# Peter W. Shor's research while affiliated with Massachusetts Institute of Technology and other places

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## Publications (187)

We introduce various measures of forward classical communication for bipartite quantum channels. Since a point-to-point channel is a special case of a bipartite channel, the measures reduce to measures of classical communication for point-to-point channels. As it turns out, these reduced measures have been reported in prior work of Wang
et al
. o...

We present a quantum compilation algorithm that maps Clifford encoders, an equivalence class of quantum circuits that arise universally in quantum error correction, into a representation in the ZX calculus. In particular, we develop a canonical form in the ZX calculus and prove canonicity as well as efficient reducibility of any Clifford encoder in...

I recount some of my memories of the early development of quantum computation, including the discovery of the factoring algorithm, of error correcting codes, and of fault tolerance.

Publicly verifiable quantum money is a protocol for the preparation of quantum states that can be efficiently verified by any party for authenticity but is computationally infeasible to counterfeit. We develop a cryptographic scheme for publicly verifiable quantum money based on Gaussian superpositions over random lattices. We introduce a verificat...

We study scenarios which arise when two spatially-separated observers, Alice and Bob, are try to identify a quantum state sampled from several possibilities. In particular, we examine their strategies for maximizing both the probability of guessing their state correctly as well as their information gain about it. It is known that there are scenario...

The out-of-time-ordered correlation (OTOC) and entanglement are two physically motivated and widely used probes of the “scrambling” of quantum information, a phenomenon that has drawn great interest recently in quantum gravity and many-body physics. We argue that the corresponding notions of scrambling can be fundamentally different, by proving an...

We introduce various measures of forward classical communication for bipartite quantum channels. Since a point-to-point channel is a special case of a bipartite channel, the measures reduce to measures of classical communication for point-to-point channels. As it turns out, these reduced measures have been reported in prior work of Wang et al. on b...

The out-of-time-ordered correlator (OTOC) and entanglement are two physically motivated and widely used probes of the "scrambling" of quantum information, which has drawn great interest recently in quantum gravity and many-body physics. By proving upper and lower bounds for OTOC saturation on graphs with bounded degree and a lower bound for entangl...

We prove that the classical capacity of an arbitrary quantum channel assisted by a free classical feedback channel is bounded from above by the maximum average output entropy of the quantum channel. As a consequence of this bound, we conclude that a classical feedback channel does not improve the classical capacity of a quantum erasure channel, and...

In this article, we investigate the additivity phenomenon in the quantum dynamic capacity region of a quantum channel for trading the resources of classical communication, quantum communication, and entanglement. Understanding such an additivity property is important if we want to optimally use a quantum channel for general communication purposes....

Recently, physicists have started applying quantum information theory to black holes. This led to the conjecture that black holes are the fastest scramblers of information, and that they scramble it in time order M log M, where M is the mass of the black hole in natural units. As stated above, the conjecture is not completely defined, as there are...

Non-Gaussian states and operations are crucial for various continuous-variable quantum information processing tasks. To quantitatively understand non-Gaussianity beyond states, we establish a resource theory for non-Gaussian operations. In our framework, we consider Gaussian operations as free operations, and non-Gaussian operations as resources. W...

In this article, we investigate the additivity phenomenon in the dynamic capacity of a quantum channel for trading classical communication, quantum communication and entanglement. Understanding such additivity property is important if we want to optimally use a quantum channel for general communication purpose. However, in a lot of cases, the chann...

Physical processes thatobtain, process, and erase information involve tradeoffs between information and energy. The fundamental energetic value of a bit of information exchanged with a reservoir at temperature T is kT ln2. This paper investigates the situation in which information is missing about just what physical process is about to take place....

We study the consequences of superquantum nonlocal correlations as represented by the PR-box model of Popescu and Rohrlich, and show that PR boxes can enhance the capacity of noisy interference channels between two senders and two receivers. PR-box correlations violate Bell and CHSH inequalities and are thus stronger—more nonlocal—than quantum mech...

Free energy is energy that is available to do work. Maximizing the free energy gain and the gain in work that can be extracted from a system is important for a wide variety of physical and technological processes, from energy harvesting processes such as photosynthesis to energy storage systems such as fuels and batteries. This paper extends recent...

Finding the optimal encoding strategies can be challenging for communication using quantum channels, as classical and quantum capacities may be superadditive. Entanglement assistance can often simplify this task, as the entanglement-assisted classical capacity for any channel is additive, making entanglement across channel uses unnecessary. If the...

We provide polylog sparse quantum codes for correcting the Erasure channel arbitrarily close to the capacity. Specifically, we provide the [[n, k, d]] quantum stabilizer codes that correct for the erasure channel arbitrarily close to the capacity if the erasure probability is at least 0.33 and with a generating set (S
<sub xmlns:mml="http://www.w3....

We study the consequences of 'super-quantum non-local correlations' as represented by the PR-box model of Popescu and Rohrlich, and show PR-boxes can enhance the capacity of noisy interference channels between two senders and two receivers. PR-box correlations violate Bell/CHSH inequalities and are thus stronger -- more non-local -- than quantum me...

Significance
We introduce a class of exactly solvable models with surprising properties. We show that even simple quantum matter is much more entangled than previously believed possible. One then expects more complex systems to be substantially more entangled. For over two decades it was believed that the area law is violated by at most a logarithm...

We give a capacity formula for the classical information transmission over a noisy quantum channel, with separable encoding by the sender and limited resources provided by the receiver's pre-shared ancilla. Instead of a pure state, we consider the signal-ancilla pair in a mixed state, purified by a "witness". Thus, the signal-witness correlation li...

We present a scheme for universal quantum computing using XY Heisenberg spin
chains. Information is encoded into packets propagating down these chains, and
they interact with each other to perform universal quantum computation. A
circuit using g gate blocks on m qubits can be encoded into chains of length
$O(g^{3+\delta} m^{3+\delta})$ for all $\de...

The problem of finding quantum-error-correcting codes is transformed into the problem of finding additive codes over the field GF (4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.

We show that universal quantum computation can be performed efficiently on
quantum networks while the fraction of controlled subsystems vanishes as the
network grows larger. We provide examples of quantum spin network families
admitting polynomial quantum gate complexity with a vanishing fraction of
controlled spins. We define a new family of graph...

The sub-volume scaling of the entanglement entropy with the system's size,
$n$, has been a subject of vigorous study in the last decade [1]. Hastings
proved "the area law" for gapped one dimensional systems [2] and it is largely
believed that, in quantum critical systems, the area law would be violated by
at most a factor of $\log\left(n\right)$.
I...

Dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one, reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones, and more generally the use of one noisy channel to simulate another. For channels of nonzero capacity, this simulation is always poss...

We present the results of a numerical study, with 20 qubits, of the
performance of the Quantum Adiabatic Algorithm on randomly generated instances
of MAX 2-SAT with a unique assignment that maximizes the number of satisfied
clauses. The probability of obtaining this assignment at the end of the quantum
evolution measures the success of the algorith...

We introduce a new relativistic orthogonal states quantum key distribution
protocol which leverages the properties of both quantum mechanics and special
relativity to securely encode multiple bits onto the spatio-temporal modes of a
single photon. If the protocol is implemented using a single photon source, it
can have a key generation rate faster...

n this work, we consider the following family of two prover one-round games. In the CHSH_q game, two parties are given x,y in F_q uniformly at random, and each must produce an output a,b in F_q without communicating with the other. The players' objective is to maximize the probability that their outputs satisfy a+b=xy in F_q. This game was introduc...

We present new constructions of codes for asymmetric channels for both binary
and nonbinary alphabets, based on methods of generalized code concatenation.
For the binary asymmetric channel, our methods construct nonlinear
single-error-correcting codes from ternary outer codes. We show that some of
the Varshamov-Tenengol'ts-Constantin-Rao codes, a c...

Frustration-free (FF) spin chains have a property that their ground state minimizes all individual terms in the chain Hamiltonian. We ask how entangled the ground state of a FF quantum spin-s chain with nearest-neighbor interactions can be for small values of s. While FF spin-1/2 chains are known to have unentangled ground states, the case s=1 rema...

In this paper we study the performance of the quantum adiabatic algorithm on
random instances of two combinatorial optimization problems, 3-regular 3-XORSAT
and 3-regular Max-Cut. The cost functions associated with these two
clause-based optimization problems are similar as they are both defined on
3-regular hypergraphs. For 3-regular 3-XORSAT the...

We construct families of high performance quantum amplitude damping codes. All of our codes are nonadditive and most modestly outperform the best possible additive codes in terms of encoded dimension. One family is built from nonlinear error-correcting codes for classical asymmetric channels, with which we systematically construct quantum amplitude...

The discipline of information theory was founded by Claude Shannon in a truly remarkable paper [Sh] which laid down the foundations of the subject. We begin with a quote from this paper which is an excellent summary of the
main concern of information theory:
The fundamental problem of communication is that of reproducing at one point either exactl...

A d-dimensional Keller graph has vertices which are numbered with each of the 4d possible d-digit numbers (d-tuples) which have each digit equal to 0, 1, 2, or 3. Two vertices are adjacent if their labels differ in at least two positions, and in at least one position the difference in the labels is two modulo four. Keller graphs are in the benchmar...

Given a single copy of an unknown quantum state, the no-cloning theorem limits the amount of information that can be extracted from it. Given a gapped Hamiltonian, in most situations it is impractical to compute properties of its ground state, even though in principle all the information about the ground state is encoded in the Hamiltonian. We show...

We study the Hamiltonian associated with the quantum adiabatic algorithm with
a random cost function. Because the cost function lacks structure we can prove
results about the ground state. We find the ground state energy as the number
of bits goes to infinity, show that the minimum gap goes to zero exponentially
quickly, and we see a localization t...

We investigate chains of d-dimensional quantum spins (qudits) on a line with generic nearest-neighbor interactions without translational invariance. We find the conditions under which these systems are not frustrated, that is, when the ground states are also the common ground states of all the local terms in the Hamiltonians. The states of a quantu...

Associated to every finite group, Kitaev has defined the quantum double model for every orientable surface without boundary. In this paper, we define boundaries for this model and characterize condensations; that is, we find all quasi-particle excitations (anyons) which disappear when they move to the boundary. We then consider two phases of the qu...

Quantum money is a cryptographic protocol in which a mint can produce a quantum state, no one else can copy the state, and anyone (with a quantum computer) can verify that the state came from the mint. We present a concrete quantum money scheme based on superpositions of diagrams that encode oriented links with the same Alexander polynomial. We exp...

This paper considers three variants of quantum interactive proof systems in
which short (meaning logarithmic-length) messages are exchanged between the
prover and verifier. The first variant is one in which the verifier sends a
short message to the prover, and the prover responds with an ordinary, or
polynomial-length, message; the second variant i...

We consider the category of finite dimensional representations of the quantum
double of a finite group as a modular tensor category. We study
auto-equivalences of this category whose induced permutations on the set of
simple objects (particles) are of the special form of PJ, where J sends every
particle to its charge conjugation and P is a transpos...

Unitary gates are interesting resources for quantum communication in part because they are always invertible and are intrinsically bidirectional. This paper explores these two symmetries: time-reversal and exchange of Alice and Bob. We present examples of unitary gates that exhibit dramatic separations between forward and backward capacities (even...

Fault-tolerant quantum computation is a basic problem in quantum computation, and
teleportation is one of the main techniques in this theory. Using teleportation on stabilizer codes, the most well-known quantum codes, Pauli gates and Clifford operators can
be applied fault-tolerantly. Indeed, this technique can be generalized for an extended set
of...

Public-key quantum money is a cryptographic protocol in which a bank can
create quantum states which anyone can verify but no one except possibly the
bank can clone or forge. There are no secure public-key quantum money schemes
in the literature; as we show in this paper, the only previously published
scheme [1] is insecure. We introduce a category...

We present a communication protocol for the erasure channel assisted by backward classical communication, which achieves a significantly better rate than the best prior result. In addition, we prove an upper bound for the capacity of the channel. The upper bound is smaller than the capacity of the erasure channel when it is assisted by two-way clas...

Graphs are closely related to quantum error-correcting codes: every
stabilizer code is locally equivalent to a graph code, and every codeword
stabilized code can be described by a graph and a classical code. For the
construction of good quantum codes of relatively large block length,
concatenated quantum codes and their generalizations play an impo...

We construct a set of instances of 3SAT which are not solved efficiently
using the simplest quantum adiabatic algorithm. These instances are obtained by
picking random clauses all consistent with two disparate planted solutions and
then penalizing one of them with a single additional clause. We argue that by
randomly modifying the beginning Hamilto...

We show how good quantum error-correcting codes can be constructed using generalized concatenation. The inner codes are quantum codes, the outer codes can be linear or nonlinear classical codes. Many new good codes are found, including both stabilizer codes as well as so-called non-additive codes.

A counterexample to the 'additivity question', the most celebrated open problem in the mathematical theory of quantum information, casts doubt on the possibility of finding a simple expression for the information capacity of a quantum channel.

The positive partial transpose test is one of the main criteria for detecting
entanglement, and the set of states with positive partial transpose is
considered as an approximation of the set of separable states. However, we do
not know to what extent this criterion, as well as the approximation, are
efficient. In this paper, we show that the positi...

We introduce the concept of generalized concatenated quantum codes. This generalized concatenation method provides a systematical way for constructing good quantum codes, both stabilizer codes and nonadditive codes. Using this method, we construct families of new single-error-correcting nonadditive quantum codes, in both binary and nonbinary cases,...

Error correction procedures are considered which are designed specifically for the amplitude damping channel. Amplitude damping errors are analyzed in the stabilizer formalism. This analysis allows a generalization of the [4,1] ldquoapproximaterdquo amplitude damping code. This generalization is presented as a class of [2(M +1), M ] codes; quantum...

The classQMA(k), introduced by Kobayashi et al., con- sists of all languages that can be verified using k unen- tangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence thatk quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besid...

Fault-tolerant quantum computation is a basic problem in quantum computation,
and teleportation is one of the main techniques in this theory. Using
teleportation on stabilizer codes, the most well-known quantum codes, Pauli
gates and Clifford operators can be applied fault-tolerantly. Indeed, this
technique can be generalized for an extended set of...

It is proved that every n n Latin square has a partial transversal of length at least n O(log2 n). The previous papers proving these results (including one by the second author) not only contained an error, but were sloppily written and quite dicult to understand. We have corrected the error and improved the clarity.

The class QMA(k). introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) pro...

A family of high rate quantum error correcting codes adapted to the amplitude damping channel is presented. These codes are nonadditive and exploit self-complementarity structure to correct all first-order errors. Their rates can be higher than 1/2. The recovery operations of these codes can be generated by a simple algorithm and have a projection...

Using random Gaussian vectors and an information-uncertainty relation, we
give a proof that the coherent information is an achievable rate for
entanglement transmission through a noisy quantum channel. The codes are random
subspaces selected according to the Haar measure, but distorted as a function
of the sender's input density operator. Using lar...

It is proved that every n × n Latin square has a partial transversal of length at least n − 5.53(log n)2.

We give a generalization to an infinite tree geometry of Vidal's infinite time-evolving block decimation (iTEBD) algorithm for simulating an infinite line of quantum spins. We numerically investigate the quantum Ising model in a transverse field on the Bethe lattice using the Matrix Product State ansatz. We observe a second order phase transition,...

We consider the clique problem in graphs, which is NP-complete, and try to find a quantum analogue of this problem. We show that, quantum clique problem can be defined as follows. Given a quantum channel, are there k states that are distinguishable, with no error, after passing through channel. This definition comes from reconsidering the clique pr...

We present a class of numerical algorithms which adapt a quantum error correction scheme to a channel model. Given an encoding and a channel model, it was previously shown that the quantum operation that maximizes the average entanglement fidelity may be calculated by a semidefinite program (SDP), which is a convex optimization. While optimal, this...

It is known that evaluating a certain approximation to the Jones polynomial
for the plat closure of a braid is a BQP-complete problem. That is, this
problem exactly captures the power of the quantum circuit model. The one clean
qubit model is a model of quantum computation in which all but one qubit starts
in the maximally mixed state. One clean qu...

We present a family of entanglement purification protocols that generalize four previous methods, namely the recurrence method, the modified recurrence method, and the two methods proposed by Maneva-Smolin and Leung-Shor. We will show that this family of protocols have improved yields over a wide range of initial fidelities F, and hence imply new l...

We present an entanglement purification protocol for Beil-diagonal mixed states and show that this protocol has improved yields over the recurrence methods and the method proposed by Maneva-Smolin. We then generalize this protocol to a family, and show that this family is also a generalization of the recurrence method, the modified recurrence metho...

In this paper, I discuss the additivity conjecture in
quantum information theory. The additivity conjecture was orig-
inally a set of at least four conjectures. These conjectures said
that certain functions of quantum states and channels were addi-
tive under tensor products. While some of these conjectures were
previously known to be stronger than...

Quantum error correction (QEC) is an essential element of physical quantum information processing systems. Most QEC efforts focus on extending classical error correction schemes to the quantum regime. The input to a noisy system is embedded in a coded subspace, and error recovery is performed via an operation designed to perfectly correct for a set...

We exhibit quantum channels whose classical and quantum capacities, when assisted by classical feedback, exceed their unassisted classical Holevo capacity. These channels are designed to be noisy in a way that can be corrected with the help of the output and a reference system entangled with part of the input. A similar construction yields quantum...

Recently, there has been growing interest in using adiabatic quantum computation as an architecture for experimentally realizable quantum computers. One of the reasons for this is the idea that the energy gap should provide some inherent resistance to noise. It is now known that universal quantum computation can be achieved adiabatically using 2-lo...

Unitary gates are interesting resources for quantum communication in part because they are always invertible and are intrinsically bidirectional. This paper explores these two symmetries: time-reversal and exchange of Alice and Bob. We present examples of unitary gates that exhibit dramatic separations between forward and backward capacities (even...

An expression is derived characterizing the set of admissible rate pairs for simultaneous transmission of classical and quantum information over a given quantum channel, generalizing both the classical and quantum capacities of the channel. Although our formula involves regularization, i.e. taking a limit over many copies of the channel, it reduces...

Local rules theory describes virus capsid self-assembly in terms of simple local binding preferences of discrete subunit conformations. The theory offered a parsimonious explanation for the complexity and regularity of virus capsids that resolved several inconsistencies between experimental observations and prior theories. Simultaneously, it provid...

We discuss the progress (or lack of it) that has been made in discovering algorithms for computation on a quantum computer.
Some possible reasons are given for the paucity of quantum algorithms so far discovered, and a short survey is given of the
state of the field.

Remote state preparation is the variant of quantum state teleportation in which the sender knows the quantum state to be communicated. The original paper introducing teleportation established minimal requirements for classical communication and entanglement but the corresponding limits for remote state preparation have remained unknown until now: p...

Virtually all of today's information technology is based on the manipulation of classical bits. Quantum systems offer the
potential of a much more powerful computing technology, however. In their Perspective,
Bennett and Shor
discuss an important aspect of quantum computing--the theoretical capacity of a quantum information channel. Although a numb...

We reduce the number of open additivity problems in quantum information theory by showing that four of them are equivalent. Namely, we show that the conjectures of additivity of the minimum output entropy of a quantum channel, additivity of the Holevo expression for the classical capacity of a quantum channel, additivity of the entanglement of form...

We give the trade-off curve showing the capacity of a quantum channel as a function of the amount of entanglement used by the sender and receiver for transmitting information. The endpoints of this curve are given by the Holevo–Schumacher–Westmoreland capacity formula and the entanglement-assisted capacity, which is the maximum over all input densi...

We investigate the capacity of three symmetric quantum states in three real dimensions to carry classical information. Several such capacities have already been defined, depending on what operations are allowed in the protocols that the sender uses to encode classical information into these quantum states, and that the receiver uses to decode it. T...

The construction of a perfectly secure private quantum channel in dimension d is known to require 2 log d shared random key bits between the sender and receiver. We show that if only near-perfect security is required, the size of the key can be reduced by a factor of two. More specifically, we show that there exists a set of roughly d log d unitary...

We discuss the progress (or lack of it) that has been made in discovering algorithms for computation on a quantum computer. Some possible reasons are given for the paucity of quantum algorithms so far discovered, and a short survey is given of the state of the field.PACS: 03.67.Lx

We calculate the entanglement assisted capacity of a multimode bosonic channel with loss. As long as the efficiency of the channel is above 50%, the superdense coding effect can be used to transmit more bits than those that can be stored in the message sent down the channel. Bounds for the other capacities of the multimode channel are also provided...

We study the communication capacities of bosonic broadband channels in the presence of different sources of noise. In particular we analyze lossy channels in presence of white noise and thermal bath. In this context, we provide a numerical solution for the entanglement assisted capacity and upper and lower bounds for the classical and quantum capac...

We reduce the number of open additivity problems in quantum information theory by showing that four of them are equivalent. We show that the conjectures of additivity of the minimum output entropy of a quantum channel, additivity of the Holevo expression for the classical capacity of a quantum channel, additivity of the entanglement of formation, a...

We survey what is known about the information transmitting capacities of quantum channels, and give a proposal for how to calculate some of these capacities using linear programming.

We show that, in a multiparty setting, two nondistillable (bound-entangled) states tensored together can make a distillable state. This is an example of true superadditivity of distillable entanglement. We also show that unlockable bound-entangled states cannot be asymptotically unentangled, providing the first proof that some states are truly boun...

We characterize the class of remote state preparation (RSP) protocols that use only forward classical communication and entanglement, deterministically prepare an exact copy of a general state, and do so obliviously-without leaking further information about the state to the receiver. We prove that any such protocol can be modified to require from t...

This paper studies the class of stochastic maps, or channels, for which (I⊗Φ)(Γ) is always separable (even for entangled Γ). Such maps are called entanglement breaking, and can always be written in the form Φ(ρ)=∑ k R k Tr F k ρ where each R k is a density matrix and F k >0. If, in addition, Φ is trace-preserving, the {F k } must form a positive op...

We report on an experimental study of the Gilmore-Gomory cutting-stock heuristic and related LP-based approaches to bin packing, as applied to instances generated according to discrete distributions. No polynomial running time bound is known to hold for the Gilmore-Gomory approach, and empirical operation counts suggest that no straightforward impl...

I examine the question of why so few classes of quantum algorithms have been discovered. I give two possible explanations for this, and some thoughts about what lines of research might lead to the discovery of more quantum algorithms.

In this paper we present a theoretical analysis of the deterministic on-line Sum of Squares algorithm (SS) for bin packing introduced and studied experimentally in [CJK 99], along with several new variants. SS is applicable to any instance of bin packing in which the bin capacity B and item sizes s(a) are integral (or can be scaled to be so), and r...

The entanglement-assisted classical capacity of a noisy quantum channel (CE) is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel, assuming that the sender and receiver have access to the resource of shared quantum entanglement, which may be used up by the communication protocol. We...

We show that for the tensor product of an entanglement-breaking quantum
channel with an arbitrary quantum channel, both the minimum entropy of an
output of the channel and the Holevo-Schumacher-Westmoreland capacity are
additive. In addition, for the tensor product of two arbitrary quantum
channels, we give a bound involving entanglement of formati...

DOI:https://doi.org/10.1103/PhysRevLett.88.099902

## Citations

... These SDPs are rendered considerably more efficient by invoking symmetry properties, in the spirit of the discussion in Section VIII.F. SDP methods in this vein can sometimes also be used to obtain strong converse bounds 22 on the classical capacity (Ding et al., 2023;Wang et al., 2018). An alternative strong converse bound is obtained from the quantum reverse Shannon theorem (Bennett et al., 2014;Berta et al., 2011): if one sends messages at a rate above the entanglement-assisted classical capacity then the error tends to unit exponentially with n. ...