Entanglement distillation

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https://en.wikipedia.org/wiki/Entanglement_distillation 

Entanglement distillation (also called entanglement purification) is the transformation of N copies of an arbitrary entangled state ${\displaystyle \rho }$ into some number of approximately pure Bell pairs, using only local operations and classical communication.

Quantum entanglement distillation can in this way overcome the degenerative influence of noisy quantum channels by transforming previously shared less entangled pairs into a smaller number of maximally entangled pairs.

History

The limits for entanglement dilution and distillation are due to C. H. Bennett, H. Bernstein, S. Popescu, and B. Schumacher, who presented the first distillation protocols for pure states in 1996; entanglement distillation protocols for mixed states were introduced by Bennett, Brassard, Popescu, Schumacher, Smolin and Wootters the same year. Bennett, DiVincenzo, Smolin and Wootters established the connection to quantum error-correction in a ground-breaking paper published in August 1996, also in the journal of Physical Review, which has stimulated a lot of subsequent research.

Quantifying entanglement

A two qubit system can be written as a superposition of possible computational basis qubit states: ${\displaystyle |00⟩,|01⟩,|10⟩,|11⟩}$, each with an associated complex coefficient ${\displaystyle \alpha }$: $${\displaystyle |\psi ⟩={\alpha }_{00}|00⟩+{\alpha }_{01}|01⟩+{\alpha }_{10}|10⟩+{\alpha }_{11}|11⟩}$$

As in the case of a single qubit, the probability of measuring a particular computational basis state ${\displaystyle |x⟩}$ is the square of the modulus of its amplitude, or associated coefficient, ${\displaystyle |{\alpha }_x{|}^2}$, subject to the normalization condition $\sum _{x\in 0,1}|{\alpha }_x{|}^2=1$. The normalization condition guarantees that the sum of the probabilities add up to 1, meaning that upon measurement, one of the states will be observed.

The Bell state is a particularly important example of a two qubit state: $\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$

Bell states possess the property that measurement outcomes on the two qubits are correlated. As can be seen from the expression above, the two possible measurement outcomes are zero and one, both with probability of 50%. As a result, a measurement of the second qubit always gives the same result as the measurement of the first qubit.

Bell states can be used to quantify entanglement. Let m be the number of high-fidelity copies of a Bell state that can be produced using local operations and classical communication (LOCC). Given a large number of Bell states the amount of entanglement present in a pure state ${\displaystyle |\psi ⟩}$ can then be defined as the ratio of ${\displaystyle n/m}$, called the distillable entanglement of a particular state ${\displaystyle |\varphi ⟩}$, which gives a quantified measure of the amount of entanglement present in a given system. The process of entanglement distillation aims to saturate this limiting ratio. The number of copies of a pure state that may be converted into a maximally entangled state is equal to the von Neumann entropy ${\displaystyle S(p)}$ of the state, which is an extension of the concept of classical entropy for quantum systems. Mathematically, for a given density matrix ${\displaystyle p}$, the von Neumann entropy ${\displaystyle S(p)}$ is ${\displaystyle S(p)=-\mathrm{T}\mathrm{r}(p\ln p)}$. Entanglement can then be quantified as the entropy of entanglement, which is the von Neumann entropy of either ${\displaystyle p_A}$ or ${\displaystyle p_B}$ as: $${\displaystyle E=-\mathrm{T}\mathrm{r}(p_A\ln p_A)=-\mathrm{T}\mathrm{r}(p_B\ln p_B),}$$

Which ranges from 0 for a product state to ${\displaystyle \ln 2}$ for a maximally entangled state (if the ${\displaystyle \ln }$ is replaced by ${\displaystyle {\log }_2}$ then maximally entangled has a value of 1).

Motivation

Suppose that two parties, Alice and Bob, would like to communicate classical information over a noisy quantum channel. Either classical or quantum information can be transmitted over a quantum channel by encoding the information in a quantum state. With this knowledge, Alice encodes the classical information that she intends to send to Bob in a (quantum) product state, as a tensor product of reduced density matrices ${\displaystyle p_1\otimes p_2\otimes \cdots }$ where each ${\displaystyle p}$ is diagonal and can only be used as a one time input for a particular channel ${\displaystyle ϵ}$.

The fidelity of the noisy quantum channel is a measure of how closely the output of a quantum channel resembles the input, and is therefore a measure of how well a quantum channel preserves information. If a pure state ${\displaystyle \psi }$ is sent into a quantum channel emerges as the state represented by density matrix ${\displaystyle p}$, the fidelity of transmission is defined as ${\displaystyle F=⟨\psi |p|\psi ⟩}$.

The problem that Alice and Bob now face is that quantum communication over large distances depends upon successful distribution of highly entangled quantum states, and due to unavoidable noise in quantum communication channels, the quality of entangled states generally decreases exponentially with channel length as a function of the fidelity of the channel. Entanglement distillation addresses this problem of maintaining a high degree of entanglement between distributed quantum states by transforming N copies of an arbitrary entangled state ${\displaystyle \rho }$ into approximately ${\displaystyle S(\rho )N}$ Bell pairs, using only local operations and classical communication. The objective is to share strongly correlated qubits between distant parties (Alice and Bob) in order to allow reliable quantum teleportation or quantum cryptography.

Entanglement concentration

Pure states

The new fidelity after one iteration of the distillation protocol for pure states.

Given n particles in the singlet state shared between Alice and Bob, local actions and classical communication will suffice to prepare m arbitrarily good copies of ${\displaystyle \varphi }$ with a yield

${\displaystyle \frac{m}{n}\to \frac{1}{E(\varphi )}}$ as ${\displaystyle n\to \mathrm{\infty }}$.

Let an entangled state ${\displaystyle |\psi ⟩}$ have a Schmidt decomposition: $${\displaystyle |\psi ⟩=\sum _x\sqrt{p(x)}|x_A⟩|x_B⟩}$$ where coefficients p(x) form a probability distribution, and thus are positive valued and sum to unity. The tensor product of this state is then, $${\displaystyle |\psi {⟩}^{\otimes m}=\sum _{x_1,x_2,\dots ,x_m}\sqrt{p(x_1)p(x_2)\dots p(x_m)}|x_{1A}{x}_{2A}\dots x_{mA}⟩|x_{1B}{x}_{2B}\dots x_{mB}⟩}$$

Now, omitting all terms ${\displaystyle x_1,\dots ,x_m}$ which are not part of any sequence which is likely to occur with high probability, known as the typical set: ${\displaystyle A_{ϵ}^{(n)}}$ the new state is $${\displaystyle |{\varphi }_m⟩=\sum _{xϵA_{ϵ}^{(n)}}\sqrt{p(x_1)p(x_2)\dots p(x_m)}|x_{1A}{x}_{2A}\dots x_{mA}⟩|x_{1B}{x}_{2B}\dots x_{mB}⟩}$$

And renormalizing, $${\displaystyle |{\varphi }_m^{\prime }⟩=\frac{|{\varphi }_m⟩}{\sqrt{⟨{\varphi }_m|{\varphi }_m⟩}}}$$

Then the fidelity

${\displaystyle F(|\psi {⟩}^{\otimes m},|{\varphi }_m^{{}^{\prime }}⟩)\to 1}$ as ${\displaystyle m\to \mathrm{\infty }}$.

Suppose that Alice and Bob are in possession of m copies of ${\displaystyle |\psi ⟩}$. Alice can perform a measurement onto the typical set ${\displaystyle A_{ϵ}^{(n)}}$ subset of ${\displaystyle p_{\psi }}$, converting the state ${\displaystyle |\psi {⟩}^{\otimes m}\to |{\varphi }_m⟩}$ with high fidelity. The theorem of typical sequences then shows us that ${\displaystyle 1-\delta }$ is the probability that the given sequence is part of the typical set, and may be made arbitrarily close to 1 for sufficiently large m, and therefore the Schmidt coefficients of the renormalized Bell state ${\displaystyle |{\varphi }_m^{\prime }⟩}$ will be at most a factor $1/\sqrt{1-\delta }$ larger. Alice and Bob can now obtain a smaller set of n Bell states by performing LOCC on the state ${\displaystyle |{\varphi }_m^{\prime }⟩}$ with which they can overcome the noise of a quantum channel to communicate successfully.

Mixed states

The new fidelity after one iteration of the distillation protocol presented here for mixed states

Many techniques have been developed for doing entanglement distillation for mixed states, giving a lower bounds on the value of the distillable entanglement ${\displaystyle D(p)}$ for specific classes of states ${\displaystyle p}$.

One common method involves Alice not using the noisy channel to transmit source states directly but instead preparing a large number of Bell states, sending half of each Bell pair to Bob. The result from transmission through the noisy channel is to create the mixed entangled state ${\displaystyle p}$, so that Alice and Bob end up sharing ${\displaystyle m}$ copies of ${\displaystyle p}$. Alice and Bob then perform entanglement distillation, producing ${\displaystyle m\cdot D(p)}$ almost perfectly entangled states from the mixed entangled states ${\displaystyle p}$ by performing local unitary operations and measurements on the shared entangled pairs, coordinating their actions through classical messages, and sacrificing some of the entangled pairs to increase the purity of the remaining ones. Alice can now prepare an ${\displaystyle m\cdot D(p)}$ qubit state and teleport it to Bob using the ${\displaystyle m\cdot D(p)}$ Bell pairs which they share with high fidelity. What Alice and Bob have then effectively accomplished is having simulated a noiseless quantum channel using a noisy one, with the aid of local actions and classical communication.

Let ${\displaystyle M}$ be a general mixed state of two spin-1/2 particles which could have resulted from the transmission of an initially pure singlet state $${\displaystyle {\psi }^{-}=(\uparrow \downarrow -\downarrow \uparrow )/\sqrt{2}}$$ through a noisy channel between Alice and Bob, which will be used to distill some pure entanglement. The fidelity of M $${\displaystyle F=⟨{\psi }^{-}|M|{\psi }^{-}⟩}$$ is a convenient expression of its purity relative to a perfect singlet. Suppose that M is already a pure state of two particles ${\displaystyle M=|\varphi ⟩⟨\varphi |}$ for some ${\displaystyle \varphi }$. The entanglement for ${\displaystyle \varphi }$, as already established, is the von Neumann entropy ${\displaystyle E(\varphi )=S(p_A)=S(p_B)}$ where $${\displaystyle p_A={\mathrm{tr}}_B^{}(|\varphi ⟩⟨\varphi |),}$$ and likewise for ${\displaystyle p_B}$, represent the reduced density matrices for either particle. The following protocol is then used:

1. Performing a random bilateral rotation on each shared pair, choosing a random SU(2) rotation independently for each pair and applying it locally to both members of the pair transforms the initial general two-spin mixed state M into a rotationally symmetric mixture of the singlet state ${\displaystyle {\psi }^{-}}$ and the three triplet states ${\displaystyle {\psi }^{+}}$ and ${\displaystyle {\varphi }^{\pm }}$: $${\displaystyle W_F=F\cdot |{\psi }^{-}⟩⟨{\psi }^{-}|+\frac{1-F}{3}|{\varphi }^{+}⟩⟨{\varphi }^{+}|+\frac{1-F}{3}|{\psi }^{+}⟩⟨{\psi }^{+}|+\frac{1-F}{3}|{\varphi }^{-}⟩⟨{\varphi }^{-}|}$$ The Werner state ${\displaystyle W_F}$ has the same purity F as the initial mixed state M from which it was derived due to the singlet's invariance under bilateral rotations.
2. Each of the two pairs is then acted on by a unilateral rotation, which we can call ${\displaystyle {\sigma }_y}$, which has the effect of converting them from mainly ${\displaystyle {\psi }^{-}}$ Werner states to mainly ${\displaystyle {\varphi }^{+}}$ states with a large component ${\displaystyle F>\frac{1}{2}}$ of ${\displaystyle {\varphi }^{+}}$ while the components of the other three Bell states are equal.
3. The two impure ${\displaystyle {\varphi }^{+}}$ states are then acted on by a bilateral XOR, and afterwards the target pair is locally measured along the z axis. The unmeasured source pair is kept if the target pair's spins come out parallel as in the case of both inputs being true ${\displaystyle {\varphi }^{+}}$ states; and it is discarded otherwise.
4. If the source pair has not been discarded it is converted back to a predominantly ${\displaystyle {\psi }^{-}}$ state by a unilateral ${\displaystyle {\sigma }_y}$ rotation, and made rotationally symmetric by a random bilateral rotation.

Repeating the outlined protocol above will distill Werner states whose purity may be chosen to be arbitrarily high ${\displaystyle F_{\text{out}}<1}$ from a collection M of input mixed states of purity $F_{\text{in}}>\frac{1}{2}$ but with a yield tending to zero in the limit ${\displaystyle F_{\text{out}}\to 1}$. By performing another bilateral XOR operation, this time on a variable number $k(F)\approx \frac{1}{\sqrt{1-F}}$ of source pairs, as opposed to 1, into each target pair prior to measuring it, the yield can be made to approach a positive limit as ${\displaystyle F_{\text{out}}\to 1}$. This method can then be combined with others to obtain an even higher yield.

Procrustean method

The Procrustean method of entanglement concentration can be used for as little as one partly entangled pair, being more efficient than the Schmidt projection method for entangling less than 5 pairs, and requires Alice and Bob to know the bias (${\displaystyle \theta }$) of the n pairs in advance. The method derives its name from Procrustes because it produces a perfectly entangled state by chopping off the extra probability associated with the larger term in the partial entanglement of the pure states: $${\displaystyle \cos \theta |{\uparrow }_A⟩\otimes |{\downarrow }_B⟩-\sin \theta |{\downarrow }_A⟩\otimes |{\uparrow }_B⟩}$$

Assuming a collection of particles for which ${\displaystyle \theta }$ is known as being either less than or greater than ${\displaystyle \pi /4}$ the Procrustean method may be carried out by keeping all particles which, when passed through a polarization-dependent absorber, or a polarization-dependent-reflector, which absorb or reflect a fraction ${\displaystyle {\tan }^2\theta }$ of the more likely outcome, are not absorbed or deflected. Therefore, if Alice possesses particles for which ${\displaystyle \theta \ne \pi /4}$, she can separate out particles which are more likely to be measured in the up/down basis, and left with particles in maximally mixed state of spin up and spin down. This treatment corresponds to a POVM (positive-operator-valued measurement). To obtain a perfectly entangled state of two particles, Alice informs Bob of the result of her generalized measurement while Bob doesn't measure his particle at all but instead discards his if Alice discards hers.

Stabilizer protocol

The purpose of an ${\displaystyle \left[n,k\right]}$ entanglement distillation protocol is to distill ${\displaystyle k}$ pure ebits from ${\displaystyle n}$ noisy ebits where ${\displaystyle 0\le k\le n}$. The yield of such a protocol is ${\displaystyle k/n}$. Two parties can then use the noiseless ebits for quantum communication protocols.

The two parties establish a set of shared noisy ebits in the following way. The sender Alice first prepares ${\displaystyle n}$ Bell states ${\displaystyle {|{\mathrm{\Phi }}^{+}⟩}^{\otimes n}}$ locally. She sends the second qubit of each pair over a noisy quantum channel to a receiver Bob. Let ${\displaystyle |{\mathrm{\Phi }}_n^{+}⟩}$ be the state ${\displaystyle {|{\mathrm{\Phi }}^{+}⟩}^{\otimes n}}$ rearranged so that all of Alice's qubits are on the left and all of Bob's qubits are on the right. The noisy quantum channel applies a Pauli error in the error set ${\displaystyle \mathcal{E}\subset {\mathrm{\Pi }}^n}$ to the set of ${\displaystyle n}$ qubits sent over the channel. The sender and receiver then share a set of ${\displaystyle n}$ noisy ebits of the form ${\displaystyle \left(\mathbf{I}\otimes \mathbf{A}\right)|{\mathrm{\Phi }}_n^{+}⟩}$ where the identity ${\displaystyle \mathbf{I}}$ acts on Alice's qubits and ${\displaystyle \mathbf{A}}$ is some Pauli operator in ${\displaystyle \mathcal{E}}$ acting on Bob's qubits.

A one-way stabilizer entanglement distillation protocol uses a stabilizer code for the distillation procedure. Suppose the stabilizer ${\displaystyle \mathcal{S}}$ for an ${\displaystyle \left[n,k\right]}$ quantum error-correcting code has generators ${\displaystyle g_1,\dots ,g_{n-k}}$. The distillation procedure begins with Alice measuring the ${\displaystyle n-k}$ generators in ${\displaystyle \mathcal{S}}$. Let ${\displaystyle \left\{{\mathbf{P}}_i\right\}}$ be the set of the ${\displaystyle 2^{n-k}}$ projectors that project onto the ${\displaystyle 2^{n-k}}$ orthogonal subspaces corresponding to the generators in ${\displaystyle \mathcal{S}}$. The measurement projects ${\displaystyle |{\mathrm{\Phi }}_n^{+}⟩}$ randomly onto one of the ${\displaystyle i}$ subspaces. Each ${\displaystyle {\mathbf{P}}_i}$ commutes with the noisy operator ${\displaystyle \mathbf{A}}$ on Bob's side so that $${\displaystyle \left({\mathbf{P}}_i\otimes \mathbf{I}\right)\left(\mathbf{I}\otimes \mathbf{A}\right)|{\mathrm{\Phi }}_n^{+}⟩=\left(\mathbf{I}\otimes \mathbf{A}\right)\left({\mathbf{P}}_i\otimes \mathbf{I}\right)|{\mathrm{\Phi }}_n^{+}⟩.}$$

The following important Bell-state matrix identity holds for an arbitrary matrix ${\displaystyle \mathbf{M}}$: $${\displaystyle \left(\mathbf{M}\otimes \mathbf{I}\right)|{\mathrm{\Phi }}_n^{+}⟩=\left(\mathbf{I}\otimes {\mathbf{M}}^T\right)|{\mathrm{\Phi }}_n^{+}⟩.}$$

Then the above expression is equal to the following: $${\displaystyle \left(\mathbf{I}\otimes \mathbf{A}\right)\left({\mathbf{P}}_i\otimes \mathbf{I}\right)|{\mathrm{\Phi }}_n^{+}⟩=\left(\mathbf{I}\otimes \mathbf{A}\right)\left({\mathbf{P}}_i^2\otimes \mathbf{I}\right)|{\mathrm{\Phi }}_n^{+}⟩=\left(\mathbf{I}\otimes \mathbf{A}\right)\left({\mathbf{P}}_i\otimes {\mathbf{P}}_i^T\right)|{\mathrm{\Phi }}_n^{+}⟩.}$$ Therefore, each of Alice's projectors ${\displaystyle {\mathbf{P}}_i}$ projects Bob's qubits onto a subspace ${\displaystyle {\mathbf{P}}_i^T}$ corresponding to Alice's projected subspace ${\displaystyle {\mathbf{P}}_i}$. Alice restores her qubits to the simultaneous +1-eigenspace of the generators in ${\displaystyle \mathcal{S}}$. She sends her measurement results to Bob. Bob measures the generators in ${\displaystyle \mathcal{S}}$. Bob combines his measurements with Alice's to determine a syndrome for the error. He performs a recovery operation on his qubits to reverse the error. He restores his qubits ${\displaystyle \mathcal{S}}$. Alice and Bob both perform the decoding unitary corresponding to stabilizer ${\displaystyle \mathcal{S}}$ to convert their ${\displaystyle k}$ logical ebits to ${\displaystyle k}$ physical ebits.

Entanglement-assisted stabilizer code

Luo and Devetak provided a straightforward extension of the above protocol (Luo and Devetak 2007). Their method converts an entanglement-assisted stabilizer code into an entanglement-assisted entanglement distillation protocol.

Luo and Devetak form an entanglement distillation protocol that has entanglement assistance from a few noiseless ebits. The crucial assumption for an entanglement-assisted entanglement distillation protocol is that Alice and Bob possess ${\displaystyle c}$ noiseless ebits in addition to their ${\displaystyle n}$ noisy ebits. The total state of the noisy and noiseless ebits is $${\displaystyle \left({\mathbf{I}}^A\otimes {\left(\mathbf{A}\otimes \mathbf{I}\right)}^B\right)|{\mathrm{\Phi }}_{n+c}^{+}⟩}$$ where ${\displaystyle {\mathbf{I}}^A}$ is the ${\displaystyle 2^{n+c}\times 2^{n+c}}$ identity matrix acting on Alice's qubits and the noisy Pauli operator ${\displaystyle {\left(\mathbf{A}\otimes \mathbf{I}\right)}^B}$ affects Bob's first ${\displaystyle n}$ qubits only. Thus the last ${\displaystyle c}$ ebits are noiseless, and Alice and Bob have to correct for errors on the first ${\displaystyle n}$ ebits only.

The protocol proceeds exactly as outlined in the previous section. The only difference is that Alice and Bob measure the generators in an entanglement-assisted stabilizer code. Each generator spans over ${\displaystyle n+c}$ qubits where the last ${\displaystyle c}$ qubits are noiseless.

We comment on the yield of this entanglement-assisted entanglement distillation protocol. An entanglement-assisted code has ${\displaystyle n-k}$ generators that each have ${\displaystyle n+c}$ Pauli entries. These parameters imply that the entanglement distillation protocol produces ${\displaystyle k+c}$ ebits. But the protocol consumes ${\displaystyle c}$ initial noiseless ebits as a catalyst for distillation. Therefore, the yield of this protocol is ${\displaystyle k/n}$.

Entanglement dilution

The reverse process of entanglement distillation is entanglement dilution, where large copies of the Bell state are converted into less entangled states using LOCC with high fidelity. The aim of the entanglement dilution process, then, is to saturate the inverse ratio of n to m, defined as the distillable entanglement.

Applications

Besides its important application in quantum communication, entanglement purification also plays a crucial role in error correction for quantum computation, because it can significantly increase the quality of logic operations between different qubits. The role of entanglement distillation is discussed briefly for the following applications.

Quantum error correction

Main article: Quantum error correction

Entanglement distillation protocols for mixed states can be used as a type of error-correction for quantum communications channels between two parties Alice and Bob, enabling Alice to reliably send mD(p) qubits of information to Bob, where D(p) is the distillable entanglement of p, the state that results when one half of a Bell pair is sent through the noisy channel ${\displaystyle ϵ}$ connecting Alice and Bob.

In some cases, entanglement distillation may work when conventional quantum error-correction techniques fail. Entanglement distillation protocols are known which can produce a non-zero rate of transmission D(p) for channels which do not allow the transmission of quantum information due to the property that entanglement distillation protocols allow classical communication between parties as opposed to conventional error-correction which prohibits it.

Quantum cryptography

Main article: Quantum cryptography

The concept of correlated measurement outcomes and entanglement is central to quantum key exchange, and therefore the ability to successfully perform entanglement distillation to obtain maximally entangled states is essential for quantum cryptography.

If an entangled pair of particles is shared between two parties, anyone intercepting either particle will alter the overall system, allowing their presence (and the amount of information they have gained) to be determined so long as the particles are in a maximally entangled state. Also, in order to share a secret key string, Alice and Bob must perform the techniques of privacy amplification and information reconciliation to distill a shared secret key string. Information reconciliation is error-correction over a public channel which reconciles errors between the correlated random classical bit strings shared by Alice and Bob while limiting the knowledge that a possible eavesdropper Eve can have about the shared keys. After information reconciliation is used to reconcile possible errors between the shared keys that Alice and Bob possess and limit the possible information Eve could have gained, the technique of privacy amplification is used to distill a smaller subset of bits maximizing Eve's uncertainty about the key.

Quantum teleportation

Main article: Quantum teleportation

In quantum teleportation, a sender wishes to transmit an arbitrary quantum state of a particle to a possibly distant receiver. Quantum teleportation is able to achieve faithful transmission of quantum information by substituting classical communication and prior entanglement for a direct quantum channel. Using teleportation, an arbitrary unknown qubit can be faithfully transmitted via a pair of maximally-entangled qubits shared between sender and receiver, and a 2-bit classical message from the sender to the receiver. Quantum teleportation requires a noiseless quantum channel for sharing perfectly entangled particles, and therefore entanglement distillation satisfies this requirement by providing the noiseless quantum channel and maximally entangled qubits.

Skeptical theism

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Skeptical_theism

Skeptical theism is the view that people should remain skeptical of their ability to discern whether their perceptions about evil can be considered good evidence against the existence of the orthodox Christian God. The central thesis of skeptical theism is that it would not be surprising for an infinitely intelligent and knowledgeable being's reasons for permitting evils to be beyond human comprehension. That is, what may seem like pointless evils may be necessary for a greater good or to prevent equal or even greater evils. This central thesis may be argued from a theistic perspective, but is also argued to defend positions of agnosticism.

Skeptical theism can be an informally held belief based on theistic doctrine, but the origin of the term skeptical theist is the 1996 paper "The Skeptical Theist" by philosopher Paul Draper. Following Draper's publication, the term skeptical theism was adopted in academic philosophy and has developed into a family of positions supporting skeptical theism's central skeptical thesis; we should remain skeptical of claims that human beings can discern God's reasons for evils. One argument is based on analogy, likening our understanding of God's motives to those of a child grasping a parent's reasons for seeking painful medical treatment, for example. Other approaches are the limitations on the human ability to understand the moral realm, and appeals to epistemic factors such as sensitivity or contextual requirements.

In the philosophy of religion, skeptical theism is not a broad skepticism toward human knowledge of God, but is instead putatively presented as a response to philosophical propositions, such as those focused on drawing "all things considered" inductive conclusions about God's motives from perceived circumstances. Additionally, skeptical theism is not a position used to defend all forms of theism, though it is most often presented in the defense of orthodox Christian theism. Moreover, skeptical theism is not supported by all theists and some who support its skeptical positions are not theists.

In philosophy, skeptical theism is a defense of theistic or agnostic positions argued to undercut a crucial premise in atheological arguments from evil, a claim that God could have no good reasons for allowing certain types of evil. It is also presented in response to other atheological arguments claiming to know God's purposes based on circumstances, such as the argument from divine hiddenness.

Draper's skeptical theism

In the philosophy of religion, skeptical theism is the position that we should be skeptical of our ability to assess God's motivations or lack of motivation from our perceptions of the circumstances we observe in the world. The view is a response to the atheological argument from evil, which asserts that some evils in the world are gratuitous, pointless, or inscrutable evils, and that they thus represent evidence against the existence of the God of orthodox Christianity. God, by the orthodox view, is thought to be omniscient (all-knowing), omnibenevolent (all-good) and omnipotent (all-powerful). Insofar as it tries to reconcile this conception of God with concerns about gratuitous evils (evils that do occur in the world, but which God is argued to have no morally sufficient reason for permitting), skeptical theism can be considered a form of theodicy. As originally proposed by agnostic philosopher Paul Draper, the view is intended to undercut a key premise in the argument from evil by suggesting that human cognitive faculties could be insufficient to permit drawing inductive inferences concerning God's reasons or lack of reasons for permitting perceived evils.

The evidential argument from evil

The evidential argument from evil asserts that the amount, types, or distribution of evils, provide an evidential basis for concluding that God's existence is improbable. The argument has a number of formulation, but can be stated in the Modus ponens logical form:

1. If an omniscient, omnibenevolent and omnipotent God exists, there should be no gratuitous evil.
2. There exists instances of gratuitous evil.
3. Therefore, an omniscient, omnibenevolent and omnipotent God does not exist.

In this logical form the conclusion (3) is true, if both the major premise (1) and minor premise (2) are true. Philosophers have challenged both premises, but skeptical theism focuses on the minor premise (2).

In 1979, philosopher William Rowe provided a defense of the minor premise (2). He argued that no state of affairs we know of is such that an omnipotent, omniscient being's obtaining it would morally justify that being's permitting some instances of horrific suffering. Therefore, Rowe concludes, it is likely that no state of affairs exists that would morally justify that being in permitting such suffering. In other words, Rowe argues that his inability to think of a good reason why God would allow a particular evil justifies the conclusion that there is no such reason, and the conclusion that God does not exist.

The "noseeum" inference

The philosophers Michael Bergmann and Michael Rea described William Rowe's justification for the second premise of the argument from evil:

Some evidential arguments from evil ... rely on a "noseeum" inference of the following sort: NI: If, after thinking hard, we can't think of any God-justifying reason for permitting some horrific evil then it is likely that there is no such reason. (The reason NI is called a 'noseeum' inference is that it says, more or less, that because we don't see 'um, they probably ain't there.)

Various analogies are offered to show that the noseeum inference is logically dubious. For example, a novice chess player's inability to discern a chess master's choice of moves cannot be used to infer that there is no good reason for the move.

The skeptical theist's response

Skeptical theism provides a defense against the evidential argument from evil, but does not take a position on God's actual reason for allowing a particular instance of evil. The defense seeks to show that there are good reasons to believe that God could have justified reasons for allowing a particular evil that we cannot discern. Consequently, we are in no position to endorse the minor premise (2) of the argument from evil because we cannot be more than agnostic about the accuracy of the premise. This conclusion would be an undercutting defeater for the premise because there would be no justification for the conclusion that evils in our world are gratuitous. To justify this conclusion, the skeptical theist argues that the limits of human cognitive faculties are grounds for skepticism about our ability to draw conclusions about God's motives or lack of motives; it is therefore reasonable to doubt the second premise. Bergmann and Rea thus concluded that Rowe's inference is unsound.

Quantum network

 

From Wikipedia, the free encyclopedia

Quantum networks form an important element of quantum computing and quantum communication systems. Quantum networks facilitate the transmission of information in the form of quantum bits, also called qubits, between physically separated quantum processors. A quantum processor is a machine able to perform quantum circuits on a certain number of qubits. Quantum networks work in a similar way to classical networks. The main difference is that quantum networking, like quantum computing, is better at solving certain problems, such as modeling quantum systems.

Basics

Quantum networks for computation

Networked quantum computing or distributed quantum computing works by linking multiple quantum processors through a quantum network by sending qubits in between them. Doing this creates a quantum computing cluster and therefore creates more computing potential. Less powerful computers can be linked in this way to create one more powerful processor. This is analogous to connecting several classical computers to form a computer cluster in classical computing. Like classical computing, this system is scalable by adding more and more quantum computers to the network. Currently quantum processors are only separated by short distances.

Quantum networks for communication

In the realm of quantum communication, one wants to send qubits from one quantum processor to another over long distances. This way, local quantum networks can be intra connected into a quantum internet. A quantum internet supports many applications, which derive their power from the fact that by creating quantum entangled qubits, information can be transmitted between the remote quantum processors. Most applications of a quantum internet require only very modest quantum processors. For most quantum internet protocols, such as quantum key distribution in quantum cryptography, it is sufficient if these processors are capable of preparing and measuring only a single qubit at a time. This is in contrast to quantum computing where interesting applications can be realized only if the (combined) quantum processors can easily simulate more qubits than a classical computer (around 60). Quantum internet applications require only small quantum processors, often just a single qubit, because quantum entanglement can already be realized between just two qubits. A simulation of an entangled quantum system on a classical computer cannot simultaneously provide the same security and speed.

Overview of the elements of a quantum network

The basic structure of a quantum network and more generally a quantum internet is analogous to a classical network. First, we have end nodes on which applications are ultimately run. These end nodes are quantum processors of at least one qubit. Some applications of a quantum internet require quantum processors of several qubits as well as a quantum memory at the end nodes.

Second, to transport qubits from one node to another, we need communication lines. For the purpose of quantum communication, standard telecom fibers can be used. For networked quantum computing, in which quantum processors are linked at short distances, different wavelengths are chosen depending on the exact hardware platform of the quantum processor.

Third, to make maximum use of communication infrastructure, one requires optical switches capable of delivering qubits to the intended quantum processor. These switches need to preserve quantum coherence, which makes them more challenging to realize than standard optical switches.

Finally, one requires a quantum repeater to transport qubits over long distances. Repeaters appear in between end nodes. Since qubits cannot be copied (No-cloning theorem), classical signal amplification is not possible. By necessity, a quantum repeater works in a fundamentally different way than a classical repeater.

Elements of a quantum network

End nodes: quantum processors

End nodes can both receive and emit information. Telecommunication lasers and parametric down-conversion combined with photodetectors can be used for quantum key distribution. In this case, the end nodes can in many cases be very simple devices consisting only of beamsplitters and photodetectors.

However, for many protocols more sophisticated end nodes are desirable. These systems provide advanced processing capabilities and can also be used as quantum repeaters. Their chief advantage is that they can store and retransmit quantum information without disrupting the underlying quantum state. The quantum state being stored can either be the relative spin of an electron in a magnetic field or the energy state of an electron. They can also perform quantum logic gates.

One way of realizing such end nodes is by using color centers in diamond, such as the nitrogen-vacancy center. This system forms a small quantum processor featuring several qubits. NV centers can be utilized at room temperatures. Small scale quantum algorithms and quantum error correction has already been demonstrated in this system, as well as the ability to entangle two and three quantum processors, and perform deterministic quantum teleportation.

Another possible platform are quantum processors based on ion traps, which utilize radio-frequency magnetic fields and lasers. In a multispecies trapped-ion node network, photons entangled with a parent atom are used to entangle different nodes. Also, cavity quantum electrodynamics (Cavity QED) is one possible method of doing this. In Cavity QED, photonic quantum states can be transferred to and from atomic quantum states stored in single atoms contained in optical cavities. This allows for the transfer of quantum states between single atoms using optical fiber in addition to the creation of remote entanglement between distant atoms.

Communication lines: physical layer

Over long distances, the primary method of operating quantum networks is to use optical networks and photon-based qubits. This is due to optical networks having a reduced chance of decoherence. Optical networks have the advantage of being able to re-use existing optical fiber. Alternately, free space networks can be implemented that transmit quantum information through the atmosphere or through a vacuum.

Fiber optic networks

Optical networks using existing telecommunication fiber can be implemented using hardware similar to existing telecommunication equipment. This fiber can be either single-mode or multi-mode, with single-mode allowing for more precise communication. At the sender, a single photon source can be created by heavily attenuating a standard telecommunication laser such that the mean number of photons per pulse is less than 1. For receiving, an avalanche photodetector can be used. Various methods of phase or polarization control can be used such as interferometers and beam splitters. In the case of entanglement based protocols, entangled photons can be generated through spontaneous parametric down-conversion. In both cases, the telecom fiber can be multiplexed to send non-quantum timing and control signals.

In 2020 a team of researchers affiliated with several institutions in China has succeeded in sending entangled quantum memories over a 50-kilometer coiled fiber cable.

Free space networks

Free space quantum networks operate similar to fiber optic networks but rely on line of sight between the communicating parties instead of using a fiber optic connection. Free space networks can typically support higher transmission rates than fiber optic networks and do not have to account for polarization scrambling caused by optical fiber. However, over long distances, free space communication is subject to an increased chance of environmental disturbance on the photons.

Free space communication is also possible from a satellite to the ground. A quantum satellite capable of entanglement distribution over a distance of 1,203 km has been demonstrated. The experimental exchange of single photons from a global navigation satellite system at a slant distance of 20,000 km has also been reported. These satellites can play an important role in linking smaller ground-based networks over larger distances. In free-space networks, atmospheric conditions such as turbulence, scattering, and absorption present challenges that affect the fidelity of transmitted quantum states. To mitigate these effects, researchers employ adaptive optics, advanced modulation schemes, and error correction techniques. The resilience of QKD protocols against eavesdropping plays a crucial role in ensuring the security of the transmitted data. Specifically, protocols like BB84 and decoy-state schemes have been adapted for free-space environments to improve robustness against potential security vulnerabilities.

Repeaters

Long-distance communication is hindered by the effects of signal loss and decoherence inherent to most transport mediums such as optical fiber. In classical communication, amplifiers can be used to boost the signal during transmission, but in a quantum network amplifiers cannot be used since qubits cannot be copied – known as the no-cloning theorem. That is, to implement an amplifier, the complete state of the flying qubit would need to be determined, something which is both unwanted and impossible.

Trusted repeaters

An intermediary step which allows the testing of communication infrastructure are trusted repeaters. Importantly, a trusted repeater cannot be used to transmit qubits over long distances. Instead, a trusted repeater can only be used to perform quantum key distribution with the additional assumption that the repeater is trusted. Consider two end nodes A and B, and a trusted repeater R in the middle. A and R now perform quantum key distribution to generate a key ${\displaystyle k_{AR}}$. Similarly, R and B run quantum key distribution to generate a key ${\displaystyle k_{RB}}$. A and B can now obtain a key ${\displaystyle k_{AB}}$ between themselves as follows: A sends ${\displaystyle k_{AB}}$ to R encrypted with the key ${\displaystyle k_{AR}}$. R decrypts to obtain ${\displaystyle k_{AB}}$. R then re-encrypts ${\displaystyle k_{AB}}$ using the key ${\displaystyle k_{RB}}$ and sends it to B. B decrypts to obtain ${\displaystyle k_{AB}}$. A and B now share the key ${\displaystyle k_{AB}}$. The key is secure from an outside eavesdropper, but clearly the repeater R also knows ${\displaystyle k_{AB}}$. This means that any subsequent communication between A and B does not provide end to end security, but is only secure as long as A and B trust the repeater R.

Quantum repeaters

Diagram for quantum teleportation of a photon

A true quantum repeater allows the end to end generation of quantum entanglement, and thus – by using quantum teleportation – the end to end transmission of qubits. In quantum key distribution protocols one can test for such entanglement. This means that when making encryption keys, the sender and receiver are secure even if they do not trust the quantum repeater. Any other application of a quantum internet also requires the end to end transmission of qubits, and thus a quantum repeater.

Quantum repeaters allow entanglement and can be established at distant nodes without physically sending an entangled qubit the entire distance.

In this case, the quantum network consists of many short distance links of perhaps tens or hundreds of kilometers. In the simplest case of a single repeater, two pairs of entangled qubits are established: ${\displaystyle |A⟩}$ and ${\displaystyle |R_a⟩}$ located at the sender and the repeater, and a second pair ${\displaystyle |R_b⟩}$ and ${\displaystyle |B⟩}$ located at the repeater and the receiver. These initial entangled qubits can be easily created, for example through parametric down conversion, with one qubit physically transmitted to an adjacent node. At this point, the repeater can perform a Bell measurement on the qubits ${\displaystyle |R_a⟩}$ and ${\displaystyle |R_b⟩}$ thus teleporting the quantum state of ${\displaystyle |R_a⟩}$ onto ${\displaystyle |B⟩}$. This has the effect of "swapping" the entanglement such that ${\displaystyle |A⟩}$ and ${\displaystyle |B⟩}$ are now entangled at a distance twice that of the initial entangled pairs. It can be seen that a network of such repeaters can be used linearly or in a hierarchical fashion to establish entanglement over great distances.

Hardware platforms suitable as end nodes above can also function as quantum repeaters. However, there are also hardware platforms specific only to the task of acting as a repeater, without the capabilities of performing quantum gates.

Error correction

Main article: Quantum error correction

Error correction can be used in quantum repeaters. Due to technological limitations, however, the applicability is limited to very short distances as quantum error correction schemes capable of protecting qubits over long distances would require an extremely large amount of qubits and hence extremely large quantum computers.

Errors in communication can be broadly classified into two types: Loss errors (due to optical fiber/environment) and operation errors (such as depolarization, dephasing etc.). While redundancy can be used to detect and correct classical errors, redundant qubits cannot be created due to the no-cloning theorem. As a result, other types of error correction must be introduced such as the Shor code or one of a number of more general and efficient codes. All of these codes work by distributing the quantum information across multiple entangled qubits so that operation errors as well as loss errors can be corrected.

In addition to quantum error correction, classical error correction can be employed by quantum networks in special cases such as quantum key distribution. In these cases, the goal of the quantum communication is to securely transmit a string of classical bits. Traditional error correction codes such as Hamming codes can be applied to the bit string before encoding and transmission on the quantum network.

Entanglement purification

Main article: Entanglement distillation

Quantum decoherence can occur when one qubit from a maximally entangled bell state is transmitted across a quantum network. Entanglement purification allows for the creation of nearly maximally entangled qubits from a large number of arbitrary weakly entangled qubits, and thus provides additional protection against errors. Entanglement purification (also known as Entanglement distillation) has already been demonstrated in Nitrogen-vacancy centers in diamond.

Applications

A quantum internet supports numerous applications, enabled by quantum entanglement. In general, quantum entanglement is well suited for tasks that require coordination, synchronization or privacy.

Examples of such applications include quantum key distribution, clock stabilization, protocols for distributed system problems such as leader election or Byzantine agreement, extending the baseline of telescopes, as well as position verification, secure identification and two-party cryptography in the noisy-storage model. A quantum internet also enables secure access to a quantum computer in the cloud. Specifically, a quantum internet enables very simple quantum devices to connect to a remote quantum computer in such a way that computations can be performed there without the quantum computer finding out what this computation actually is (the input and output quantum states can not be measured without destroying the computation, but the circuit composition used for the calculation will be known).

Secure communications

When it comes to communicating in any form the largest issue has always been keeping these communications private. Quantum networks would allow for information to be created, stored and transmitted, potentially achieving "a level of privacy, security and computational clout that is impossible to achieve with today’s Internet."

By applying a quantum operator that the user selects to a system of information the information can then be sent to the receiver without a chance of an eavesdropper being able to accurately be able to record the sent information without either the sender or receiver knowing. Unlike classical information that is transmitted in bits and assigned either a 0 or 1 value, the quantum information used in quantum networks uses quantum bits (qubits), which can have both 0 and 1 value at the same time, being in a state of superposition. This works because if a listener tries to listen in then they will change the information in an unintended way by listening, thereby tipping their hand to the people on whom they are attacking. Secondly, without the proper quantum operator to decode the information they will corrupt the sent information without being able to use it themselves. Furthermore, qubits can be encoded in a variety of materials, including in the polarization of photons or the spin states of electrons.

Current status

Further information: Timeline of quantum computing and communication

Quantum internet

One example of a prototype quantum communication network is the eight-user city-scale quantum network described in a paper published in September 2020. The network located in Bristol used already deployed fibre-infrastructure and worked without active switching or trusted nodes.

In 2022, Researchers at the University of Science and Technology of China and Jinan Institute of Quantum Technology demonstrated quantum entanglement between two memory devices located at 12.5 km apart from each other within an urban environment.

In the same year, Physicist at the Delft University of Technology in Netherlands has taken a significant step toward the network of the future by using a technique called quantum teleportation that sends data to three physical locations which was previously only possible with two locations.

In 2024, researchers in the U.K and Germany achieved a first by producing, storing, and retrieving quantum information. This milestone involved interfacing a quantum dot light source and a quantum memory system, paving the way for practical applications despite challenges like quantum information loss over long distances.

Quantum networks for computation

In 2021, researchers at the Max Planck Institute of Quantum Optics in Germany reported a first prototype of quantum logic gates for distributed quantum computers.

Experimental quantum modems

A research team at the Max-Planck-Institute of Quantum Optics in Garching, Germany is finding success in transporting quantum data from flying and stable qubits via infrared spectrum matching. This requires a sophisticated, super-cooled yttrium silicate crystal to sandwich erbium in a mirrored environment to achieve resonance matching of infrared wavelengths found in fiber optic networks. The team successfully demonstrated the device works without data loss.

Mobile quantum networks

In 2021, researchers in China reported the successful transmission of entangled photons between drones, used as nodes for the development of mobile quantum networks or flexible network extensions. This could be the first work in which entangled particles were sent between two moving devices. Also, it has been researched the application of quantum communications to improve 6G mobile networks for joint detection and data transfer with quantum entanglement, where there are possible advantages such as security and energy efficiency.

Quantum key distribution networks

Several test networks have been deployed that are tailored to the task of quantum key distribution either at short distances (but connecting many users), or over larger distances by relying on trusted repeaters. These networks do not yet allow for the end to end transmission of qubits or the end to end creation of entanglement between far away nodes.

Major quantum network projects and QKD protocols implemented

Quantum network	Start	BB84	BBM92	E91	DPS	COW
DARPA Quantum Network	2001	Yes	No	No	No	No
SECOCQ QKD network in Vienna	2003	Yes	Yes	No	No	Yes
Tokyo QKD network	2009	Yes	Yes	No	Yes	No
Hierarchical network in Wuhu, China	2009	Yes	No	No	No	No
Geneva area network (SwissQuantum)	2010	Yes	No	No	No	Yes

DARPA Quantum Network

Starting in the early 2000s, DARPA began sponsorship of a quantum network development project with the aim of implementing secure communication. The DARPA Quantum Network became operational within the BBN Technologies laboratory in late 2003 and was expanded further in 2004 to include nodes at Harvard and Boston Universities. The network consists of multiple physical layers including fiber optics supporting phase-modulated lasers and entangled photons as well free-space links.

SECOQC Vienna QKD network

From 2003 to 2008 the Secure Communication based on Quantum Cryptography (SECOQC) project developed a collaborative network between a number of European institutions. The architecture chosen for the SECOQC project is a trusted repeater architecture which consists of point-to-point quantum links between devices where long distance communication is accomplished through the use of repeaters.

Chinese hierarchical network

In May 2009, a hierarchical quantum network was demonstrated in Wuhu, China. The hierarchical network consists of a backbone network of four nodes connecting a number of subnets. The backbone nodes are connected through an optical switching quantum router. Nodes within each subnet are also connected through an optical switch and are connected to the backbone network through a trusted relay.

Geneva area network (SwissQuantum)

The SwissQuantum network developed and tested between 2009 and 2011 linked facilities at CERN with the University of Geneva and hepia in Geneva. The SwissQuantum program focused on transitioning the technologies developed in the SECOQC and other research quantum networks into a production environment. In particular the integration with existing telecommunication networks, and its reliability and robustness.

Tokyo QKD network

In 2010, a number of organizations from Japan and the European Union setup and tested the Tokyo QKD network. The Tokyo network build upon existing QKD technologies and adopted a SECOQC like network architecture. For the first time, one-time-pad encryption was implemented at high enough data rates to support popular end-user application such as secure voice and video conferencing. Previous large-scale QKD networks typically used classical encryption algorithms such as AES for high-rate data transfer and use the quantum-derived keys for low rate data or for regularly re-keying the classical encryption algorithms.

Beijing-Shanghai Trunk Line

In September 2017, a 2000-km quantum key distribution network between Beijing and Shanghai, China, was officially opened. This trunk line will serve as a backbone connecting quantum networks in Beijing, Shanghai, Jinan in Shandong province and Hefei in Anhui province. During the opening ceremony, two employees from the Bank of Communications completed a transaction from Shanghai to Beijing using the network. The State Grid Corporation of China is also developing a managing application for the link. The line uses 32 trusted nodes as repeaters. A quantum telecommunication network has been also put into service in Wuhan, capital of central China's Hubei Province, which will be connected to the trunk. Other similar city quantum networks along the Yangtze River are planned to follow.

In 2021, researchers working on this network of networks reported that they combined over 700 optical fibers with two QKD-ground-to-satellite links using a trusted relay structure for a total distance between nodes of up to ~4,600 km, which makes it Earth's largest integrated quantum communication network.

IQNET

IQNET (Intelligent Quantum Networks and Technologies) was founded in 2017 by Caltech and AT&T. Together, they are collaborating with the Fermi National Accelerator Laboratory, and the Jet Propulsion Laboratory. In December 2020, IQNET published a work in PRX Quantum that reported a successful teleportation of time-bin qubits across 44 km of fiber. For the first time, the published work includes a theoretical modelling of the experimental setup. The two test beds for performed measurements were the Caltech Quantum Network and the Fermilab Quantum Network. This research represents an important step in establishing a quantum internet of the future, which would revolutionise the fields of secure communication, data storage, precision sensing, and computing.