QuantumInfo.Finite.Capacity

Quantum Capacity

This focuses on defining and proving theorems about the quantum capacity, the maximum asymptotic rate at which quantum information can be coherently transmitted. The precise definition is not consistent in the literature, see [Capacity_doc](./QuantumInfo/Finite/Capacity_doc.html) for a note on what has been used and how that was used to arrive at the following definition:

1. A channel A `Emulates` another channel B if there are D and E such that D∘A∘E = B. 2. A channel A `εApproximates` channel B (of the same dimensions) if the for every state ρ, the fidelity F(A(ρ), B(ρ)) is at least 1-ε. 3. A channel A `AchievesRate` R:ℝ if for every ε>0, n copies of A emulates some channel B such that log2(dimout(B))/n ≥ R, and that B is εApproximately the identity. 4. The `quantumCapacity` of the channel A is the supremum of the achievable rates, i.e. `sSup { R : ℝ | AchievesRate A R }`.

The most basic facts: * `emulates_self`: Every channel emulates itself. * `emulates_trans`: If A emulates B and B emulates C, then A emulates C. (That is, emulation is an ordering.) * `εApproximates A B ε` is equivalent to the existence of some δ (depending ε and dims(A)) so that |A-B| has diamond norm at most δ, and δ→0 as ε→0. * `achievesRate_0`: Every channel achievesRate 0. So, the set of achievable rates is Nonempty. * If a channel achievesRate R₁, it also every achievesRate R₂ every R₂ ≤ R₁, i.e. it is an interval extending left towards -∞. Achievable rates are `¬BddBelow`. * `bddAbove_achievesRate`: A channel C : dimX → dimY cannot achievesRate R with `R > log2(min(dimX, dimY))`. Thus, the interval is `BddAbove`.

The nice lemmas we would want: * Capacity of a replacement channel is zero. * Capacity of an identity channel is `log2(D)`. * Capacity is superadditive under tensor products. (That is, at least additive. Showing that it isn't _exactly_ additive, unlike classical capacity which is additive, is a much harder task.) * Capacity of a kth tensor power is exactly k times the capacity of the original channel. * Capacity does not decrease under tensor sums. * Capacity does not increase under composition.

Then, we should show that our definition is equivalent to some above. Most, except (3), should be not too hard to prove.

Then the LSD theorem establishes that the single-copy coherent information is a lower bound. This is stated in `coherentInfo_le_quantumCapacity`. The corollary, that the n-copy coherent information converges to the capacity, is `quantumCapacity_eq_piProd_coherentInfo`.

TODO

The only notion of "capacity" here currently is "quantum capacity" in the usual sense. But there are several non-equal capacities relevant to quantum channels, see e.g. [Watrous's notes](https://cs.uwaterloo.ca/~watrous/TQI/TQI.8.pdf) for a list: * Quantum capacity (`quantumCapacity`) * Quantum 1-shot capacity * Entanglement-assisted classical capacity * Qss, the quantum side-channel capacity * Holevo capacity, aka Holevo χ. The Holevo–Schumacher–Westmoreland theorem as a major theorem * Entanglement-assisted Holevo capacity * Entanglement-assisted quantum capacity * One- and two-way distillable entanglement

And other important theorems like superdense coding, nonadditivity, superactivation

17 declarations

definition

$\Lambda_1$ emulates $\Lambda_2$ via encoding and decoding maps

#Emulates

A quantum channel (CPTP map) $\Lambda_1: \text{Mat}_{d_1}(\mathbb{C}) \to \text{Mat}_{d_2}(\mathbb{C})$ emulates another quantum channel $\Lambda_2: \text{Mat}_{d_3}(\mathbb{C}) \to \text{Mat}_{d_4}(\mathbb{C})$ if there exist an encoding CPTP map $E: \text{Mat}_{d_3}(\mathbb{C}) \to \text{Mat}_{d_1}(\mathbb{C})$ and a decoding CPTP map $D: \text{Mat}_{d_2}(\mathbb{C}) \to \text{Mat}_{d_4}(\mathbb{C})$ such that their composition satisfies $D \circ \Lambda_1 \circ E = \Lambda_2$ .

definition

Channel $\mathcal{A}$ $\epsilon$ -approximates $\mathcal{B}$ if $\forall \rho, F(\mathcal{A}(\rho), \mathcal{B}(\rho)) \geq 1 - \epsilon$

#εApproximates

Let $\mathcal{A}$ and $\mathcal{B}$ be two completely positive trace-preserving (CPTP) maps (quantum channels) from a system of dimension $d_1$ to a system of dimension $d_2$ . We say that $\mathcal{A}$ $\epsilon$ -approximates $\mathcal{B}$ if for every input quantum state (density matrix) $\rho \in \text{MState}(d_1)$ , the fidelity between the output states $F(\mathcal{A}(\rho), \mathcal{B}(\rho))$ is at least $1 - \epsilon$ , where $\epsilon \in \mathbb{R}$ .

definition

$R$ is an achievable rate for CPTP map $\mathcal{A}$

#AchievesRate

Let $\mathcal{A}: \mathcal{B}(\mathcal{H}_{in}) \to \mathcal{B}(\mathcal{H}_{out})$ be a completely positive trace-preserving (CPTP) map. A rate $R \in \mathbb{R}$ is said to be achievable by $\mathcal{A}$ if for every $\epsilon > 0$ , there exists an integer $n > 0$ and a CPTP map $\mathcal{B}: \mathcal{B}(\mathbb{C}^d) \to \mathcal{B}(\mathbb{C}^d)$ such that: 1. The $n$ -fold tensor product channel $\mathcal{A}^{\otimes n}$ emulates $\mathcal{B}$ . That is, there exist encoding and decoding CPTP maps $\mathcal{E}$ and $\mathcal{D}$ such that $\mathcal{D} \circ \mathcal{A}^{\otimes n} \circ \mathcal{E} = \mathcal{B}$ . 2. The transmission rate satisfies $\frac{\log_2 d}{n} \geq R$ (or equivalently, $\log_2 d \geq R \cdot n$ ). 3. The channel $\mathcal{B}$ is an $\epsilon$ -approximation of the identity channel $\text{id}_d$ , meaning that for every state $\rho$ , the fidelity $F(\mathcal{B}(\rho), \rho) \geq 1 - \epsilon$ .

definition

Quantum capacity $Q(\mathcal{A})$ of a channel $\mathcal{A}$ as the supremum of achievable rates.

#quantumCapacity

For a quantum channel $\mathcal{A}$ mapping states between dimensions $d_1$ and $d_2$ (represented as a completely positive trace-preserving map), its quantum capacity $Q(\mathcal{A})$ is defined as the supremum of all achievable rates $R \in \mathbb{R}$ . A rate $R$ is achievable if for every $\varepsilon > 0$ , there exists an integer $n$ such that $n$ copies of the channel $\mathcal{A}^{\otimes n}$ can emulate a channel $\mathcal{B}$ with $\frac{1}{n}\log_2(\text{dim}(\mathcal{B})) \geq R$ , where $\mathcal{B}$ is $\varepsilon$ -approximately the identity channel.

theorem

Every CPTP map emulates itself

#emulates_self

For any completely positive trace-preserving (CPTP) map $\Lambda$ , the map $\Lambda$ emulates itself. According to the definition of emulation, this implies there exist CPTP maps $\mathcal{D}$ and $\mathcal{E}$ such that $\mathcal{D} \circ \Lambda \circ \mathcal{E} = \Lambda$ .

theorem

Emulation of quantum channels is transitive

#emulates_trans

Let $\Lambda_1, \Lambda_2,$ and $\Lambda_3$ be completely positive trace-preserving (CPTP) maps (quantum channels). If $\Lambda_1$ emulates $\Lambda_2$ and $\Lambda_2$ emulates $\Lambda_3$ , then $\Lambda_1$ emulates $\Lambda_3$ . Here, a channel $A$ is said to emulate a channel $B$ if there exist CPTP maps $\mathcal{D}$ and $\mathcal{E}$ such that $\mathcal{D} \circ A \circ \mathcal{E} = B$ .

theorem

Every CPTP map $\Lambda$ satisfies $\varepsilon\text{-Approximates}(\Lambda, \Lambda, 0)$

#εApproximates_self

Let $\Lambda$ be a completely positive trace-preserving (CPTP) map from the space of matrix states $\mathcal{M}_{d_1}$ to $\mathcal{M}_{d_2}$ . Then $\Lambda$ $0$ -approximates itself. Specifically, for any state $\rho$ , the fidelity between $\Lambda(\rho)$ and $\Lambda(\rho)$ satisfies $F(\Lambda(\rho), \Lambda(\rho)) \geq 1 - 0$ .

theorem

$\epsilon$ -approximation of quantum channels is monotone in $\epsilon$

#εApproximates_monotone

Let $\mathcal{A}$ and $\mathcal{B}$ be quantum channels (completely positive trace-preserving maps) from dimensions $d_1$ to $d_2$ . If $\mathcal{A}$ is an $\epsilon_0$ -approximation of $\mathcal{B}$ (meaning for every state $\rho$ , the fidelity $F(\mathcal{A}(\rho), \mathcal{B}(\rho)) \ge 1 - \epsilon_0$ ), then for any $\epsilon_1 \in \mathbb{R}$ such that $\epsilon_0 \le \epsilon_1$ , $\mathcal{A}$ is also an $\epsilon_1$ -approximation of $\mathcal{B}$ .

theorem

Every CPTP map achieves rate $0$

#achievesRate_0

For any completely positive trace-preserving (CPTP) map $\Lambda$ between finite-dimensional Hilbert spaces, the channel $\Lambda$ achieves a transmission rate of $R = 0$ in the sense of quantum capacity.

theorem

The identity channel achieves a rate of $\log_2(\text{dim})$

#id_achievesRate_log_dim

Let $d_1$ be a finite type representing the dimension of a Hilbert space, and let $I$ be the identity completely positive trace-preserving (CPTP) map on that space. Then the map $I$ achieves a quantum communication rate $R = \log_2(|d_1|)$ , where $|d_1|$ denotes the cardinality of the type $d_1$ .

theorem

Quantum Achievable Rate $R \le \log_2(d_{\text{in}})$

#not_achievesRate_gt_log_dim_in

Let $\Lambda: \mathcal{L}(\mathbb{C}^{d_1}) \to \mathcal{L}(\mathbb{C}^{d_2})$ be a Completely Positive Trace-Preserving (CPTP) map (a quantum channel) where $d_1$ and $d_2$ are the dimensions of the input and output Hilbert spaces respectively. If a rate $R \in \mathbb{R}$ satisfies $R > \log_2(d_1)$ , then the channel $\Lambda$ cannot achieve rate $R$ .

theorem

A CPTP map $\Lambda$ cannot achieve a rate $R > \log_2(\text{dim}_{\text{out}}(\Lambda))$

#not_achievesRate_gt_log_dim_out

Let $\Lambda$ be a completely positive trace-preserving (CPTP) map from a system with dimension $d_1$ to a system with dimension $d_2$ . For any rate $R \in \mathbb{R}$ , if $R > \log_2(d_2)$ , then $\Lambda$ does not achieve rate $R$ (i.e., $\neg \text{AchievesRate}(\Lambda, R)$ ), where $d_2$ is the cardinality of the output type.

theorem

The set of achievable rates of a quantum channel is bounded above

#bddAbove_achievesRate

Let $\Lambda$ be a completely positive trace-preserving (CPTP) map from a system of dimension $d_1$ to a system of dimension $d_2$ . The set of all achievable rates $R \in \mathbb{R}$ for the channel $\Lambda$ is bounded above.

theorem

The quantum capacity of a CPTP map is non-negative ( $0 \le Q(\Lambda)$ )

#zero_le_quantumCapacity

For any quantum channel (completely positive trace-preserving map) $\Lambda$ with input dimension $d_1$ and output dimension $d_2$ , its quantum capacity $Q(\Lambda)$ is non-negative, i.e., $0 \le Q(\Lambda)$ .

theorem

Quantum Capacity of $\Lambda$ is bounded by $\log_2(\text{dim}_{\text{in}})$

#quantumCapacity_ge_log_dim_in

Let $\Lambda$ be a Completely Positive Trace-Preserving (CPTP) map with input dimension $|d_1|$ . The quantum capacity $Q(\Lambda)$ of the channel satisfies: $Q(\Lambda) \le \log_2(|d_1|)$ where $|d_1|$ denotes the cardinality of the finite type $d_1$ representing the input Hilbert space basis.

theorem

The LSD Theorem: $I_c(\rho, \Lambda) \leq Q(\Lambda)$

#coherentInfo_le_quantumCapacity

For any quantum channel $\Lambda$ (represented as a completely positive trace-preserving map from dimension $d_1$ to $d_2$ ) and any input state $\rho$ in the space of states of dimension $d_1$ , the single-copy coherent information $I_c(\rho, \Lambda)$ is a lower bound for the quantum capacity $Q(\Lambda)$ . That is, $I_c(\rho, \Lambda) \leq Q(\Lambda)$ .

theorem

Quantum Capacity $Q(\Lambda)$ Equals the Limit of Coherent Information of $n$ -copy Channels

#quantumCapacity_eq_piProd_coherentInfo

Let $\Lambda: \mathcal{B}(\mathcal{H}_{d_1}) \to \mathcal{B}(\mathcal{H}_{d_2})$ be a completely positive trace-preserving (CPTP) map. The quantum capacity of $\Lambda$ , denoted $Q(\Lambda)$ , is equal to the supremum of the set of rates $r$ such that there exists a natural number $n \in \mathbb{N}$ and an input state $\rho$ on the $n$ -fold tensor product space $\mathcal{H}_{d_1}^{\otimes n}$ for which $r = \frac{1}{n} I_c(\rho, \Lambda^{\otimes n})$ , where $I_c$ denotes the coherent information. Specifically, $Q(\Lambda) = \sup \left\{ r \in \mathbb{R} \mid \exists n \in \mathbb{N}, \exists \rho \in \mathcal{S}(\mathcal{H}_{d_1}^{\otimes n}), r = \frac{1}{n} I_c(\rho, \Lambda^{\otimes n}) \right\},$ where $\Lambda^{\otimes n}$ is the $n$ -copy tensor product channel $\bigotimes_{i=1}^n \Lambda$ .

QuantumInfo.Finite.Capacity

Quantum Capacity

TODO

Λ1\Lambda_1Λ1​ emulates Λ2\Lambda_2Λ2​ via encoding and decoding maps

Channel A\mathcal{A}A ϵ\epsilonϵ-approximates B\mathcal{B}B if ∀ρ,F(A(ρ),B(ρ))≥1−ϵ\forall \rho, F(\mathcal{A}(\rho), \mathcal{B}(\rho)) \geq 1 - \epsilon∀ρ,F(A(ρ),B(ρ))≥1−ϵ

RRR is an achievable rate for CPTP map A\mathcal{A}A

Quantum capacity Q(A)Q(\mathcal{A})Q(A) of a channel A\mathcal{A}A as the supremum of achievable rates.

Every CPTP map emulates itself

Emulation of quantum channels is transitive

Every CPTP map Λ\LambdaΛ satisfies ε-Approximates(Λ,Λ,0)\varepsilon\text{-Approximates}(\Lambda, \Lambda, 0)ε-Approximates(Λ,Λ,0)

ϵ\epsilonϵ-approximation of quantum channels is monotone in ϵ\epsilonϵ

Every CPTP map achieves rate 000

The identity channel achieves a rate of log⁡2(dim)\log_2(\text{dim})log2​(dim)

Quantum Achievable Rate R≤log⁡2(din)R \le \log_2(d_{\text{in}})R≤log2​(din​)

A CPTP map Λ\LambdaΛ cannot achieve a rate R>log⁡2(dimout(Λ))R > \log_2(\text{dim}_{\text{out}}(\Lambda))R>log2​(dimout​(Λ))

The set of achievable rates of a quantum channel is bounded above

The quantum capacity of a CPTP map is non-negative (0≤Q(Λ)0 \le Q(\Lambda)0≤Q(Λ))

Quantum Capacity of Λ\LambdaΛ is bounded by log⁡2(dimin)\log_2(\text{dim}_{\text{in}})log2​(dimin​)

The LSD Theorem: Ic(ρ,Λ)≤Q(Λ)I_c(\rho, \Lambda) \leq Q(\Lambda)Ic​(ρ,Λ)≤Q(Λ)

Quantum Capacity Q(Λ)Q(\Lambda)Q(Λ) Equals the Limit of Coherent Information of nnn-copy Channels

$\Lambda_1$ emulates $\Lambda_2$ via encoding and decoding maps

Channel $\mathcal{A}$ $\epsilon$ -approximates $\mathcal{B}$ if $\forall \rho, F(\mathcal{A}(\rho), \mathcal{B}(\rho)) \geq 1 - \epsilon$

$R$ is an achievable rate for CPTP map $\mathcal{A}$

Quantum capacity $Q(\mathcal{A})$ of a channel $\mathcal{A}$ as the supremum of achievable rates.

Every CPTP map $\Lambda$ satisfies $\varepsilon\text{-Approximates}(\Lambda, \Lambda, 0)$

$\epsilon$ -approximation of quantum channels is monotone in $\epsilon$

Every CPTP map achieves rate $0$

The identity channel achieves a rate of $\log_2(\text{dim})$

Quantum Achievable Rate $R \le \log_2(d_{\text{in}})$

A CPTP map $\Lambda$ cannot achieve a rate $R > \log_2(\text{dim}_{\text{out}}(\Lambda))$

The quantum capacity of a CPTP map is non-negative ( $0 \le Q(\Lambda)$ )

Quantum Capacity of $\Lambda$ is bounded by $\log_2(\text{dim}_{\text{in}})$

The LSD Theorem: $I_c(\rho, \Lambda) \leq Q(\Lambda)$

Quantum Capacity $Q(\Lambda)$ Equals the Limit of Coherent Information of $n$ -copy Channels