Quantum Max-Flow Min-Cut theorem

10/03/2021
by   Nengkun Yu, et al.
0

The max-flow min-cut theorem is a cornerstone result in combinatorial optimization. Calegari et al. (arXiv:0802.3208) initialized the study of quantum max-flow min-cut conjecture, which connects the rank of a tensor network and the min-cut. Cui et al. (arXiv:1508.04644) showed that this conjecture is false generally. In this paper, we establish a quantum max-flow min-cut theorem for a new definition of quantum maximum flow. In particular, we show that for any quantum tensor network, there are infinitely many n, such that quantum max-flow equals quantum min-cut, after attaching dimension n maximally entangled state to each edge as ancilla. Our result implies that the ratio of the quantum max-flow to the quantum min-cut converges to 1 as the dimension n tends to infinity. As a direct application, we prove the validity of the asymptotical version of the open problem about the quantum max-flow and the min-cut, proposed in Cui et al. (arXiv:1508.04644 ).

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1 Introduction

The max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the total weight of the edges in a minimum cut [FF56]. This beautiful theorem has nontrivial applications in network connectivity, network reliability, security of statistical data, data mining, distributed computing, and many many more [FF10].

A tensor network associated with a graph is constructing tensors in large spaces from smaller building-block tensors. Calegari et al. introduced the quantum max-flow min-cut conjecture in [CFW10]. This conjecture studies the rank of a tensor network by analysing the maximal classical flow (or minimal cut) on the graph corresponding to the tensor network. Cui et al. considered two versions of the quantum max-flow min-cut conjecture and provided explicit counter-examples for both versions in [CFS16]. They left an open problem about the asymptotical version of the quantum max-flow min-cut. They considered an interesting scenario with parameter

as the degrees of freedom on the network’s edges by attaching dimension

maximally entangled state to each edge as ancilla for any quantum tensor network. They conjectured that the quantum max-flow becomes quantum min-cut as tends to infinity. Their intuition of the above equation is that a quantum phenomenon (quantum max-flow is strictly less than quantum min-cut) disappears in a more extensive system. Hastings proved that the quantum max-flow min-cut conjecture–version II is asymptotically true [Has16]. More precisely, Hastings showed that the ratio of the quantum max-flow to the quantum min-cut converges to 1, in the limit of large dimension . Hastings also suggested a weaker version of the quantum max-flow min-cut theorem: There exists such that version II quantum max-flow equals min-cut theorem. By verifying a numerical prediction from [Has16], Gesmundo et al. proved that there are infinitely many such that the version II quantum max-flow is strictly less than the quantum min-cut, even after attaching -dimensional ancilla in [GLW18]. Regarding quantum tensor network as an entanglement network, we studied the quantum capacities for such quantum tensor network in [CJYZ16].

This paper defines a new version of quantum maximum flow, motivated by the study of quantum capacity for the quantum repeater network with free classical communication. Due to the natural structure of quantum mechanics, the problem of finding a maximum flow for a quantum tensor network becomes integer programming with monomial constraints. We establish quantum max-flow min-cut theorem for this new definition of quantum maximum flow: For any quantum tensor network, there always exist infinitely many , such that quantum max-flow equals quantum min-cut, after attaching dimension maximally entangled state to each edge as ancilla. This theorem implies that the ratio of the newly defined quantum max-flow and quantum min-cut converges to as the dimension tends to infinity. By connecting this quantum maximum flow and the version I quantum maximum flow in [CFS16], we show that the ratio of the version I quantum maximum flow to the quantum min-cut converges to as tends to infinity.

In Section 2, we present definitions of quantum maximum flows and minimum cut. In Section 3, we first prove the existence of rational flow equals minimum cut, then use that to show the quantum maximum flow minimum cut theorem. In section 4, we provide an operational meaning by interpreting our quantum maximum flow minimum cut theorem as the existence of entanglement distribution using quantum teleportation. That interpretation implies the validity of the asymptotical version of the conjecture in [CFS16].

2 Preliminaries and Notations

In this section, we provide the definitions of the quantum maximum flows and minimum cut after recalling the max-flow min-cut theorem of classical flow networks. Then we

2.1 Classical max-flow min-cut theorem

The max-flow min-cut theorem relates two quantities: the maximum flow through a network and the minimum capacity of a cut of the network. That is, the flow achieves the minimum capacity. To state the theorem, we recall the definitions of these quantities.

Let be a directed graph, where denotes the set of vertices, and is the set of edges. Let and be the source and the sink of , respectively. We only need to consider networks with one source vertex and one sink vertex . The general case for transmitting information from to reduces to this simple case by viewing all / as one source/sink.

The capacity of an edge is a mapping denoted by where . It represents the maximum amount of flow that can pass through an edge.

The flow of a flow network is defined as follows. We will call it a classical flow.

Definition 1.

A flow is a mapping denoted by, subject to the following two constraints:

  • Capacity Constraint: For every edge , .

  • Conservation of Flows: For each vertex apart from and (i.e. the source and sink, respectively), the following equality holds:

The value of a flow is defined by

The maximum flow problem asks for the largest flow on a given network, denoted as .

The cut of a flow network is the smallest total weight of the edges which if removed would disconnect the source from the sink. We will call it a classical cut.

Definition 2.

A cut is a partition of into , where and . The cut set of the cut is the set of edges with and . For a given flow network, and a given cut set separating from , its capacity is defined as

The min-cut of the network, is the minimum capacity over all cuts.

The following result links the maximum flow through a network with the minimum cut of the network. It was published in 1956 by L. R. Ford Jr. and D. R. Fulkerson.

Theorem 1.

[FF56] The maximum value of a flow is equal to the minimum capacity overall cuts. Moreover, if the capacity for all edges in the graph are integers, then the Ford-Fulkerson method finds a max flow in which every flow value is an integer.

We call the integer part of this theorem the Flow Integrality Theorem.

2.2 Quantum maximum flows and min cut

A tensor network associated with a graph is constructing tensors in large spaces from smaller building-block tensors. Given an undirected graph with a capacity function and a resource (resp. sink ), we can define a quantum tensor network associated to as an entanglement network, where each pair of connected nodes of share a maximally entangled state .

Usually, the dimension of the maximally entangled state is chosen to be the same on each edge as studied in [Has16]. However, the more general case, where the s may be different, is also known to be interesting, which has demonstrated connections to the topological quantum field theory and the theory of quantum gravity [CFS16, RT06, HHLR14, HNQ16].

We can define the quantum minimum cut as follows.

Definition 3.

A cut is a partition of into , where and . The cut set of the cut is the set of edges with and . For a given graph, and a given cut set separating from , define to equal the product of the capacities of the edges in the cut set. In other words,

The quantum min-cut of the network, is the minimum of over all cuts.

Different from classcial min-cut, the quantum min-cut is defined as the product of the capacities of the edges in the cut set. This quantum feature can be understood by observing

An interesting type of entanglement networks we focus on is the tensor network, which transports linear algebraic things like rank and entanglement [Bau11, CFS16]. Given a network , let , and for each , let be the set of edges containing . We define , and define . One can assign a set of tensors to , where is an arbitrary tensor in , i.e., each index of corresponds to an edge containing . Then contracting the tensors along all internal edges results in a linear map . With the notations above,

Definition 4.

The quantum max-flow, , is the the maximum rank of over all tensor assignments .

One can interpret the quantum max-flow as the maximum rank of entanglement between the source and the sink that can be generated by applying stochastic local operation assisted by classical communication protocols. One can implement in stochastically.

It is proved in [CFS16] that

Theorem 2.

For any quantum tensor network ,

Unfortuantely, the equality is not generally saturated. For given quantum tensor network, they studied new networks by attaching an additional maximally entangled state on each edge.

Definition 5.

Given the network , and , we define . In other words, the capacity of each edge multuples .

Motivated by properties of frustration free local hamiltonian, Cui et al. proposed the open question

Conjecture 1.

For any quantum tensor network ,

To track this problem, we consider the quantum information transmission from the source vertex to the sink vertex through the network via local quantum operations and classical communications. More precisely, we use quantum teleportation for transmitting quantum information in the network. One can implement quantum teleportation by allowing quantum operations at each vertex, and classical communications are free. The goal is then to establish maximum bipartite entanglement between and via teleportation protocols.

Since we allow unlimited classical communications, the direction of quantum communication on each edge can be arbitrary given a maximally entangled state associated with an edge . Therefore, we allow both directions of quantum communication for any edge . The only restriction is that the product of both capacities is at most .

This motivates us to provide the following definition of the quantum flow.

Definition 6.

A flow is a map that satisfies the following:

  • Capacity constraint. The flow of an edge cannot exceed its capacity, in other words: for all .

  • Conservation of flows. The product of the flows entering a node must equal the product of the flows exiting that node, except for the source and the sink. In other words, :

    (1)

We allow the incoming flow of the source and outgoing flow of the sink. In other words, there may exists such that or .

The value of a flow is defined as follows.

Definition 7.

The value of a flow is the amount of flow passing from the source to the sink. Formally for a flow it is given by

(2)

The maximum flow problem is to route as much flow as possible from the source to the sink. In other words, the maximum flow of the network, , is the flow with maximum value.

A particular flow class with no incoming flow of the source and no outgoing flow of the sink is of great interest.

Definition 8.

A flow is a strict flow if for all . The maximum strict flow of the network, , is the flow with maximum value.

Here is undirected graph, therefore, implies .

Compute the quantum maximum flow and strict quantum maximum flow are both

To prove our main results, we need the following definitions of flow and flow. For or , we define the quantum flow over .

Definition 9.

For quantum tensor network , a flow over is a map that satisfies the following:

  • Single direction. or for all .

  • Capacity constraint. The flow of an edge cannot exceed its capacity, in other words: for all .

  • Conservation of flows. The product of the flows entering a node must equal the product of the flows exiting that node, except for the source and the sink. In other words, :

    (3)

The value of a flow it is given by

(4)

The maximum (or supremum) flow over of the network, , is the flow with maximum (or supremum) value over .

For , the constraints are all closed, so the maximum is achievable. For , the definition uses supremum.

3 Quantum maximum flow minimum cut theorem

Before presenting the main result of this paper, we first observe the following lemma.

Lemma 1.

For any flow over of the quantum tensor network , and any cut and , we have

Proof.

Let .

In the second line, we use the conservation of flows. In the third line, we use the following fact: The flow of each edge except those between and appears twice: One in the numerator and the other in the denominator. ∎

Using this lemma, we can show

Lemma 2.

For any ,

Proof.

Observe that a flow over is also a flow over , we know that

To bound , we choose a minimum cut with . Using Lemma 1, we know that any flow over satsifies

where in the last step, we use the Single direction condition and the capacity constraint of a flow: if , then ; otherwise , then . ∎

This proof of the above lemma shows the following property of flows which saturates the minimum cut.

Lemma 3.

If a flow over in quantum tensor network such that , then for any minimum cut with , we have or for and .

Interestingly, there is always a rational flow to achieve the quantum minimum cut.

Lemma 4.

For any quantum tensor network ,

Moreover, there is a rational flow which achieves the maximum flow.

Proof.

We first prove that

To show this, we only need to provide a flow over satisfies . We consider a classical flow network from quantum tensor network with undirected graph : is a directed graph where

  • ,

  • the capacity .

In other words, the source has only outgoing edges, the sink has only incoming edges, and other edges in are duplicated in both directions.

For any cut of , with and as defined in Definition 2. The value of this cut is

It is clear to observe that the following relation between the min-cut of and .

We consider classical flow over as Definition 1. Theorem 1, the max-flow min-cut theorem, implies that there exists a flow such that

It is possible that there exist such that both . We can simplify the flow to as following: For such that both , we choose to let and . We have

Now, we have for all , or .

Define where . According to the properties of , we know that and ,

In other words, is a flow over in quantum tensor network and satisfies

If is also a rational flow, then the statement of this lemma is already true.

Otherwise, there exists at least one such that . We observe that there exists such that or . To see this, we consider three cases:

Case 1: : Since is a flow, according to the conservation condition on

(5)

The product of the right handside contains . If the product , then there is an such that ; If the product , then there is an such that .

Case 2: . According to

there exists such that .

Case 3: . According to Lemma 1, we have

Thus, there exists such that .

By repreating this procedure, we can obtain a circle since the graph is finite. More precisely, we have

For any , we can define by

and leaving the rest of unchanged. By using this method to modify and for each , we can obtain a flow satisfies

  • or for all .

  • for all .

  • holds .

  • .

  • such that for all .

where . In otherwords, we allow incoming edges of and outgoing edges of . According to lemma 3, we know that any edge of are not in any minmimum cut.

Then, for all .

By choose some such that , we design by

and leave the rest of unchange. One can verify that is still a flow over with according to Lemma 1.

Moreover, each run of this procedure decreases the number of non-rational edge asignement by at least .

By repeating this procedure, we can obtain a flow such that , and ,

This proves

Using Lemma 4, we are able to prove the following lemma.

Lemma 5.

For any , there are infinitely many such that

Proof.

For any cut , the value of the cut in is

where denotes the size of .

Then, there exists such that for any , a minimum cut of is also a minimum cut of . Let us choose a minimum cut of , denoted by . Therefore, for ,

with being the size of .

According to Lemma 4, there exists a flow satisfies and ,

Let satisfies that for all

In the following, we will prove that for any sufficient large , there exists a flow of such that

We first define a classical flow network from with if . In such flow network as Definition 1, the min-cut of is .

The integrality theorem of Theorem 1 states that if all arc capacities are integer, the maximum flow problem has an integer maximum flow.

Applying the integrality theorem on , we know that there is a flow with value . We can first simplify that into an equivalent flow . In other words, we can choose disjoint paths, consisting of undirected edges, from to where we call two paths are disjoint if they do not share edges. We can assign directions for these paths, from to . Now we have disjoint paths from to . We use to denoted the set of directed edges in these paths,

We use the information of to update the flow .

We know that for any directed edge , if , we can change the assignement of and by letting , and and do not change the rest. is still a rational flow and .

For any sufficiently large , we construct a flow of .

If the directed edge , we let

For , and , , we let

One can cerify that is a flow by observing

  • since .

  • Capacity constraint: and .

  • The conservation condition is preserved during the process: for .

The conservation condition follows from the following observation: If is not in , then the conservation condition follows from the conservation condition of ; If is in , then

can be obtained from

by multiplying from each handsides for approperate integer due to the face that is consisting of directed paths.

One can verify that

This complete the proof. ∎

Lemma 6.

Given quantum tensor network , if there exist a flow such that

then there exists a strict flow such that

Proof.

The difference between flow and strict flow is that in the strict flow, there is no incoming flow of the source and no outgoing flow of the sink .

Let us fix a minimum cut of , denoted by . According to Lemma 1,