The resource theory of entanglement BDSW96 has been one of the richest contributions to quantum information theory H12 ; H06book ; W17 ; Wat16 , and these days, the seminal ideas coming from it are influencing diverse areas of physics CG18 . A fundamental question in entanglement theory is to determine the smallest rate at which Bell states (or ebits) are needed, along with the assistance of free classical communication, in order to generate copies of an arbitrary bipartite state reliably (in this introduction, should be understood to be an arbitrarily large number) BDSW96 . The optimal rate is known as the entanglement cost of BDSW96 , and a formal expression is known for this quantity in terms of a regularization of the entanglement of formation HHT01 . An upper bound in terms of entanglement of formation has been known for some time BDSW96 ; HHT01 , while a lower bound has been determined recently WD17 . Conversely, a related fundamental question is to determine the largest rate at which one can distill ebits reliably from copies of , again with the assistance of free classical communication BDSW96 . This optimal rate is known as the distillable entanglement, and various lower bounds DW05 and upper bounds Rai99 ; Rai01 ; CW04 ; WD16pra are known for it.
The above resource theory is quite rich and interesting, but soon after learning about it, one might immediately question its operational significance. How are the bipartite states established in the first place? Of course, a quantum communication channel, such as a fiber-optic or free-space link, is required. Consequently, in the same paper that introduced the resource theory of entanglement BDSW96 , the authors there appreciated the relevance of this point and proposed that the distillation question could be extended to quantum channels. The distillation question for channels is then as follows: given uses of a quantum channel connecting a sender Alice to a receiver Bob, along with the assistance of free classical communication, what is the optimal rate at which these channels can produce ebits reliably BDSW96 ? By invoking the teleportation protocol BBC+93
and the fact that free classical communication is allowed, this rate is also equal to the rate at which arbitrary qubits can be reliably communicated by using the channeltimes BDSW96 . The optimal rate is known as the distillable entanglement of the channel BDSW96 , and various lower bounds DW05 and upper bounds TGW14 ; TGW14b ; W16 ; BW17 are now known for it, strongly related to the bounds for distillable entanglement of states, as given above.
Some years after the distillable entanglement of a channel was proposed in BDSW96 , the question converse to it was proposed and addressed in BBCW13 . The authors of BBCW13 defined the entanglement cost of a quantum channel as the smallest rate at which entanglement is required, in addition to the assistance of free classical communication, in order to simulate uses of . Key to their definition of entanglement cost is the particular notion of simulation considered. In particular, the goal of their simulation protocol is to simulate parallel uses of the channel, written as . Furthermore, they considered a simulation protocol to have the following form:
where is an arbitrary input state, is a free channel, whose implementation is restricted to consist of local operations and classical communication (LOCC) BDSW96 ; CLM+14 , and is a maximally entangled resource state. For , the simulation is then considered -distinguishable from if the following condition holds
where denotes the diamond norm Kit97 . The physical meaning of the above inequality is that it places a limitation on how well any discriminator can distinguish the channel from the simulation in a guessing game. Such a guessing game consists of the discriminator preparing a quantum state , the referee picking or at random and then applying it to the systems of , and the discriminator finally performing a quantum measurement on the systems . If the inequality in (2
) holds, then the probability that the discriminator can correctly distinguish the channel from its simulation is bounded from above by, regardless of the particular state and final measurement chosen for his distinguishing strategy Kit97 ; H69 ; H73 ; Hel76 . Thus, if is close to zero, then this probability is not much better than random guessing, and in this case, the channels are considered nearly indistinguishable and the simulation thus reliable.
In parallel to the above developments in entanglement theory, there have indubitably been many advances in the theory of quantum channel discrimination CDP08b ; CDP08a ; GDP09 ; DFY09 ; Harrow10 ; CMW16 and related developments in the theory of quantum interactive proof systems GW07 ; G09 ; G12 ; GAJ18 . Notably, the most general method for distinguishing a quantum memory channel from another one consists of a quantum-memory-assisted discrimination protocol CDP08a ; GDP09 . In the language of quantum interactive proof systems, memory channels are called strategies and memory-assisted discrimination protocols are called co-strategies GW07 ; G09 ; G12 . For a visual illustration of the physical setup, please consult (CDP08a, , Figure 2) or (GW07, , Figure 2). In subsequent work after GW07 ; CDP08a , a number of theoretical results listed above have been derived related to memory channel discrimination or quantum strategies.
The aforementioned developments in the theory of quantum channel discrimination indicate that the notion of channel simulation proposed in BBCW13 is not the most general notion that could be considered. In particular, if a simulator is claiming to have simulated uses of the channel , then the discriminator should be able to test this assertion in the most general way possible, as given in GW07 ; CDP08a ; GDP09 . That is, we would like for the simulation to pass the strongest possible test that could be performed to distinguish it from the uses of . Such a test allows for the discriminator to prepare an arbitary state , call the first channel use or its simulation, apply an arbitrary channel , call the second channel use or its simulation, etc. After the th call is made, the discriminator then performs a joint measurement on the remaining quantum systems. See Figure 1 for a visual depiction. If the simulation is good, then the probability for the discriminator to distinguish the channels from the simulation should be no larger than , for small .
In this paper, I propose a new definition for the entanglement cost of a channel , such that it is the smallest rate at which ebits are needed, along with the assistance of free classical communication, in order to simulate uses of , in such a way that a discriminator performing the most stringest test, as described above, cannot distinguish the simulation from actual calls of (Section II.2). Here I denote the optimal rate by , and the prior quantity defined in BBCW13 by , given that the simulation there was only required to pass a less stringent parallel discrimination test, as discussed above. Due to the fact that it is more difficult to pass the simulation test as specified by the new definition, it follows that (discussed in more detail in what follows). After establishing definitions, I then prove a general upper bound on the entanglement cost of a quantum channel, using the notion of teleportation simulation (Section III.1). I prove that the entanglement cost of certain “resource-seizable,” teleportation-simulable channels takes on a particularly simple form (Section III.2), which allows for concluding single-letter formulas for the entanglement cost of dephasing, erasure, three-dimensional Werner–Holevo channels, and epolarizing channels (complements of depolarizing channels), as detailed in Section IV. Note that the result about entanglement cost of dephasing channels solves an open question from BBCW13 . I then extend the results to the case of bosonic Gaussian channels (Section V), proving single-letter formulas for the entanglement cost of fundamental channel models, including pure-loss and pure-amplifier channels (Theorem 2 in Section V.7). These examples lead to the conclusion that the resource theory of entanglement for quantum channels is not reversible. I also prove that the entanglement cost of thermal, amplifier, and additive-noise bosonic Gaussian channels is bounded from below by the entanglement cost of their “Choi states.” In Section VI, I discuss how to generalize the basic notions to other resource theories. Finally, Section VII concludes with a summary and some open questions.
Ii Notions of quantum channel simulation
In this section, I review the definition of entanglement cost of a quantum channel, as detailed in BBCW13 , and I also review the main theorem from BBCW13 . After that, I propose the revised definition of a channel’s entanglement cost.
Before starting, let us define a maximally entangled state of Schmidt rank as
where and are orthonormal bases. An LOCC channel is a bipartite channel that can be written in the following form:
where and are sets of completely positive, trace-non-increasing maps, such that the sum map is a quantum channel (completely positive and trace preserving) CLM+14 . The diamond norm of the difference of two channels and is defined as Kit97
where the optimization is with respect to all pure bipartite states with system isomorphic to system and the trace norm of an operator is defined as . The operational interpretation of the diamond norm is that it is related to the maximum success probability for any physical experiment, of the kind discussed after (2), to distinguish the channels and :
ii.1 Entanglement cost of a quantum channel from Bbcw13
Let us now review the notion of entanglement cost from BBCW13 . Fix , , and a quantum channel . According to BBCW13 , an (parallel) LOCC-assisted channel simulation code consists of an LOCC channel and a maximally entangled resource state of Schmidt rank , such that together they implement a simulation channel , as defined in (1). In this model, to be clear, we assume that Alice has access to all systems labeled by , Bob has access to all systems labeled by , and they are in distant laboratories. The simulation is considered -distinguishable from parallel calls of the actual channel if the condition in (2) holds. Note here again that the condition in (2) corresponds to a discriminator who is restricted to performing only a parallel test to distinguish the calls of from its simulation. Let us also note here that the condition in (2) can be understood as the simulation providing an approximate teleportation simulation of , in the language of the later work of KW17a .
A rate is said to be achievable for (parallel) channel simulation of if for all , , and sufficiently large , there exists an LOCC-assisted channel simulation code. The (parallel) entanglement cost of the channel is equal to the infimum of all achievable rates, with the superscript indicating that the test of the simulation is restricted to be a parallel discrimination test.
The main result of BBCW13 is that the channel’s entanglement cost is equal to the regularization of its entanglement of formation. To state this result precisely, recall that the entanglement of formation of a bipartite state is defined as BDSW96
where the infimum is with respect to all convex decompositions of into pure states and
is the quantum entropy of the marginal state . The entanglement of formation does not increase under the action of an LOCC channel BDSW96 . A channel’s entanglement of formation is then defined as
where , and it suffices to take the optimization with respect to a pure state input , with system isomorphic to system , due to purification, the Schmidt decomposition theorem, and the LOCC monotonicity of entanglement of formation BDSW96 . We can now state the main result of BBCW13 described above:
The regularized formula on the right-hand side may be difficult to evaluate in general, and thus can only be considered a formal expression, but if the additivity relation holds for a given channel for all , then it simplifies significantly as .
ii.2 Proposal for a revised notion of entanglement cost of a channel
Now I propose the new or revised definition for entanglement cost of a channel. As motivated in the introduction, a parallel test of channel simulation is not the most general kind of test that can be considered. Thus, the new definition proposes that the entanglement cost of a channel should incorporate the most stringent test possible.
To begin with, let us fix , , and a quantum channel . We define an (sequential) LOCC-assisted channel simulation code to consist of a maximally entangled resource state of Schmidt rank and a set
of LOCC channels. Note that the systems of the final LOCC channel can be taken trivial without loss of generality. As before, Alice has access to all systems labeled by , Bob has access to all systems labeled by , and they are in distant laboratories. The structure of this simulation protocol is intended to be compatible with a discrimination strategy that can test the actual channels versus the above simulation in a sequential way, along the lines discussed in CDP08a ; GDP09 and G12 . I later show how this encompasses the parallel tests discussed in the previous section.
A discrimination strategy consists of an initial state , a set of adaptive channels, and a quantum measurement . Let us employ the shorthand to abbreviate such a discrimination strategy. Note that, in performing a discrimination strategy, the discriminator has a full description of the channel and the simulation protocol, which consists of and the set in (11). If this discrimination strategy is performed on the uses of the actual channel , the relevant states involved are
for . If this discrimination strategy is performed on the simulation protocol discussed above, then the relevant states involved are
for , where , and
for . The discriminator then performs the measurement and guesses “actual channel” if the outcome is and “simulation” if the outcome is . Figure 1 depicts the discrimination strategy in the case that the actual channel is called times and in the case that the simulation is performed.
If the a priori probabilities for the actual channel or simulation are equal, then the success probability of the discriminator in distinguishing the channels is given by
where the latter inequality is well known from the theory of quantum state discrimination H69 ; H73 ; Hel76 . For this reason, we say that the calls to the actual channel are -distinguishable from the simulation if the following condition holds for the respective final states
If this condition holds for all possible discrimination strategies , i.e., if
then the simulation protocol constitutes an channel simulation code. It is worthwhile to remark: If we ascribe the shorthand for the uses of the channel and the shorthand for the simulation, then the condition in (18) can be understood in terms of the -round strategy norm of CDP08a ; GDP09 ; G12 :
As before, a rate is achievable for (sequential) channel simulation of if for all , , and sufficiently large , there exists an (sequential) channel simulation code for . We define the (sequential) entanglement cost of the channel to be the infimum of all achievable rates. Due to the fact that this notion is more general, we sometimes simply refer to as the entanglement cost of the channel in what follows.
ii.3 LOCC monotonicity of the entanglement cost
Let us note here that if a channel can be realized from another channel via a preprocessing LOCC channel and a postprocessing LOCC channel as
then it follows that any protocol for sequential channel simulation of realizes an protocol for sequential channel simulation of . This is an immediate consequence of the fact that the best strategy for discriminating from its simulation can be understood as a particular discrimination strategy for , due to the structural decomposition in (20). Following definitions, a simple consequence is the following LOCC monotonicity inequality for the entanglement cost of these channels:
Thus, it takes more or the same entanglement to simulate the channel than it does to simulate . Furthermore, the decomposition in (20) and the bound in (21) can be used to bound the entanglement cost of a channel from below. Note that the structure in (20) was discussed recently in the context of general resource theories (CG18, , Section III-D-5).
ii.4 Parallel tests as a special case of sequential tests
A parallel test of the form described in Section II.1 is a special case of the sequential test outlined above. One can see this in two seemingly different ways. First, we can think of the sequential strategy taking a particular form. The state is prepared, and here we identify systems with system of in an adaptive protocol and system of with system of . Then the channel or its simulation is called. After that, the action of the first adaptive channel is simply to swap in system of to the second call of the channel or its simulation, while keeping systems as part of the reference of the state . Then this iterates and the final measurement is performed on all of the remaining systems.
The other way to see how a parallel test is a special kind of sequential test is to rearrange the simulation protocol as has been done in Figure 2. Here, we see that the simulation protocol has a memory structure, and it is clear that the simulation protocol can accept as input a state and outputs a state on systems , which can subsequently be measured.
As a consequence of this reduction, any sequential channel simulation protocol can serve as an parallel channel simulation protocol. Furthermore, if is an achievable rate for sequential channel simulation, then it is also an achievable rate for parallel channel simulation. Finally, these reductions imply the following inequality:
It is an interesting question (not addressed here) to determine if there exists a channel such that the inequality in (22) is strict.
If desired, it is certainly possible to obtain a non-asymptotic, weak-converse bound that implies the above bound after taking limits. Let us state this bound as follows:
Let be a quantum channel, and let and . Set , i.e., the minimum of the input and output dimensions of the channel . Then the following bound holds for any sequential channel simulation code:
where is understood as the non-asymptotic entanglement cost of the protocol and the bosonic entropy function is defined for as
Proof. To see this, suppose that there exists an protocol for sequential channel simulation. Then by the above reasoning (also see Figure 2), it can be thought of as a parallel channel simulation protocol, such that the criterion in (2) holds. Suppose that is a test input state, with , leading to when the actual channels are applied and when the simulation is applied. Then we have that
The first inequality follows from the condition in (18), as well as from the continuity bound for entanglement of formation from (Winter15, , Corollary 4). The second inequality follows from the LOCC monotonicity of the entanglement of formation BDSW96 , here thinking of the person who possesses systems to be in the same laboratory as the one possessing the systems , while the person who possesses the systems is in a different laboratory. The first equality follows from the fact that
is in tensor product with, so that by a local channel, one may remove or append it for free. The final equality follows because the entanglement of formation of the maximally entangled state is equal to the logarithm of its Schmidt rank. Since the bound holds uniformly regardless of the input state , after an optimization and a rearrangement we conclude the stated lower bound on the non-asymptotic entanglement cost of the protocol.
Let us note here that the entanglement cost of a quantum channel is equal to zero if and only if the channel is entanglement-breaking HSR03 ; Holevo2008 . The “if-part” follows as a straightforward consequence of definitions and the fact that these channels can be implemented as a measurement followed by a preparation HSR03 ; Holevo2008 , given that this measure-prepare procedure is a particular kind of LOCC and thus allowed for free (without any cost) in the above model. The “only-if” part follows from (22) and (BBCW13, , Corollary 18), the latter of which depends on the result from YHHS05 .
Iii Bounds for the entanglement cost of teleportation-simulable channels
iii.1 Upper bound on the entanglement cost of teleportation-simulable channels
The most trivial method for simulating a channel is to employ the teleportation protocol BBC+93 directly. In this method, Alice and Bob could use the teleportation protocol so that Alice could transmit the input of the channel to Bob, who could then apply the channel. Repeating this times, this trivial method would implement an simulation protocol in either the parallel or sequential model. Alternatively, Alice could apply the channel first and then teleport the output to Bob, and repeating this times would implement an simulation protocol in either the parallel or sequential model. Thus, they could always achieve a rate of using this approach, and this reasoning establishes a simple dimension upper bound on the entanglement cost of a channel:
In this context, also see (KW17a, , Proposition 9).
A less trivial approach is to exploit the fact that some channels of interest could be teleportation-simulable with associated resource state , in which the resource state need not be a maximally entangled state (see (BDSW96, , Section V) and (HHH99, , Eq. (11))). Recall from these references that a channel is teleportation-simulable with associated resource state if there exists an LOCC channel such that the following equality holds for all input states :
If a channel possesses this structure, then we arrive at the following upper bound on the entanglement cost:
Let be a quantum channel that is teleportation-simulable with associated resource state , as defined in (28). Let and . Then there exists an sequential channel simulation code satisfying the following bound
Definition 1 (Smooth EoF Bd11 )
Let and be a bipartite state. Let denote a pure-state ensemble decomposition
of , meaning that , where
is a pure state and is a probability distribution.
Define the conditional entropy of order zero
is a probability distribution. Define the conditional entropy of order zeroof a bipartite state as
where denotes the projection onto the support of and is a density operator. Then the -smooth entanglement of formation of is given by
where the minimization is with respect to all pure-state ensemble decompositions of , is a labeled pure-state extension of , and the -ball of cq states for a cq state is defined as
The -smooth entanglement of formation has the property that, for a tensor-power state , the following limit holds (BD11, , Theorem 2)
where the latter quantity denotes the entanglement cost of the state HHT01 .
Proof of Proposition 2. The approach for an sequential channel simulation consists of the following steps:
First, employ the one-shot entanglement cost protocol from (BD11, , Theorem 1), which consumes a maximally entangled state of Schmidt rank along with an LOCC channel to generate approximate copies of the resource state . In particular, using the maximally entangled state with
one can achieve the following approximation (BD11, , Theorem 1):
Next, at the first instance in which the channel should be simulated, Alice and Bob apply the LOCC channel from (28) to the and systems of . For the second instance, they apply the LOCC channel from (28) to the and systems of . This continues for the next rounds of the sequential channel simulation.
By the data processing inequality for trace distance, it is guaranteed that the following bound holds on the performance of this protocol for sequential channel simulation:
This follows because the distinguishability of the simulation from the actual channel uses is limited by the distinguishability of the states and , due to the assumed structure of the channel in (28), as well as the structure of the sequential channel simulation.
Let be a quantum channel that is teleportation-simulable with associated resource state , as defined in (28). Then the entanglement cost of the channel is never larger than the entanglement cost of the resource state :
The above corollary captures the intuitive idea that if a single instance of the channel can be simulated via LOCC starting from a resource state , then the entanglement cost of the channel should not exceed the entanglement cost of the resource state. The idea of the above proof is simply to prepare a large number of copies of and then use these to simulate uses of the channel , such that the simulation could not be distinguished from uses of the channel in any sequential test.
iii.2 The entanglement cost of resource-seizable, teleportation-simulable channels
In this section, I define teleportation-simulable channels that are resource-seizable, meaning that one can seize the channel’s underlying resource state by the following procedure:
prepare a free, separable state,
input one of its systems to the channel, and then
post-process with a free, LOCC channel.
This procedure is indeed related to the channel processing described earlier in (20). After that, I prove that the entanglement cost of a resource-seizable channel is equal to the entanglement cost of its underlying resource state.
Definition 2 (Resource-seizable channel)
Let be a teleportation-simulable channel with associated resource state , as defined in (28). Suppose that there exists a separable input state to the channel and a postprocessing LOCC channel such that the resource state can be seized from the channel as follows:
Then we say that the channel is a resource-seizable, teleportation-simulable channel.
In Appendix A, I discuss how resource-seizable channels are related to those that are “implementable from their image,” as defined in (CM17, , Appendix A). In Section VI, I also discuss how to generalize the notion of a resource-seizable channel to an arbitrary resource theory.
The main result of this section is the following simplifying form for the entanglement cost of a resource-seizable channel (as defined above), establishing that its entanglement cost in the asymptotic regime is the same as the entanglement cost of the underlying resource state. Furthermore, for these channels, the entanglement cost is not increased by the need to pass a more stringest test for channel simulation as required in a sequential test.
Let be a resource-seizable, teleportation-simulable channel with associated resource state , as given in Definition 2. Then the entanglement cost of the channel is equal to its parallel entanglement cost, which in turn is equal to the entanglement cost of the resource state :
Proof. Consider from (22) that
Let be an arbitrary pure input state to consider at the input of the tensor-power channel , leading to the state
From the assumption that the channel is teleportation-simulable with associated resource state , we have from (28) that
where the inequality follows from LOCC monotonicity of the entanglement of formation. Since the bound holds for an arbitrary input state, we conclude that the following inequality holds for all :
Now taking the limit , we conclude that
To see the other inequality, let a decomposition of the separable input state be given by
Considering that is a particular input to the tensor-power channel , we conclude that
Since this holds for all , we have that
where the equality follows because introducing a product state locally does not change the entanglement, the second inequality follows from convexity of entanglement of formation BDSW96 , and the last inequality follows from the assumption in (40) and the LOCC monotonicity of the entanglement of formation. Since the inequality holds for all , we can divide by and take the limit to conclude that
and in turn, from (48), that
The equality in Theorem 1 provides a formal expression for the entanglement cost of any resource-seizable, teleportation-simulable channel, given in terms of the entanglement cost of the underlying resource state . Due to the fact that the entanglement cost of a state is generally not equal to its entanglement of formation H09 , it could still be a significant challenge to compute the entanglement cost of these special channels. However, for some special states, the equality