantum information theory is quantum error correction, which in- volves building circuits to correct errors that occur during transmission. One of the most insurmountable engineering challenges of building a quantum computer is the fact that quantum information will interact with its environment, leading to decoherence (a loss of information). This is one source by which error is introduced to the quantum message. A quantum error correcting code (QECC) aims to recover the original message by map- ping k qubits into n qubits (a map from Hilbert space of dimension 2k to one of dimension 2n) where n > k)[5]. The k qubits represent the message we want to encode, so we add redundancy in the form of the remaining n-k qubits to minimize the chance that the errors are made on the k important bits. In addition to classical errors, called bit flip errors, where j0i $ j1i, there are also phase errors in the quantum case, such as j0i $ j0i, j1i $ − j1i. However, quantum errors are continuous, so really a bit flip or phase shift could be by any number of intermediate angles between 0 and 360 degrees. This makes the task of quantum error correction non-trivial, and there are many codes out there to correct quantum errors. It would be beyond the scope of this paper to go through all possible QECCs, but to illustrate this particular application of quantum entropy, we will calculate the entropy of one of the simplest QECCs: the 3-qubit bit-flip code[6]. This code only considers qubit flips, and not the phase changes, so it is not a full quantum code, but it is sufficient for our purposes. We start with the two basis states j000i and j111i. We can map any arbitrary single qubit state α j0i + β j1i to α j000i + β j111i using the quantum circuit shown in Fig.2.2: If jΨi = α j0i, then the input state is α j000i, and the application of the CNOT gates means that nothing would happen to the second and third qubit (they would stay 0). If the 1 The process just described relates to sending classical information. Quantum data transmission has been done as well, but the process is less standardized (they differ more significantly between experiments), so we will skip a discussion of equivalent quantum methods of data transmission. The Concept of Entropy: Quantum Communication 3 Fig. 1. 3 Qubit Bit-Flip Circuit: Mapping α j0i + β j1i to α j000i + β j111i [6] input jΨi = β j1i, then the input state is β j100i but the CNOT gates would flip both of the second two qubits and the output state is β j111i. To correct errors using this code, we would add two ancilla qubits that extract in- formation about possible errors. The circuit shown in Fig.2.2 includes both the encoding component as shown in Fig.2.2, and a correction component. For the sake of simplicity, we consider no errors to occur during encoding (those CNOT gates are sound), and only between the encoding and correction steps. Fig. 2. 3 Qubit Bit-Flip Circuit with Ancilla Qubits for Measurement and Error Correction [6] 4 Medha Goyal The first ancilla is connected by a CNOT gate to the input j i. If jΨi = α j0i, then the ancilla stays j0i. If jΨi = β j1i, then ancilla flips to from j0i to j1i. Next a CNOT gate connects the same ancilla to the second qubit of the original code. If the original jΨi = α j0i, and there was no error, then the second qubit would still be α j0i, and the first ancilla would stay j0i. If there was an error, the second qubit would be β j1i and the ancilla would flip to j1i. It is then measured at the end of the circuit. A similar process occurs if the original jΨi = 0, and with the second ancilla. The results can be summarized in this table. Error Location Final State: jdatai jancillai No Error α j000i j00i + β j111i j00i Qubit 1 Flip α j100i j11i + β j011i j11i Qubit 2 Flip α j010i j10i + β j101i j10i Qubit 3 Flip α j001i j01i + β j110i j01i Note that each ancilla combination is different for each possible scenario. Knowing what the ancilla values are, we can now apply a "correction" on the qubit with a bit flip error by applying a X gate to that qubit. So, if we measure the ancilla values to be j11i, we then know to apply an X gate to qubit 1. Unfortunately this QECC only works for a maximum of one qubit error. If we get a bit flip error in qubits 1 and 2, then the ancilla measurement becomes j01i, and the assumption is that qubit 3 is erroneous. The X gate will be applied to qubit 3, and in fact all three qubits will have been flipped, meaning that our final result is the exact opposite of what we wanted it to be. 2.3 The Uncertainty Inherent to Data Transmission Based on our description in sections 2.1-2.3 about the way data transmission works, we see that since a communication channel may distort and add noise to messages, there is a lot of uncertainty associated with the process.