UUnconstrained Summoning for relativistic quantum information processing
Adrian Kent
Centre for Quantum Information and Foundations, DAMTP, Centre for Mathematical Sciences,University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, U.K. andPerimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada. (Dated: December 18, 2018)We define a summoning task to require propagating an unknown quantum state to a point in space-timebelonging to a set determined by classical inputs at points in space-time. We consider the classical analogue, inwhich a known classical state must be returned at precisely one allowed point. We show that, when the inputsare unconstrained, any summoning task that is possible in the classical case is also possible in the quantum case.
INTRODUCTION
In the not necessarily distant future, many or most significant economic decisions may depend on algorithms that processinformation arriving around the world as efficiently and quickly as possible, given light speed signalling constraints. Suchalgorithms will need to decide not only whether to make a trade, but also when and where. Quantum money enables efficienttransactions, by allowing a delocalized quantum money token effectively to be “summoned” to a given space-time point asa result of information distributed through space-time. There are also alternative technological solutions [1] in the form oftoken schemes that do not require long term quantum state storage and in some embodiments are based entirely on classicalcommunication.This paper defines types of summoning task that naturally arise in relativistic economies, and shows that for an importantand natural class of these tasks the two technologies have the same functionality. That is, if quantum money can be summonedto the correct space-time point in a given scenario, then so can classical tokens, and conversely. This has implications for thefoundations of relativistic quantum theory, since summoning has proved a very fruitful way of understanding the properties ofquantum information in space-time [2–5] and we also discuss these.
SUMMONING
Summoning [2, 3] is a task defined between two parties, Alice and Bob, who each have networks of collaborating agents.Each party trusts their own agents but not those of the other party. Summoning tasks discussed in the literature to date take thefollowing form. In some fixed background space-time, Bob creates a quantum state (say, a qudit) in pre-agreed physical form,keeping its classical description private among his agents. One of his agents gives the state to one of Alice’s at some point P .At one of a number of pre-agreed later points { c i } i ∈ I , Bob’s local agent may ask for the state back from Alice’s local agent. Thepre-agreed set I may be finite or infinite; the original discussion of the task [2] allowed I to contain every point in the future of P and also considered the possibility that I contains just two space-like separated points.Alice must then return the state at some point or within some region related to c i in a pre-agreed way. For example, in oneversion considered in Ref. [3], a request requires the state to be returned at the same point in space at a slightly later time,with respect to a given frame. In Ref. [4], this was generalized so that a request at c i requires the state to be returned at somepre-agreed point r i ≻ c i . We say such a summoning task is possible in relativistic quantum theory if there is an algorithmthat guarantees that Alice will comply with any allowed request (assuming she has ideal devices and instantaneous computingoperations). Alice is allowed to return only a single state (here a qudit), so that compliance means guaranteeing that this is thestate originally received.Another interesting version of summoning allows any non-zero number of valid return points [5]. In this case we say the taskis possible if there is an algorithm that guarantees that Alice will return the state to some valid return point. For example, onecan consider a variant of Hayden-May’s version of summoning in which requests may be made at any non-zero number of callpoints { c i } i ∈ I ′ , where I ′ ⊆ I , and the state must be returned at one of the corresponding return points, i.e. at some r i where i ∈ I ′ . Interestingly, this is a strictly harder version of the task [5]. EXTENDING THE DEFINITION OF SUMMONING
For definiteness, we focus on the case of Minkowski space-time, which is a good local approximation in the region of the solarsystem. All the variants of summoning considered to date [2–5] are examples of a general class of quantum tasks in Minkowski a r X i v : . [ qu a n t - ph ] D ec space-time [3], in which Alice receives some number of classical and/or quantum inputs at a set of points in space-time and mustproduce classical and/or quantum outputs at another (possibly identical or overlapping) set of points, where the output points andinformation depend in some prescribed way on the input points and information. For the summoning tasks considered in Refs.[4, 5], the classical information consists of single bits ( or corresponding respectively to “call” or “no call”) at a finite set of“call points” { c i } ni = , with a corresponding set of “return points” { r i } ni = . An input at c i gives the instruction that returning thestate at r i is required [4] or is one of the required set of possibilities [5]. The summoning tasks considered in Refs. [2, 3] alsohave single bit inputs. However, more general types of input and more general rules for defining allowed return points are alsointeresting. The key feature common to all summoning tasks considered to date is that an unknown quantum state is suppliedand must be returned at some later point belonging to a set that is identified by information supplied during the task. It is naturalto extend the term summoning to cover all such tasks.We focus here on the case in which all the information supplied, other than the original unknown state, is classical, the sets ofinput points and possible return points are both finite and known in advance, and finite bounds on the information to be suppliedat each input point are also known in advance. Our terminology here is to be understood modulo these restrictions, so thatwe speak here of “summoning” rather than “finite-input-point bounded-classical-input finite-output-point summoning”. This ispurely for brevity: more general types of summoning are also interesting. The bounds on classical inputs are imposed to simplifythe notation; our results extend to general classical inputs, including unbounded integer or real number inputs.We thus define a summoning task with one return point in a given space-time to be a task set by one party, Bob, for anotherparty, Alice, who has an arbitrarily dense network of collaborating agents. Bob gives Alice (i.e. Bob’s local agent gives Alice’slocal agent) an unknown quantum state at the start point P . Alice must produce the state at some return point Q ≻ P . Here Q ∈ { Q , . . . , Q N } . Its identity depends, according to pre-agreed rules, on classical information supplied at some set of inputpoints { P i } Mi = . The point P and the sets { P i } and { Q i } are known to both parties in advance of the task. The sets may overlap,and may include P . We take the sets of input and return points to be finite, although the infinite case is also interesting. Theclassical information Bob supplies at P i is an integer m i in the range ≤ m i ≤ ( n i − ) . Alice must return the state to a point Q ( m , . . . , m M ) ∈ { Q , . . . , Q N } . Alice knows the functional dependence of Q and the set { n i } Mi = in advance of the task; shelearns the values of the m i only at the points P i . To exclude trivial cases that complicate the notation, we assume that everyreturn point may be designated by some set of inputs: i.e. for each i there is at least one set of allowed inputs m , . . . , m M such that Q i = Q ( m , . . . , m M ) . We also assume that every input takes more than one possible value, i.e. that n i ≥ foreach i . We say the task is possible if, given unlimited predistributed classical and/or quantum resources, ideal devices andinstantaneous classical and quantum computational power, there is an algorithm by which Alice can guarantee to return the stateto Q ( m , . . . , m M ) for any allowed set of inputs { m , . . . , m M } .We define a summoning task with at most one return point similarly, allowing that Q ( m , . . . , m M ) may be any point inthe pre-agreed set { Q i } Ni = or may be the empty set ∅ , in which case the state should not be returned at any point. Such tasksare possible only if there is an algorithm by which Alice can guarantee that the state is returned to a point Q i if and only if Q i = Q ( m , . . . , m M ) , and is not returned anywhere if Q ( m , . . . , m M ) = ∅ .We may also define a summoning task with multiple return points , for which Q ( m , . . . , m M ) may be any subset of { Q i } Mi = ,and ∣ Q ( m , . . . , m M )∣ > for at least one set of inputs.[12] Here we assume that for each i there is at least one set of allowedinputs m , . . . , m M such that Q i ∈ Q ( m , . . . , m M ) . Such tasks are possible only if the algorithm guarantees that the stateis returned to a point Q i ∈ Q ( m , . . . , m M ) if and only if Q ( m , . . . , m M ) is non-empty, and is not returned anywhere if Q ( m , . . . , m M ) = ∅ .We distinguish between summoning tasks with constrained inputs and those with unconstrained inputs . In the former, Alice isguaranteed that some non-trivial constraint holds on the possible inputs. That is, there is at least one set of inputs { m , . . . , m M } ,in the prescribed ranges ≤ m i ≤ ( n i − ) , that is guaranteed never to arise. Hayden-May [4] define a summoning task withconstrained inputs, since only one input is allowed. The version of multi-call summoning [5] in which it is also allowed thatno call may be made at any call point is a task with unconstrained inputs. CLASSICAL SUMMONING TASKS
It is also interesting to consider classical versions of these summoning tasks. Here Alice is given a classical state at point P . We assume she can perfectly determine its classical description instantaneously, broadcast the description everywhere atlight speed, and reconstruct a perfect copy instantaneously anywhere she receives the broadcast description. She thus may makearbitrarily many perfect copies of the state at P or anywhere within its causal future. At each point Q j ( j ∈ J ) she must declareif she is returning the state; if she does, she must supply a copy of it. We assume she only has classical resources. We saythe task is classically possible if there is an algorithm which guarantees that she will return a copy of the state at precisely onevalid return point (if there are any) and at no other point. (So, if there are no valid return points, she does not return the stateanywhere.) xt 𝑃 𝑃 𝑃 𝑀 𝑚 𝑚 𝑄 𝑄 𝑄 𝑄 𝑄 𝑁 𝑃 𝑚 𝑚 𝑀 P FIG. 1: A summoning task with at most one return point. Alice is given an unknown quantum state at the point P and classical data inthe form of integers m i at the points P i . The dashed lines indicate the future light cone of P . She is required to send the quantum state to Q ( m , . . . , m M ) ∈ { Q . . . Q M } , or to retain it if Q ( m , . . . , m M ) = ∅ . Not all classically possible summoning tasks are also possible when a quantum state is being summoned. For example, in theoriginal setting for the no-summoning theorem [2], Alice knows that precisely one call will be made, but does not know where.If she is required to return a classical state, she may simply broadcast its description everywhere, and return it in response to thecall, wherever it is made. The same is true of Hayden-May summoning [4]. So long as all the return points are in the causalfuture of the start point, and the call points are in the causal past of each return point, Alice knows at each return point whetheror not the state should be returned. Again, she may simply broadcast the description of the classical state to all the return points,and return it in response to the call, regardless of the relationship between the causal diamonds defined by the call and returnpoints. In both cases, the task is thus classically possible, but quantum summoning is not generally possible.These examples show that classical summoning may be possible while quantum summoning is impossible for a summoningtask with constrained inputs. We now focus on summoning tasks with unconstrained inputs.
SUMMONING TASKS WITH UNCONSTRAINED INPUTS AND AT MOST ONE RETURN POINT
Consider now a classically possible summoning task with unconstrained inputs and at most one return point.
Lemma 1.
Each return point Q i is in the causal future of the start point P , i.e. Q i ⪰ P for each i .Proof. Each return point Q i may be designated by some set of allowed inputs. If Q i is so designated, Alice must propagate thequdit from P to Q i . Unless Q i ⪰ P , this would require superluminal signalling. ∎ Lemma 2.
For every pair of return points ( Q i , Q j ) the set of common past input points S ij = { P k ∶ P k ⪯ Q i & P k ⪯ Q j } isnon-empty.Proof. Both Q i and Q j are designated return points for some (different) sets of inputs. Since the task is classically possible, theinputs at points in S i = { P k ∶ P k ⪯ Q i } must determine whether or not Q i is a (and hence the) valid return point. Since both Q i and Q j are valid return points for some sets of inputs, and the task is classically possible, S ij cannot be empty, otherwise somesets of inputs on S i and S j would be consistent with both Q i and Q j being valid return points. ∎ Lemma 3.
For every pair of return points Q i , Q j , any possible set of inputs at their common past input points S ij must logicallyexclude at least one of the pair as the designated return point.Proof. If there is a set of inputs at points in S ij that is consistent with both Q i and Q j being a valid return point, these inputsmust form part of a complete set of inputs that is consistent with both Q i and Q j being valid return points, since the inputs areunconstrained and it must be knowable at each Q k whether Q k is a valid return point. This contradicts the assumption of at mostone valid return point. ∎ The next result relies on the technique of distributed non-local computation [6], which relies on using pre-shared entangle-ment to implement a series of teleportation operations, with the classical teleportation data broadcast so that it is available toreconstruct the required quantum state at the appropriate point.In the simplest example, Alice is given a quantum state ψ at point P and inputs m , m at points P , P respectively, andis required to return ψ at point Q ( m , m ) ∈ { Q , Q } , where the function Q ( m , m ) is known in advance. Suppose that Q i ⪰ P and Q i ⪰ P j for i = , and j = , . She may then carry out a teleportation operation on ψ at P using a predistributedentangled state shared between P and (say) P , broadcasting the classical teleportation data. This produces a quantum state ρ at P from which ψ may later be reconstructed. Now suppose that Alice has pre-shared n labelled entangled states between P and P . At P , she carries out a teleportation operation on ρ , using the entangled state with label m , and broadcasts theclassical teleportation data and the value of m . At P , she sends the entangled state with label m to the point Q ( m, m ) , andbroadcasts the value of m . At Q and Q , she receives both m and m , and so knows which is the required return point. Shealso receives all the relevant teleportation data and the required quantum state at that return point, and so can reconstruct andreturn ψ there.By iterating this technique, we obtain the next lemma. This requires a large amount of pre-shared entanglement in exampleswith many call points; we are concerned here only with feasibility in principle, rather than resource optimization. Lemma 4.
If a state ψ ij is initially located at the start point P , it may be propagated in such a way that it arrives at Q i if theinputs on S ij preclude Q j (but not Q i ) as a valid return point and at Q j if the inputs on S ij preclude Q i (but not Q j ).Proof. A non-local computation [6], taking into account the inputs at all points P k ∈ S ij , can be carried out by iterative tele-portations from P to one of these points and through a complete sequence of the remaining points, such that the teleportationoutput containing the quantum information linked with ψ ij is propagated appropriately to Q i or Q j . We have that P ⪯ Q i and P k ⪯ Q i for all P k ∈ S ij , and similarly for Q j . The state may thus be reconstructed at the relevant return point from the classicalteleportation data. ∎ This algorithm could be extended to give rules as to where the state goes if both Q i and Q j are precluded; however this isirrelevant for our purposes.The next lemma uses techniques of and results about quantum secret sharing, developed in Refs. [7, 8]. A quantum secretsharing scheme for quantum states in a given Hilbert space H is defined by a quantum operation A ∶ H → H ⊗ . . . ⊗ H n from H to a tensor product of component Hilbert spaces H i , together with quantum operations A S ∶ ⊗ i ∈ S H i → H for some subsets S ⊆ { , . . . , n } , such that A S ⋅ A = I . The access structure of the secret sharing scheme is the list L of subsets S for whichsuch an operation A S is defined. Effectively, a quantum secret sharing scheme allows an unknown quantum state ψ to be sharedamong n parties in such a way that any subset S of them belonging to the access structure may reconstruct ψ . Quantum secretsharing schemes can be constructed with any access structure that is monotonic (so that if S ∈ L and S ⊆ S then S ∈ L ) anddoes not violate the no-cloning theorem (so that if S ∩ S = ∅ and S ∈ L then S ∉ L ). Here again we are presently interestedonly in whether a scheme exists in principle; we do not consider resource optimization. Lemma 5.
There is a quantum secret sharing scheme for a general state ψ , with components labelled by { ψ ij } ≤ i < j ≤ N , with theproperty that ψ may be reconstructed from any subset of the form { ψ ij } ≤ j ≤ N ; j ≠ i , where we set ψ ji = ψ ij if j > i .Proof. No pair of these subsets are disjoint, and so they satisfy the conditions to generate an access structure for a quantumsecret sharing scheme [7, 8]. ∎ Theorem 1.
A classically possible summoning task with unconstrained inputs and at most one return point is also a possiblequantum summoning task.Proof.
This follows by construction from the preceding lemmas. ∎ SUMMONING TASKS WITH UNCONSTRAINED INPUTS AND MULTIPLE RETURN POINTS
Consider now a classically possible summoning task with unconstrained inputs and multiple return points. Since the task isclassically possible, there must be some classical algorithm that allows Alice to decide, based on the inputs, to exclude returningat one of any pair Q i and Q j of return points when both are valid. Without loss of generality, we may assume this algorithmis deterministic, since Alice has only classical resources, and any classical randomness in a probabilistic algorithm may beprecomputed and predistributed. The algorithm must be consistent: i.e. it must identify a valid return point when there is one.The determination of whether Q i or Q j is excluded can only depend on the inputs at points in S ij , by causality.We can thus incorporate the algorithm within a refined definition of the task, producing a summoning task with unconstrainedinputs and at most one return point. We may delete any return points that are never used by the algorithm. This defines a possiblequantum summoning task, by the previous discussion. Hence we have: Theorem 2.
A classically possible summoning task with unconstrained inputs and multiple return points is also a possiblequantum summoning task.
CLASSICAL AND QUANTUM POSSIBILITY
We have shown that classically possible summoning tasks with unconstrained inputs are also possible quantum summoningtasks, whether they have at most one return point or multiple return points.Conversely, consider a quantum summoning task that can be solved with a deterministic quantum algorithm, by which wemean an algorithm that always returns the state to the same return point Q ( m , . . . , m M ) for a given set of inputs m , . . . , m M .By including appropriate ancillae, we can describe any such quantum algorithm as a deterministic sequence of unitary operations,in which unitaries U Q act on the collective state at point Q and propagate outputs along secure channels to further points R i ≻ Q .All the operations that might be performed in this algorithm can be described classically: “apply unitary U Q at Q ”, “prepare state φ at Q ′ ”, and so on, where each operation depends in a prescribed way on inputs received at the relevant point or in its causalpast. If the algorithm is deterministic in the sense above, then the success of a summoning task that is supposed to propagatea state from the start point P to a valid return point Q j is determined by, and deducible from, the subset of these operationsapplied in the past light cone of Q j .Hence any deterministic quantum algorithm that guarantees success may be simulated by classical communications, describ-ing the relevant operations and state preparations, broadcast from the corresponding space-time points. This broadcast simulationallows Alice’s agent at the valid return point Q j (if there is a valid return point), to deduce that the quantum algorithm wouldhave returned the quantum state there, and thus that she should return a copy of the classical state there. It also allows Alice’sagents at all other return points Q k ≠ Q j to deduce that the quantum algorithm would not have returned the quantum state attheir locations, and hence that they should not return copies of the classical states. This gives us: Theorem 3.
A quantum summoning task with unconstrained inputs and multiple return points that can be solved with a deter-ministic quantum algorithm is also classically possible.
DISCUSSION
Summoning can be thought of as a type of distributed quantum computation, or more generally a sub-routine within such acomputation, in which quantum states need to be propagated in response to incoming classical information. This informationcould come from nature (for example detected photon fluxes near given spacetime points), from human activity (for examplelocal market prices at given points in time on a distributed financial network), or as outputs from other computations or otherparts of the same computation (which may themselves use natural and/or human-made inputs).Our results further illustrate (cf. [3–6, 9–11]) the power of quantum information in a relativistic context. Roughly speaking,they show that if a classical observer at a given spacetime point can know, from available classical data that has no knownconstraints, that a given unknown quantum state should have been propagated to them, then there was an infallible algorithmthat could have done so, and vice versa. We take this as further support for viewing summoning as a key primitive of relativisticquantum information theory.The algorithm defined is almost certainly far from optimal for most interesting unconstrained summoning tasks. It would bevery interesting to understand better how to optimize the use of entanglement and other resources for these tasks. One strongmotivation for doing so comes from financial and other applications of distributed algorithms on networks where relativisticsignalling constraints are significant. As noted earlier, token or money schemes that prevent illegitimate duplication can inprinciple be based on quantum money, but can also be implemented by alternative techniques that require no long term quantumstate storage, and in some cases no quantum information processing at all [1]. Unconstrained summoning tasks are naturalproblems in this context: a market agent wants to be able to respond as flexibly and fast as possible to incoming market dataacross the network, and to be able to present their money token at the optimal point in space-time, according to some appropriatefinancial metric. An unconstrained quantum summoning scheme in which quantum money tokens are summoned and presentedis an elegant theoretical solution. However if, as seems very plausible, such schemes generally require unfeasible amounts ofentanglement to solve realistic problems, then rival technologies [1] are likely to prove advantageous.
Acknowledgments
This work was partially supported by UK Quantum Communications Hub grant no. EP/M013472/1and by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. [1] Adrian Kent. Summonable supermoney: virtual tokens for a relativistic economy. arXiv preprint arXiv:1806.05884 , 2018.[2] Adrian Kent. A no-summoning theorem in relativistic quantum theory.
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SIAM Journal on Computing , 43(1):150–178, 2014.[12] A fully exhaustive terminology would again distinguish between the cases where Q ( m , . . . , m M ))