We find a strong-converse bound on the private capacity of a quantum channel assisted by unlimited two-way classical communication. The bound is based on the max-relative entropy of entanglement and ...its proof uses a new inequality for the sandwiched Rényi divergences based on complex interpolation techniques. We provide explicit examples of quantum channels where our bound improves upon both the transposition bound (on the quantum capacity assisted by classical communication) and the bound based on the squashed entanglement. As an application, we study a repeater version of the private capacity assisted by classical communication and provide an example of a quantum channel with high private capacity but negligible private repeater capacity.
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, ODKLJ, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, SIK, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We prove that the quantum relative entropy decreases monotonically under the application of any positive trace-preserving linear map, for underlying separable Hilbert spaces. This answers in the ...affirmative a natural question that has been open for a long time, as monotonicity had previously only been shown to hold under additional assumptions, such as complete positivity or Schwarz-positivity of the adjoint map. The first step in our proof is to show monotonicity of the sandwiched Renyi divergences under positive trace-preserving maps, extending a proof of the data processing inequality by Beigi (J Math Phys 54:122202,
2013
) that is based on complex interpolation techniques. Our result calls into question several measures of non-Markovianity that have been proposed, as these would assess all positive trace-preserving time evolutions as Markovian.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Fault-Tolerant Coding for Quantum Communication Christandl, Matthias; Muller-Hermes, Alexander
IEEE transactions on information theory,
2024-Jan., 2024-1-00, Volume:
70, Issue:
1
Journal Article
Peer reviewed
Open access
Designing encoding and decoding circuits to reliably send messages over many uses of a noisy channel is a central problem in communication theory. When studying the optimal transmission rates ...achievable with asymptotically vanishing error it is usually assumed that these circuits can be implemented using noise-free gates. While this assumption is satisfied for classical machines in many scenarios, it is not expected to be satisfied in the near term future for quantum machines where decoherence leads to faults in the quantum gates. As a result, fundamental questions regarding the practical relevance of quantum channel coding remain open. By combining techniques from fault-tolerant quantum computation with techniques from quantum communication, we initiate the study of these questions. We introduce fault-tolerant versions of quantum capacities quantifying the optimal communication rates achievable with asymptotically vanishing total error when the encoding and decoding circuits are affected by gate errors with small probability. Our main results are threshold theorems for the classical and quantum capacity: For every quantum channel <inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula> and every <inline-formula> <tex-math notation="LaTeX">\epsilon >0 </tex-math></inline-formula> there exists a threshold <inline-formula> <tex-math notation="LaTeX">p(\epsilon,T) </tex-math></inline-formula> for the gate error probability below which rates larger than <inline-formula> <tex-math notation="LaTeX">C-\epsilon </tex-math></inline-formula> are fault-tolerantly achievable with vanishing overall communication error, where <inline-formula> <tex-math notation="LaTeX">C </tex-math></inline-formula> denotes the usual capacity. Our results are not only relevant in communication over large distances, but also on-chip, where distant parts of a quantum computer might need to communicate under higher levels of noise than affecting the local gates.
Channel capacities quantify the optimal rates of sending information reliably over noisy channels. Usually, the study of capacities assumes that the circuits which the sender and receiver use for ...encoding and decoding consist of perfectly noiseless gates. In the case of communication over quantum channels, however, this assumption is widely believed to be unrealistic, even in the long-term, due to the fragility of quantum information, which is affected by the process of decoherence. Christandl and Müller-Hermes have therefore initiated the study of fault-tolerant channel coding for quantum channels, i.e. coding schemes where encoder and decoder circuits are affected by noise, and have used techniques from fault-tolerant quantum computing to establish coding theorems for sending classical and quantum information in this scenario. Here, we extend these methods to the case of entanglement-assisted communication, in particular proving that the fault-tolerant capacity approaches the usual capacity when the gate error approaches zero. A main tool, which might be of independent interest, is the introduction of fault-tolerant entanglement distillation. We furthermore focus on the modularity of the techniques used, so that they can be easily adopted in other fault-tolerant communication scenarios.
Annihilating Entanglement Between Cones Aubrun, Guillaume; Müller-Hermes, Alexander
Communications in mathematical physics,
06/2023, Volume:
400, Issue:
2
Journal Article
Peer reviewed
Open access
Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels ...with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform the
k
-fold maximal tensor product of a cone
C
1
into the
k
-fold minimal tensor product of a cone
C
2
, and the pair
(
C
1
,
C
2
)
is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that
(
C
1
,
C
2
)
is resilient if either
C
1
or
C
2
is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, ODKLJ, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, SIK, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Genuine high-dimensional entanglement, i.e., the property of having a high Schmidt number, constitutes an instrumental resource in quantum communication, overcoming limitations of low-dimensional ...systems. States with a positive partial transpose (PPT) are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question of whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e., linear, scaling in the local dimension should be possible, albeit without providing insight into the possible slope. We provide the first explicit construction of a family of PPT states that achieves linear scaling in the local dimension and we prove that random PPT states typically share this feature. Our construction also allows us to prove a recent conjecture of Chen et al. on the existence of PPT states whose Schmidt number increases by an arbitrarily large amount upon partial transposition. Finally, we link the Schmidt number to entangled sub-block matrices of a quantum state. We use this connection to prove that quantum states that are either (i) invariant under partial transposition on the smallest of their two subsystems, or (ii) absolutely PPT cannot have a maximal Schmidt number. Overall, our findings shed new light on some fundamental problems in entanglement theory.
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CMK, CTK, FMFMET, IJS, NUK, PNG, UL, UM
Let
ϕ
be a positive map from the
n
×
n
matrices
M
n
to the
m
×
m
matrices
M
m
. It is known that
ϕ
is 2-positive if and only if for all
K
∈
M
n
and all strictly positive
X
∈
M
n
,
ϕ
(
K
∗
X
-
1
K
)
⩾
...ϕ
(
K
)
∗
ϕ
(
X
)
-
1
ϕ
(
K
)
. This inequality is not generally true if
ϕ
is merely a Schwarz map. We show that the corresponding tracial inequality
Tr
ϕ
(
K
∗
X
-
1
K
)
⩾
Tr
ϕ
(
K
)
∗
ϕ
(
X
)
-
1
ϕ
(
K
)
holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results.
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
10.
Cutting Cakes and Kissing Circles Müller-Hermes, Alexander
The Mathematical intelligencer,
09/2021, Volume:
43, Issue:
3
Journal Article
Peer reviewed
Open access
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, ODKLJ, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, SIK, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
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