This page gives plain-language summaries of physics papers. We draw no link to Dost or to any other text; reading side by side is the reader's own work.
Experiment: measured. Theory: calculated or proved. Interpretation: its meaning is disputed. The labels are our own classification; a paper may have more than one nature.
Main question: Why can quantum states not be copied?
In quantum mechanics the state of a system (for example the polarization of a photon) is described by a state vector. Classical information can be copied without limit; duplicating a file does not damage the original. The question is this: can a device be built that makes an exact copy, without damaging the original, of a quantum system whose state we do not know? The papers of Wootters and Zurek and of Dieks, published in the same year, show from the theory's own rules that there can be no such device. This result is a precondition for subjects such as quantum cryptography and error correction.
On this page11. No-Cloning · Wootters 1982 — A single quantum cannot · Dieks 1982 — Communication by EPR device …
A short two-page paper showing that a device copying an unknown quantum state would contradict the linearity of quantum mechanics.
The paper's question is this: if we have a single quantum system whose state we do not know, can we, without measuring and damaging it, prepare a second system in the same state? In the classical world this is an ordinary operation; we put a sheet of paper in a photocopier. The authors show that for quantum states this is in general not possible.
The basic idea is the linearity of the theory. In quantum mechanics, the change of a system over time obeys a rule such that, applied to a sum of states (their superposition), it equals applying it to each component separately and adding up the results. Let us think of a copying device: if we give it the state "0" it outputs two "0"s, if we give it the state "1" it outputs two "1"s. By linearity, when "the superposition of 0 and 1" is given, the device's output is "the superposition of 00 and 11". Yet a true copy is two copies of the superposition itself; these two are different states. That is, the same device cannot correctly copy both two separate basis states and their mixtures.
The result is read as follows: a device can copy one of the states that are known in advance and can be told apart from each other with certainty (mutually orthogonal); but it cannot copy an unknown arbitrary state. If such copying were possible, the limits of learning a state by measuring it on a single system could be exceeded. The paper is short; it presents the theorem only with the linearity argument, without a detailed model of a device.
It is important to read the scope of the result correctly: what is forbidden is a perfect copy of an arbitrary, unknown state. Producing approximate copies of the state or preparing a known state as many times as desired is not forbidden; because in the latter the state is already known. Likewise, transferring the state of a qubit to another system (with the first system being destroyed) is not copying.
This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.
Afterwards: Dieks independently published a similar result the same year. In the following years no-cloning was used as a basic constraint in the design of quantum key distribution and quantum error correction.
A short paper, appearing independently in the same year as Wootters–Zurek, that takes up, on grounds of linearity, whether EPR setups can be used for communication.
The paper's title ("Communication by EPR devices") gives the question: can the correlations in the joint (entangled) state of two particles be used to carry information between a sender and a receiver? The answer to this question is tied to whether there is a device that can duplicate an unknown quantum state.
An EPR setup is an arrangement producing entangled particle pairs of the kind discussed by Einstein, Podolsky and Rosen in 1935. When the two parts of such a pair are measured separately at a distance, the results are correlated with each other. The question of whether this correlation can be used by the person making the measurement to send a faster-than-light message to the distant person is the paper's starting point.
Dieks's approach proceeds from the linear structure of the theory. When a measurement or copying arrangement is written as a linear operation on states, a device that copies an unknown state falls into contradiction on superpositions. This is the same reasoning as the result Wootters and Zurek published in the same year; the two papers arose independently of each other.
The importance of the paper lies in that it does not leave no-cloning as merely a mathematical detail but relates it to the question that EPR correlations carry no message. If copying were possible, the distant party could distinguish the other side's choice of measurement by measuring copies of his own particle; quantum mechanics does not make this possible.
The common point of this paper and the Wootters–Zurek paper is that the result comes not from an experiment but from the mathematical structure of the theory. Therefore no-cloning is not a technological shortcoming but a consequence of the theory itself. That copying is forbidden is cited as one of the first basic examples showing in what respect quantum information differs from classical information.
This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.
Afterwards: The Wootters–Zurek and Dieks results are together known as the no-cloning theorem.
12. Quantum Teleportation
Main question: Can a quantum state be transported?
If a quantum state cannot be copied, how can it be sent from one place to another? Teleportation is a method that, using an entangled particle pair shared in advance by sender and receiver and an ordinary communication channel, destroys an unknown quantum state in one place and rebuilds it in another. What is transferred is information; neither matter nor energy is carried. The subject is treated with two papers: the 1993 theoretical paper proposing the method and one of the first experimental demonstrations.
The paper proposing the method that transfers a quantum state by means of a shared entangled pair and a classical communication channel.
The question is whether it is possible to transfer an unknown quantum state to a distant receiver without learning the state beforehand and without sending the particle carrying it. The authors show with a method that this is possible.
In the method there are two persons: the sender (Alice) and the receiver (Bob). Beforehand two particles are prepared as an entangled pair of the kind treated by Einstein, Podolsky and Rosen (EPR); one is given to Alice, the other to Bob. Alice measures together the particle carrying the unknown state she wants to transfer and her own EPR particle. This joint measurement has four possible results. Alice informs Bob of the result through an ordinary channel, for example by telephone. Bob applies to his own particle the operation corresponding to the result reported, and his particle becomes an exact copy of Alice's original state.
As said in the paper's abstract, the unknown state is first decomposed into "purely classical information and purely quantum EPR correlations", and then rebuilt from these. Alice's joint measurement destroys the original state she holds; therefore the no-cloning principle is not violated, and the same state is not in two places at the same time. Before receiving the classical message, Bob finds in his particle only a state that looks random; for this reason the arrangement does not provide faster-than-light communication.
The distinction to keep in mind here is this: Bob's particle becomes equal to Alice's state only after the classical message has arrived and the correction has been applied. Since the classical message cannot travel faster than light, the speed of transfer is limited by the speed of this channel. Moreover the method does not require knowing the state beforehand; Alice too does not learn the state.
Afterwards: In later years the method became a building block used in quantum networks and for gate implementations in quantum computing.
One of the first experimental demonstrations of the teleportation proposal: the transfer of photon polarization states.
This paper seeks an answer to the question of whether the 1993 proposal can be realized in the laboratory. The experiment is done with photons: the state to be transferred is the polarization of a photon, and the entangled pair is a photon pair produced in a special crystal.
In the arrangement, the joint measurement made by Alice rests on bringing two photons together at a beam splitter and catching them separately. This measurement can distinguish only one of the four possible results; that is, teleportation is not completed in every trial, but only when the relevant result is obtained. The polarization of the photon on Bob's side is measured and its agreement with the state aimed to be transferred is checked.
The authors report that the transferred states came out in agreement with the expected polarization on Bob's side. What the paper puts forward is an experimental demonstration that a state can be physically transferred. Since in this demonstration not all four results are distinguished, the transfer is complete only in a selected subset of trials; this should be read as a limit of the arrangement.
This summary was written without looking at the text of the paper, within the known framework of the subject; for details one must return to the paper itself.
In experiments, teleportation is not the transport of light, matter or energy but the writing of a state onto another particle. The criterion of such experiments is therefore that the measurement statistics of the target particle agree with the source state. For the details of the paper (beam splitter, crystal, measurement results) one must look at the paper itself. In a demonstration of this kind the comparison criterion is that the measurement statistics of the transferred state coincide with the statistics of the source state; a single measurement result does not show this, the distribution of many trials is needed.
This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.
Afterwards: In later experiments the transfer distance and the success rate were increased; the details are outside the scope of the papers in this file.
13. Quantum Cryptography (BB84)
Main question: Is absolutely secure communication possible?
Can two people, even if they have never seen each other, create a shared secret key that an eavesdropper cannot learn? Most classical encryption is secure because some mathematical problem is hard. In Bennett and Brassard's 1984 proposal, security rests on the principle that measuring a quantum system changes it: if an eavesdropper looks at the photons, he leaves a trace. This paper, which appears in the topic list as "BB84", has been chosen.
IEEE Int. Conf. on Computers, Systems and Signal Processing, Bangalore, 1984; yeniden basım: Theoretical Computer Science 560 (2014) 7–11 theory
The paper proposing secret key distribution with polarized photons (the BB84 protocol) and quantum coin tossing.
The paper's starting point is this: when digital information is carried with basic quantum systems such as polarized photons, the uncertainty principle gives rise to cryptographic possibilities that do not exist in conventional media. In particular, because measurement disturbs the system, someone listening in on the transmission channel is noticed with high probability.
In the protocol the sender (Alice) encodes each bit on a photon with one of two "bases" that she chooses at random; for example vertical–horizontal polarization or 45° diagonal polarization. The receiver (Bob), not knowing which basis was used, measures each photon by choosing a basis at random. When the right basis is chosen the result is Alice's bit; when the wrong one is chosen the result comes out random. Then the two sides compare, over an ordinary channel open to eavesdropping, only which bases they chose, not the bits; the bits of the photons whose bases agree form the key.
If an eavesdropper (Eve) measures the photons and sends them on, she disturbs the state in the photons she measured in the wrong basis. Alice and Bob openly compare a part of their keys and check whether there are more errors than expected; if there are errors, they discard the key.
The paper also presents a "coin tossing" protocol by quantum messaging. The abstract says that this protocol is secure against the conventional kinds of cheating, even against an opponent with unlimited computing power. The authors also note, ironically, that the protocol could be broken by using the EPR phenomenon.
The strength of the protocol rests on this: since the eavesdropper does not know which basis was used, he can learn the state neither without disturbing it nor by copying it. For this reason the no-cloning principle is a building block of the protocol's security argument. In real devices the imperfections of the photon source and the detector are a subject that must be treated separately.
Afterwards: In the paper's own abstract it is recorded that the coin tossing protocol could be broken by the EPR phenomenon. Formal security analyses of key distribution are outside the scope of this file.
14. Entanglement Entropy and the Measurement of Entanglement
Main question: How is entanglement measured?
Entanglement is a quantum state in which the parts of a whole cannot be described separately. The question is whether this "amount" can be measured with a single number. Two papers by Bennett and colleagues from 1996 lay the foundation of this subject. One shows that for pure (fully known) states entanglement can be measured by entropy; the other shows that purifying entanglement in noisy (mixed) states is equivalent to quantum error correction. The expression "Bennett et al. 1996" in the list has been read so as to cover both papers.
A paper showing that shared pure, partially entangled pairs can be converted by local operations into maximally entangled pairs, and that the rate is determined by entropy.
The question is this: if two distant observers hold many copies of the same partially entangled particle pair, can they, by applying only local operations to their own particles and talking to each other classically, concentrate this entanglement into a smaller number of maximally entangled (EPR "singlet") pairs?
The authors show that this is possible. As the number of pairs grows, the number of maximally entangled pairs obtained approaches the "entanglement entropy" of the initial state. The entanglement entropy is the von Neumann entropy of the reduced density matrix seen by one side. Intuitively: even though the whole state is completely known, it measures how much uncertainty remains for a party who looks only at its own particle. It is computed in base two and its unit is the bit. The entropy of a maximally entangled pair is one bit; the entropy of a pair with no entanglement is zero. For a partially entangled pair the value lies between the two, and from a large number of pairs, on average that many maximally entangled pairs can be extracted.
The converse is also shown: by communicating classically and applying local operations, and by spending maximally entangled pairs as a resource, the two parties can produce any pure or mixed entangled state. In this way maximally entangled pairs act as a common "currency" for entanglement. The result means that, for pure states, entanglement can be expressed by a single number.
A notable result of the paper is that, for pure states, entanglement is measured by the same number (the entanglement entropy) in both directions; that is, in these cases entanglement behaves like a "reversible" resource. In mixed states the situation is more complicated; the group's second paper takes this up.
Afterwards: How the same question is to be treated for mixed states is taken up in the second paper of the same year.
A paper showing that sending a quantum state through a noisy channel is equivalent to distilling maximal entanglement from mixed entangled states.
This paper examines what can be done when entanglement is mixed with noise. Two ways of protection are compared: entanglement purification (extracting a small number of maximally entangled pairs from the mixed state held by the two parties) and quantum error-correcting codes (sending an arbitrary quantum state through a noisy channel without corruption).
The authors show that a purification method working with one-way classical communication corresponds to an error-correcting code working at the same rate, and that the converse also holds. Purification requires classical communication; error correction does not. Adding one-way communication does not increase the error-correction rate. By contrast, two-way communication can improve both purification and error correction. For example, a channel that is fifty percent depolarizing can be used for quantum state transfer if two-way communication is available; it cannot be used if only one-way communication is available.
The paper also takes up the amount of entanglement needed to prepare a given mixed state by local operations (today called "entanglement of formation") and gives an exact expression for states diagonal in the Bell basis. Finally it presents a family of codes based on universal hashing and a simple five-bit block code that corrects a single error. The authors also show that, subject to the condition that an error-correcting code works with high fidelity in the error-free case, the encoder can be written as the matrix inverse of the decoder.
The paper's results are cited for bringing two separate headings of quantum information theory (entanglement theory and error correction) together in a single framework. This summary relays the claims in the paper's abstract; the details of the proofs are in the paper itself.
Afterwards: The five-bit code in this paper is one of the examples frequently cited in the quantum error-correction literature.
15. Quantum Error Correction
Main question: How does one compute with noisy systems?
Quantum information is very easily corrupted: every contact with the environment disperses the information in the state (this is called decoherence). In classical computers the way to correct errors is to copy the information and take a majority vote; this does not work for quantum states, because they cannot be copied and change when measured. In 1995-96 Shor and Steane showed that by spreading the information over several physical qubits, it is possible to detect and correct errors without learning the state.
A paper proposing a quantum counterpart of classical error-correcting codes in order to reduce the effect of decoherence in quantum memory.
The paper's question concerns one of the greatest obstacles to quantum computation: decoherence corrupts the information in a superposition and makes long computations impossible. The author begins by saying that interest in quantum computation has recently grown rapidly and that it has been noticed that certain computations can be dramatically sped up by the properties of quantum mechanics.
The proposed scheme reduces the effect of decoherence on information stored in memory. The assumption is that the decoherence process acts on each bit in memory independently of the others. In that case, a quantum analogue of classical error-correcting codes is used. The idea is to spread the information of a single qubit over the joint state of several qubits instead of a single physical qubit. Classical majority voting does not work, because a quantum state cannot be copied and is corrupted if measured directly. Instead, the presence of an error is detected by measurements that probe the correlations between the qubits, not the carried information itself (today called "syndrome measurement"). In this way the information is not learned, and therefore not corrupted; only which qubit has which kind of error is inferred, and it is corrected.
The paper's own limit is clear: the scheme rests on the independent-error assumption and aims to protect information stored in memory; carrying out the computation itself in a fault-tolerant way is not the subject of this paper.
The paper is cited as an important turning point in the debate on the feasibility of quantum computation, in that it shows that error correction is possible for quantum states as well. However, the paper's abstract presents this as a scheme, not as an implementation. A short reminder for the reader: the word "bit" here is the paper's own usage; today the memory unit is called a qubit.
Afterwards: In the same period Steane took up the same problem with a different code construction; in the following years came works making error correction fault tolerant together with gate operations.
Proceedings of the Royal Society A 452, 2551–2577 theory
A paper presenting quantum error-correcting codes and their limits by treating multi-particle interference in the light of classical coding theory.
The paper brings two things together: the mathematics of classical error-correcting codes and many-particle quantum interference. Using this connection, the author explains how error correction can be done in quantum communication channels and quantum computers, and evaluates its limits.
The basic result puts a lower and an upper bound on the amount of information that can be encoded. If K qubits of information are encoded into n qubits, then for protection against arbitrary decoherence of x qubits the rate must lie in the following range: it may exceed 1 − 2H(2x/n), but cannot exceed 1 − 2H(x/n); here H is the binary entropy function. Correcting more errors requires more physical qubits; the rate is bounded.
The author's conclusion is this: as the code length grows, the effect of decoherence can be reduced exponentially, whereas the required resources grow only polynomially. The author interprets this to mean that, at physically realistic decoherence levels, quantum computation can be carried out free of errors; this is a theoretical inference, not a demonstration of a device.
The methods can also be used to isolate communication from noise and from an eavesdropper (quantum privacy amplification). Because the paper translates the language of classical coding theory into the quantum setting, it is counted among the papers that prepared the ground for later methods of deriving quantum codes from classical codes.
The paper's importance lies in setting out the limits of quantum error correction numerically and in showing that the resource cost of error-correcting codes grows only polynomially. This is one of the theoretical underpinnings of the debate on the scalability of the quantum computer. These limits depend on the type of error and on the structure of the code; the conditions under which the formulas hold are defined in the paper's text, and this summary conveys only the form of the result.
Afterwards: The five-bit code in Bennett et al. 1996 belongs to the same period (topic 14 in this file).