We introduce a class of channels called heralded channels, which generalize the quantum erasure channel composed with an arbitrary other quantum channel. We show that monogamy of squashed entanglement limits the superadditivity of Holevo capacity of heralded channels in the regime of rare success (frequent erasure). We prove that in the limit of near-zero success probability, the classical capacity of the heralded channel converges to its Holevo information, which is equal to success probability times the single-letter Holevo information of the channel with which erasure is composed. We also show how entanglement monogamy applies to non-classicality in quantum games, and we consider how faithful monogamous entanglement measures may bound other entanglement-dependent quantities in many-party scenarios.
I. INTRODUCTION
Entanglement powers many of the core non-classical capabilities in the quantum world.1 Unlike classical correlations, entanglement is monogamous. A finite-dimensional quantum system can hold only a finite amount of entanglement, preventing it from entangling strongly with many others simultaneously. This property has some benefits, including natural privacy of quantum correlations, provided one can verify their quantum nature. By contrast, it also seems a severe limitation. If we require entanglement between subsets of disjoint subsystems but do not know in advance which will be relevant, monogamy limits the potential advantage from entanglement.
Our main example comes from the theory of communication over quantum channels.2 A challenge and opportunity lies in the phenomenon of superadditivity, in which the capacity (maximum possible rate) to transmit information over a collection of quantum channels may exceed the sum of their individual capacities.3–6 For classical channels, the Shannon capacity is additive, allowing a single-letter entropy expression—a formula defined for one use of the channel. By contrast, so far there is no such single-letter expression for the classical capacity of a quantum channel, and an open question as to the general tractability of capacity calculations.7 By the HSW (Holevo, Schumacher, and Westmoreland) theorem,8,9 the classical capacity C(Φ) of a quantum channel Φ is given by regularization of Holevo information χ(Φ), which maximizes the entropy expression for m copies of the channel at once, taking m to infinity.
In many laboratory settings, we expect that not every run of a protocol will succeed. This is a common paradigm in linear optics: photon sources,10 protocols,11,12 and gates13 all have a probability or amplitude of failing, in such a way that we might only discover after a measurement, or at the end of the experiment. Many of these situations can be described by quantum erasure channels, which have some probability of outputting a flagged error state. Sometimes the probabilistic failure may be composed with successful but imperfect transmission. Keeping with the experimental and photonics literature, we call these heralded channels14–18 (Fig. 1). These are a special case of flagged quantum channels, as discussed in Refs. 5, 19, and 20, in which the classical output is alerted to the outcome of some non-deterministic channel selection process. A flagged channel randomly selects among a fixed ensemble of quantum channels to apply to its input states and reports which was applied along with the output state. Some cases of these channels are mentioned in Refs. 21, 22, and 4. Flagged channels differ from the compound channels discussed in Refs. 23–25, which consistently apply an unknown channel taken from a known set. Flagged channels draw randomly from a set of known channels at each use, revealing the result of this random drawing at the output. A heralded channel is a special case of flagged channel, which chooses between outcomes interpreted as success or failure.
Diagram of outcomes of a flagged channel that sometimes applies Φ and other times applies Ψ. We do not know for each turn which will be applied at preparation time, only at the output.
Diagram of outcomes of a flagged channel that sometimes applies Φ and other times applies Ψ. We do not know for each turn which will be applied at preparation time, only at the output.
In these situations, the potential contribution of superadditivity is limited. Superadditivity relies on entangling input states to send over multiple uses of the channel. Entanglement as a quantum phenomenon is monogamous.26–28 Thus unpredictably assigning nearly additive channels to many of the inputs dilutes the available entanglement for superadditivity of the desired ones. For example, even if one could efficiently prepare a large number of entangled photons at the input to a set of optical quantum channels, losing many of these photons means the entanglement after transmission is small. In this situation, an entangled scheme has limited or no advantage over a probably simpler non-entangled input, because entanglement monogamy limits the average entanglement between potentially superadditive outputs.
We directly apply the monogamy of squashed entanglement and its faithfulness with the trace distance.29 Our results suggest one circumstance in which we can avoid the full difficulty of regularization. Surprisingly, post-selecting on success may not produce an exactly equivalent situation to having a channel without the failure probability, as it limits the phenomenon of superadditivity. We also show that with limited entangled blocksize, the number of inputs that are not separable in a quantum code, a wide variety of quantities become nearly additive for heralded channels.
In addition to the experimental motivations for understanding heralding, we derive a new technique of fundamental interest in estimating quantum channel capacity and other entropy expressions. Instead of relying on the details of the channel, we show that entanglement monogamy limits the contribution of superadditivity when considering a small subset of outputs randomly chosen from a larger set. Going beyond the case of capacity and based on the faithfulness of squashed entanglement,30 any entanglement-dependent quantity that is faithful in the trace distance should also be faithful in the squashed entanglement, admitting a bound from entanglement monogamy. We briefly consider the consequences in a general form of quantum game (Sec. VII) to illustrate the principle.
The faithfulness of squashed entanglement with trace distance implies a very general bound on a wide variety of quantities. For example, all entanglement measures that are convex, faithful, and continuous in trace distance are weakly monogamous. One may derive this by using the original entanglement measure’s implied lower bound on the trace distance to a separable state, the implied lower bound on squashed entanglement, and finally the monogamy of squashed entanglement. As claimed in Ref. 29, “squashed entanglement is about the distance from highly extendible states.” These extendible states are bipartite, for which it is possible to consider one of the parties to be replicated a large number of times in the environment. It is not possible to entangle a large network of low-dimensional, identical parties, in such a way that the majority of small, arbitrary subnetworks are individually entangled. We expect these ideas to go beyond the applications in this paper to many scenarios, including multipartite systems.
A. Main contributions
Let Φ be a quantum channel. That is, a completely positive trace preserving map which sends density operators (positive and trace one) of one Hilbert space HA to another HB. The HSW theorem8,9 proves that the classical capacity of a quantum channel C(Φ) is given by regularization of its Holevo information χ(Φ) as follows:
where S(σ) = −tr(σ log σ) is the von Neumann entropy and I(X; B) = S(X) + S(B) − S(XB) is the mutual information. The supremum runs over all classical-quantum inputs
and σ = Φ(ρ), σx = Φ(ρx) are the outputs.
We define for λ ∈ [0, 1] the generalized quantum erasure channel with 1 − λ erasure probability,
where Y is the heralding/flagging classical output and σ is some fixed state. Using monogamy and faithfulness of squashed entanglement, we have the following estimate of its classical capacity:
Here the asymptotic expression is taken for small λ, noting the significant dimension-dependence. Our theorem is most effective at describing small channels that erase often. We emphasize that as λ → 0, the Holevo capacity approaches an additive regime regardless of Φ. As discussed in Sec. VI, the existence of this asymptotic regime does not follow directly from convexity or continuity of the Holevo capacity. The technical derivation of this result appears in Sec. IV, in which we also consider the heralded channels with fixed success number, , which erase n − k random locations out of the n inputs (k being the success number). The main consequence is that there exists an approximately additive regime in the limit of low success probability or heavy post-selection. In particular, we emphasize the following corollary.
This notion is motivated by a common experimental scenario: we perform an experiment many times with some success rate λ, but we only wish to consider the instances in which the experiment was run successfully. For example, we may post-select on having the expected number of photons at the end of an optics experiment, discarding a much larger number of trials in which probabilistic photon sources did not produce exactly one photon each. This is equivalent to projecting the total outcome onto the heralding signal |0⟩ ⟨0|Y subspace. Even though we discard the failures, they still matter for superadditive quantities such as classical capacity. In the limit as λ → 0, χ(Φ) takes on the operational meaning of post-selected classical capacity. We suspect that this will be true for other superadditive quantities as well.
Going beyond capacity, we construct a form of quantum game in which Alice prepares a joint state via free communication with many “B” players simultaneously, though the dimension of her system is fixed and does not depend on the number of B players. After this initial state is prepared, the referee randomly selects one B player, and Alice plays a bipartite no-communication game with that B player. When there are many B players, monogamy of entanglement limits the extent to which the average value of these games with a quantum pre-shared state may exceed that with a classical pre-shared state. See Sec. VII B for the details of this setup.
We organize this work as follows: Sec. II introduces some notation and reviews channel capacity, squashed entanglement, and continuity of entropies. In Sec. III, we explain our main technique, which we call “heralded averaging.” Section IV applies this technique with monogamy of squashed entanglement to a variety of Holevo information quantities for heralded channels. Section V shows Theorem 1.1 and relates heralded switch channels to erasure channels. Section VI compares our result to additivity bounds derived by convexity and continuity of Holevo information. Section VII discusses other applications of our principles to other capacities and quantum games. We end with discussion and conclusions in Sec. VIII.
II. PRELIMINARY
A. Channels and capacity
Let {Φ1, …, Φn} and {Ψ1, …, Ψn} be two classes of quantum channels. For k < n, we define the flagged switch channel as follows:
The flagged switch channel is a convex combination of tensor products of “” and “” channels, randomly selecting positions at which to apply the corresponding “” channel, and applying “” channels at all others. Here specifies a set of positions., and denotes the tensor product on subsytems , where . In other words, specifies the set of outputs containing “” channels. is its complement. is an extra classical output revealing . We assume that the flagging process produces such a classical signal Y, such as by detection of a secondary photon at the output of a source10 or the outcome of an attempted quantum experiment. That is, by the time it reaches the output, the flagging signal has already been converted into a classical register Y of value R, the states of which we denote . Note that we do not switch the positions of input channels, only choosing between alternatives at each position. If all Ψj are erasure operations which have a fixed output Ψj(ρ) = Θ(ρ) = σ for some state σ, we call
the heralded channel with a fixed success number. We consider the fixed-output positions to represent failure. To shorten our notation, in the following, we will write
where the classes of channel Φ = {Φ1, …, Φn} and Ψ = {Ψ1, …, Ψn} are clear. Some cases of these channels are mentioned in Refs. 21, 22, and 4. We also consider the quantum erasure channel combined with another channel, described in Sec. V, to be a heralded channel. This is actually more common in physical scenarios: many processes randomly fail in a detectable way.
Our notion of the flagged channel differs from that of the compound channels discussed in Refs. 23–25, as they consider an unknown channel taken from a known set that is applied consistently, while we consider a new channel that draws randomly from a set of known channels at each use, but which reveals the result of this random drawing at the output.
In addition to the classical capacity, we will also discuss the potential Holevo capacity introduced in Ref. 31. The potential Holevo capacity is the maximum Holevo information gain of a channel when assisted with another channel,
It is clear that χ(pot)(Φ) ≥ C(Φ) ≥ χ(Φ), and we say that the channel’s Holevo information is additive if C(Φ) = χ(Φ) and strongly additive if χ(pot)(Φ) = χ(Φ). In both cases, its classical capacity C(Φ) is fully characterized by the single-letter Holevo information χ(Φ). We refer to the book by Wilde2 for more information about quantum channel capacity and other basics in quantum Shannon theory.
B. Squashed entanglement
The squashed entanglement of a bipartite state ρAB, defined in Ref. 30, is
where the infimum runs over all extensions ρABC of ρAB. We will use the following properties of the squashed entanglement in this paper:
Convexity: Let be a convex combination of states {ρx}. Then
- (ii)
Monogamy: Let be a (k + 1)-partite state. Then
- (iii)
1-norm faithfulness:
where ∥·∥1 is the trace class norm.The convexity and monogamy of squashed entanglement, as well as Property (iv), are basic properties.30 The faithfulness of Schatten norms are proved by Brandão et al.32 and Li and Winter.29 The readers are also referred to the survey paper33 for more information about measures of quantum entanglement.
C. Continuity of entropic expressions
Given a bipartite state ρAB, its conditional entropy (conditional on B) is given by S(A|B)ρ = S(AB)ρ − S(B)ρ. The continuity of conditional entropy is characterized by the Alicki-Fannes inequality, as introduced in Ref. 34 and later refined by Winter in Ref. 35. For two bipartite states ρ and σ and δ ∈ [0, 1],
where h(p) = −p log p − (1 − p) log(1 − p) is the binary entropy function and |A| is the dimension of A. Note that for 0 ≤ ϵ ≤ 1,
We may also use the following (weaker but simpler) variant of (7):
For simplicity, we will rewrite this with δ = 2ϵ, obtaining
III. HERALDED AVERAGING
For a channel Φ: A → B and a bipartite state ρ ∈ B0 ⊗ A, we introduce the following notation of squashed entanglement:
and similarly for conditional entropy and mutual information
Any function that depends on entanglement and is faithful in the 1-norm can be bounded by the method of Theorem 3.4. This generality comes at the cost of strong dimension-dependence. While in Sec. IV we use the Holevo superadditivity to illustrate applications of this result, it is really a demonstration that the monogamy of a faithful entanglement measure carries through to all entanglement-dependent quantities. The ultimate implication is that in the asymptotic limit of rare success and fixed dimensions, all quantities that depend on entanglement between outputs should approach their separable values.
IV. ITERATIVE BOUNDS ON HOLEVO INFORMATION
A. Holevo information estimates for heralded channels
Our main technical theorem of this section considers tensor products of heralded channels or flagged switch channels, showing that heralding removes the superadditivity from a channel by diluting the available entanglement. It is the building block we use to prove the simpler corollaries and Theorem 1.1. In the following, we always assume that are pairs of families of channels such that for each (i, j), Φi,j, Ψi,j: Ai,j → Bi,j share the same input and output system.
The above theorem bounds the additivity violation between heralded channels and an arbitrary extra channel Φ0. The form is analogous to an approximate strong additivity for heralded channels with small , but it depends on the dimension of output system B0. So it is not truly a strong additivity, as the bound becomes trivial when Φ0 is a channel with a sufficiently large output space. Based off of Theorem 4.1, we prove the following approximate additivity in the regime of small .
Unlike Theorem 4.1, this corollary depends only on the dimension of the constituent channels Φi,j. Therefore, it is an approximate additivity when is small enough.
Finally, we derive a bound analogous to strong additivity entirely in terms of single-letter expressions.
This corollary combines Theorem 4.1 and Corollary 4.2 to bound all superadditivity present, allowing us to write the Holevo information of a product of heralded channels enhanced by an arbitrary Φ0 as a sum of single-letter expressions and correction terms. We say that this bound is merely analogous to strong additivity (rather than an actual strong additivity) due to the dimension-dependence on the channel Φ0, which prevents us from bounding the full potential capacity of a heralded channel.
B. Now the proofs
We start with a lemma showing that Holevo information is not increasing when conditioning on a separable extra system.
The next lemma replaces each Ψ channel by its potential capacity in an upper bound, which reduces the discussion of Theorem 4.1 to heralded channels.
V. ERASURE CHANNELS COMPOSED WITH OTHER CHANNELS
We define a generalized erasure channel as an erasure channel composed with some channel Φ in its successful trial,
where λ ∈ [0, 1] and σ is some fixed state. We call the successes those cases in which Φ was applied with a classical signal of |0⟩ ⟨0|. Zλ(Φ) can be viewed as a heralded channel in terms of success probability, rather than imposing a definite number of successes. Indeed, tensor products of the erasure channel can be expressed as a probabilistic sum over heralded channels,
This is easy to see by noting that each erasure channel is a binary heralding with probability λ, so the overall distribution of successes is binomial. In the following, we denote
as the heralded failure channel with n identical Φ’s and fixed success number k.
We prepare our proof with a lemma.
Corollary 5.3 follows from Theorem 5.1 and Corollary 4.2. It is the technical version of Theorem 1.1.
VI. COMPARISON WITH EXISTING BOUNDS ON SUPERADDITIVITY
It is known in Ref. 36, Proposition 4.3, that the Holevo information and its potential analog are convex functions of quantum channels. For a d-dimensional generalized erasure, we know the convexity bound as follows:
where the last inequality is the trivial bound χpot(Φ) ≤ log d. Suppose Φ has output dimension d. Our monogamy bounds in Corollary 5.3 are nontrivial when
If χ(Φ) < log d (the channel is not perfect), there is always a region of small λ such that our monogamy bound is better.
It is possible to obtain stronger bounds than this convexity by using knowledge of the channel. Leditzky et al. in Ref. 37 estimated the classical capacity of a channel Φ by the smallest diamond norm distance to a weakly additive channel.
Our correction term in Corollary 5.3 decays asymptotically as , going beyond (16) and convexity bound when λ ≪ 1/|B|. This implies that the super-additivity decays faster than the linear term in the asymptotic limit of low success probability. As a consequence, we obtain Corollary 1.2 that when the success rate λ → 0, the classical capacity of the heralded channel Zλ(Φ) averaged by λ converges to the Holevo information of Φ. This cannot be obtained by the bound (16) from the diamond norm.
VII. OTHER APPLICATIONS
A. Capacities with small entangled blocksize coding
While we would like to immediately extend our results to quantum and private capacities, these quantities have a more complicated interaction with heralding and erasure. The quantum erasure channel, for instance, has zero quantum capacity if the success probability is less than 1/2,38 subsuming the values for which our main near-additivity results would apply. It is possible that the quantum capacity has a different monogamy phenomenon from Holevo information, but that is beyond the scope of this paper. We might instead consider the quantum and private capacities with feedback, but these are much more complicated to estimate with entropy techniques.39 See Refs. 2 and 31 for more information on different capacities.
In practice, the realities of quantum hardware often limit the size of entangled blocks. When the success probability of a heralding process is small compared with the entangled blocksize of inputs, most successes will be the only success in their respective block. In this case, we show that superadditivity is limited by a simpler argument. For example, recent proposals for trapped ion computing schemes mention heralded photonic Bell state analysis as a method for connecting trapped ion arrays, a necessity due to the number-limiting interactions of trapped ions in the same array.40 The array size restricts the entangled input blocksize for inter-array communication, so lossy optical links may exhibit superadditivity-limiting effects even with 2-way classical assistance. The generation of large-scale entangled states between many individually controllable qubits is at the time of this writing an ongoing challenge in experimental quantum information. Realistic calculations of attainable quantum and private rates with today’s hardware probably should assume small entangled blocksizes.
The intuition behind Proposition 7.1 is that when the erasure probability of an erasure channel is sufficiently high compared to the maximum entangled blocksize of the encoder, it is unlikely that more than one non-trivial channel will appear in any block. This result is not based on entanglement monogamy.
Proposition 7.1 applies to the block-limited coherent information Q(m), but when λ ≪ 1, the unassisted quantum capacity vanishes anyway due to antidegradability of the erasure channel.38 Proposition 7.1 applies to the Holevo information χ as well as the classical capacity with limited entanglement assistance, as discussed in Ref. 41. We conjecture that it will apply to the quantum capacity with classical feedback and 2-way classical communication, as discussed in Ref. 39, but we cannot yet confirm the necessary properties to apply our results, as we do not have a one-shot entropy expression for these quantities.
B. Entanglement monogamy in quantum games
Going beyond the case of communication, the idea that entanglement monogamy implies bounds on entanglement-based super-additivity also applies to non-local games. Vidick et al. used monogamy in tripartite systems to show that entangled multi-prover games have values close to their classical counterparts, making their values similarly difficult to approximate.42 Several previous authors have noted the monogamy of quantum correlations43,44 and the connection between symmetric extendibility and hidden variable explanations.45 Here we show how the monogamy of squashed entanglement and k-extendibility yield a very general bound on multiplayer quantum games.
A two-player game G = (A, B, X, Y, π, ) is played between a referee and two isolated players, Alice and Bob, who communicate only with the referee and not between themselves. The referee chooses a question pair (x, y) according to some probability distribution π on the question alphabets X × Y, sending x to Alice and y to Bob. The two players respond with answers a and b, respectively, from answer sets A and B. They win the game if (x, y, a, b) = 1 for the verification function : X × Y × A × B → {0, 1} and lose otherwise. The classical value of the game
is the maximum winning probability when Alice and Bob are allowed to use optimal deterministic strategies ∑aax(ω) = ∑bby(ω) = 1 based on some classical correlation (common randomness) . The entangled or quantum value of the game
allows Alice and Bob to answer the question by performing positive-operator valued measures (POVMs) with some auxiliary bipartite entangled state ρ. It is clear that for all games, (G) ≤ *(G). A bound on the value with classical states, such as Bell’s inequality, can be evidence of quantum non-locality (see Ref. 46 for more information on non-local games).
We consider the application of entanglement monogamy in the following scenario. Suppose the player Alice has a single system A, which can share classical correlation or quantum entanglement with a large number of “Bob” players B1, …, Bn simultaneously. The referee randomly (with equal probability) selects a “Bob” player Bi and plays the corresponding bipartite game Gi = (A, Bi, Xi, Yi, πi, i) with Alice and Bi. This is analogous to the heralding process because although the two players A and Bi know the game Gi after Bi is selected by the referee, the strategy and auxiliary resource, classical or quantum, has to be prepared between Alice and B1, …, Bn before the game is played. In this situation, the non-classicality in entangled games will be bounded by entanglement monogamy if the size of Alice’s system is fixed because the average amount of entanglement Alice’s system can have with each Bi is limited by the number of “Bob” players.
For , we define the average entangled value when Alice having an at most d-dimensional quantum system as follows:
where for each x and y, are POVMs on A, B1, …, Bn, respectively, and is a multipartite state with |A| being at most d. Also, the average classical value is given by
because the classical correlation used for different Gi can be combined.
We can see that with large n and fixed d, the classicality violation decays to 0. In keeping with the results of Ref. 47, however, increasing the dimension of Alice and Bobs’ systems simultaneously with the number of measurements can compensate for a decline in entanglement. For this reason, we do not expect a bound that is independent of the dimension of Alice’s system.
This simple application in quantum games may give some intuition behind the phenomenon in Theorem 7.1, and possibly those in Sec. IV A. The existence of a monogamous, faithful entanglement measure (like squashed entanglement) necessarily constrains entanglement-dependent quantities to have a sort of monogamy of their own. This technique can be applied to any situation in which one seeks entanglement-dependent effects in the presence of a heralding-like process.
VIII. DISCUSSION AND CONCLUSION
It follows intuitively that the monogamy of a faithful entanglement measure would also imply monogamy for entanglement-dependent quantities. Recent advances in the theory of squashed entanglement29 have combined these properties in a sufficiently quantitative way as to directly derive bounds on entropy expressions. As a potential consequence of entangled coding, we expect superadditivity of channel capacity to be monogamous. If many entangled systems enhance the capacity of a single channel, monogamy would intuitively imply that the enhancement due to each is correspondingly limited. While this alone might be an interesting anecdote and of potential interest to routines designed to optimize quantum input states, the heralded channel provides a direct operational application of the monogamy of superadditivity. When an arbitrary channel is randomly distributed amongst a larger number of copies of a strongly additive channel, entanglement in the input state goes to waste, and the combined channel loses superadditivity.
One application is to situations encountered in experimental physics, such as photonics, where the non-trivial channel created by decoherence combines with the erasure channel arising from photon loss. Due to the difficulty of creating and maintaining large-scale entanglement in transmitted photons, and the likelihood that entangling inputs with eventually destroyed copies sacrifice some capacity, many real-world quantum information systems will probably transmit in the nearly additive realm of Theorem 7.1 and not gain much by attempting to exploit superadditivity.
Our bound on the superadditivity of Holevo information is probably not tight, mostly due to the dimension constant and fourth-root power. The weakness of this bound in high dimension may stem from the generality of the continuity estimates, which do not exploit any detailed model of the Holevo capacity or the specific channel under study. The monogamy of squashed entanglement is itself fairly strong. The proof of faithfulness discussed in Ref. 29 first converts entanglement into symmetric extendibility and then to 1-norm distance. Finally, we convert this back to an entropic quantity. Each of these conversions comes with a penalty in constants or polynomial order. Future work may consider a path that either directly uses extendibility of the heralded output or examines whether extendibility directly limits superadditivity. It should be possible to improve these bounds for some cases if one is more specific about the channels composed with erasure.
Finally, we note that this technique is not limited to capacity or entropy expressions. Any quantity that depends on entanglement and is faithful with the trace norm may show a monogamy-like effect due to the comparison with squashed entanglement. For example, all entanglement measures that are faithful and comparable to the trace distance are at least monogamous up to a 4th-root dependence. This broader implication of entanglement monogamy bounding entanglement-dependent quantities may also be significant.
ACKNOWLEDGMENTS
We thank Mark Wilde and Debbie Leung for helpful conversations. We thank Paul Kwiat and the Kwiat lab at UIUC for conversations and experiences that gave us a sense of the importance of heralded channels in practical applications. We thank the anonymous referees for the careful reading and constructive suggestions. This material is based upon work supported by NSF Graduate Research Fellowship Program DGE-1144245, and partially supported by NSF-DMS 1501103.
