52、Quantum Communication: Coherence, Trade - offs, and Privacy

Quantum Communication: Coherence, Trade - offs, and Privacy

1. Coherent Communication with Noisy Resources

1.1 Maintaining Quantum Coherence

Quantum coherence maintenance is a key concept. When Alice and Bob perform encoding and decoding in superposition, they can execute powerful protocols. In entanglement - assisted coherent communication and coherent state transfer, Alice uses controlled gates. Bob conducts coherent measurements, storing outcomes in an ancilla register without destroying superpositions. His final step in both protocols is a controlled decoupling unitary, ensuring the environment’s state is independent of their final state.

These two tasks can generate other protocols when combined with entanglement distribution, teleportation, and super - dense coding. The resulting protocols include entanglement - assisted quantum communication, quantum communication, quantum - assisted state transfer, and classical - assisted state transfer.

1.2 Trade - off Coding

Trade - off coding addresses the question of how much of one resource can be generated given a fixed amount of another. Noisy quantum channels are fundamental for information propagation, so it’s crucial to use them effectively. A protocol was determined for entanglement - assisted communication of classical and quantum information by combining existing protocols for classical communication and entanglement - assisted coherent communication.

2. Private Classical Communication

2.1 The Information - Processing Task

The information - processing task for private classical communication involves defining an $(n, P, \varepsilon)$ private classical code.
1. Message Selection and Transmission : Alice selects a message $m$ from a set $M$. She prepares a state $\rho_m^{A’n}$ as input for multiple uses of the quantum channel $N_{A’\to B}$. The state at Bob’s end is $N_{A’n\to B^n}(\rho_m^{A’n})$, where $N_{A’n\to B^n}\equiv(N_{A’\to B})^{\otimes n}$.
2. Decoding and Error Probability : Bob uses a decoding POVM ${\Lambda_m}$ to detect the message. The probability of error for message $m$ is $p_e(m) = Tr{(I - \Lambda_m)N_{A’n\to B^n}(\rho_m^{A’n})}$, and the maximal probability of error is $p_e^ =\max_{m\in M}p_e(m)$, with $p_e^ \leq\varepsilon$ for an $(n, P, \varepsilon)$ code. The rate $P$ of the code is $P\equiv\frac{1}{n}\log|M|$.
3. Privacy Condition : Let $U_{A’\to BE}^N$ be an isometric extension of the channel $N_{A’\to B}$, and the complementary channel to the environment Eve is $\tilde{N} {A’\to E}(\sigma)\equiv Tr_B{U {A’\to BE}^N(\sigma)}$. If Alice sends message $m$, Eve’s state is $\omega_m^{E^n}\equiv\tilde{N} {A’n\to E^n}(\rho_m^{A’n})$. The $\varepsilon$-privacy condition is $\forall m\in M:\frac{1}{2}|\omega_m^{E^n}-\sigma {E^n}|_1\leq\varepsilon$, which means Eve can’t learn about the message.

A rate $P$ of private classical communication is achievable for $N_{A’\to B}$ if there exists an $(n, P - \delta, \varepsilon)$ private classical code for all $\varepsilon\in(0, 1)$, $\delta > 0$, and large enough $n$. The private classical capacity $C_P(N)$ of a channel $N_{A’\to B}$ is the supremum of all achievable rates.

2.2 Mutual Information of Eve

The privacy condition implies that Eve has little mutual information about the transmitted message. From the privacy condition, we have $\varepsilon\geq\frac{1}{2}\sum_{m\in M}\frac{1}{\vert M\vert}|\omega_m^{E^n}-\sigma_{E^n}| 1=\frac{1}{2}|\omega {ME^n}-\pi_M\otimes\sigma_{E^n}| 1$. This implies that Eve’s Holevo information with $M$ is small:
$I(M; E^n) {\omega}=H(M) {\omega}-H(M|E^n) {\omega}=H(M|E^n) {\pi\otimes\sigma}-H(M|E^n) {\omega}\leq\varepsilon\log|M|+(1 + \varepsilon)h_2(\frac{\varepsilon}{1+\varepsilon})$

2.3 The Private Classical Capacity Theorem

The private classical capacity $C_P(N)$ of a quantum channel $N_{A’\to B}$ is equal to the regularized private information of the channel:
$C_P(N)=P_{reg}(N)$, where $P_{reg}(N)\equiv\lim_{k\to\infty}\frac{1}{k}P(N^{\otimes k})$. The private information $P(N)$ is defined as $P(N)\equiv\max_{\rho}[I(X; B) {\sigma}-I(X; E) {\sigma}]$, with $\rho_{XA’}$ being a classical - quantum state $\rho_{XA’}\equiv\sum_{x}p_X(x)\vert x\rangle\langle x\vert_X\otimes\rho_x^{A’}$, and $\sigma_{XBE}\equiv U_{A’\to BE}^N(\rho_{XA’})$.

For degradable channels, the regularization is not necessary as the private information is additive.

2.4 The Direct Coding Theorem

2.4.1 Intuition

Alice aims to create a doubly - indexed codebook ${x^n(m, k)}_{m\in M,k\in K}$ with two properties:
1. Bob should detect the message $m$ and the “junk” variable $k$ with high probability, and $|M||K|\approx2^{nI(X;B)}$.
2. Randomizing over $k$ should cover Eve’s typical subspace, making Eve’s state independent of the message $m$. The size of $K$ should be at least $|K|\approx2^{nI(X;E)}$.

2.4.2 Dimensionality Arguments

Suppose Alice has an ensemble ${p_X(x), \rho_x^{A’}}$. After Alice inputs $\rho_x^{A’}$, the joint state of Bob and Eve is $\rho_x^{BE}\equiv U_{A’\to BE}^N(\rho_x^{A’})$. The local density operators for Bob and Eve are $\rho_x^{B}\equiv Tr_E{\rho_x^{BE}}$ and $\rho_x^{E}\equiv Tr_B{\rho_x^{BE}}$, and the expected density operators are $\rho_B=\sum_{x}p_X(x)\rho_x^{B}$ and $\rho_E=\sum_{x}p_X(x)\rho_x^{E}$.

The following table lists relevant mathematical quantities:
| Party | Quantity | Typical Set/Subspace | Projector |
| — | — | — | — |
| Alice | $X$ | $T_{X^n}^{\delta}$ | N/A |
| Bob | $\rho_{B^n}$ | $T_{\delta}^{B^n}$ | $\Pi_{\delta}^{B^n}$ |
| Bob conditioned on $x^n$ | $\rho_{x^n}^{B^n}$ | $T_{\delta}^{B^n\vert x^n}$ | $\Pi_{\delta}^{B^n\vert x^n}$ |
| Eve | $\rho_{E^n}$ | $T_{\delta}^{E^n}$ | $\Pi_{\delta}^{E^n}$ |
| Eve conditioned on $x^n$ | $\rho_{x^n}^{E^n}$ | $T_{\delta}^{E^n\vert x^n}$ | $\Pi_{\delta}^{E^n\vert x^n}$ |

The packing lemma conditions for Bob’s states and the covering lemma conditions for Eve’s states hold. Consider two sets $M$ and $K$ with sizes $|M| = 2^{n[I(X;B)-I(X;E)-6c\delta]}$ and $|K| = 2^{n[I(X;E)+3c\delta]}$, so $|M\times K| = 2^{n[I(X;B)-3c\delta]}$. We can use $M\times K$ for sending classical information and $|M|$ “privacy amplification” sets of size $|K|$ to reduce Eve’s knowledge.

2.4.3 Random Code Construction

If Alice randomly selects a code according to the ensemble ${p’ {X’^n}(x^n), \rho {x^n}^{A’^n}}$, where $p’ {X’^n}(x^n)$ is the pruned distribution, we choose $|M||K|$ random variables $X^n(m, k)$ according to $p’ {X’^n}(x^n)$. The code $C = {x^n(m, k)} {m\in M,k\in K}$ is then determined.
1. Bob’s Decoding : By the packing lemma, the expectation of the average error probability over all codes for Bob to distinguish $(m, k)$ is low.
2. Eve’s Obfuscation : Divide the code $C$ into $|M|$ privacy amplification sets $C_m\equiv{\rho {X^n(m,k)}^{E^n}} {k\in K}$. The fake density operator of each set is $\hat{\rho}_m^{E^n}\equiv\frac{1}{|K|}\sum {k\in K}\rho_{X^n(m,k)}^{E^n}$, and the obfuscation error $oe(C_m)\equiv|\hat{\rho} m^{E^n}-\rho {E^n}|_1$. The covering lemma shows that the obfuscation error for each set has a high probability of being small when $n$ is large.

2.4.4 Derandomization

We want to find a code that has good classical communication and privacy. Let (E_0) be the event that the random code (C) is (\varepsilon) - good for classical communication ((E_0={\overline{p} e(C)\leq\varepsilon})), and (E_m) be the event that the (m)th message in the random code is (\varepsilon) - private ((E_m = {oe(C_m)\leq\varepsilon})). We want the event (E {priv}\equiv E_0\cap\bigcap_{m\in M}E_m) to occur.

We consider the complementary event (E_{priv}^c=E_0^c\cup\bigcup_{m\in M}E_m^c). Using the union bound, we have:

(Pr\left(E_0^c\cup\bigcup_{m\in M}E_m^c\right)\leq Pr{E_0^c}+\sum_{m\in M}Pr{E_m^c})

• Bounding (Pr{E_0^c}) : By Markov’s inequality, since (\overline{p}_e(C)) is non - negative, (Pr{E_0^c}=Pr\left[\overline{p}_e(C)\geq(\varepsilon’)^{3/4}\right]\leq\frac{E_C{\overline{p}_e(C)}}{(\varepsilon’)^{3/4}}\leq\frac{\varepsilon’}{(\varepsilon’)^{3/4}}=\sqrt[4]{\varepsilon’})
• Bounding (Pr{E_m^c}) : From previous results, (Pr{E_m^c}=Pr\left[oe(C_m)>\varepsilon + 4\sqrt{\varepsilon}+24\sqrt[4]{\varepsilon}\right]<\frac{\varepsilon}{|M|}), so (\sum_{m\in M}Pr{E_m^c}<|M|\frac{\varepsilon}{|M|}=\varepsilon)

Then (Pr\left[E_{priv}^c\right]\leq\sqrt[4]{\varepsilon’}+\varepsilon), and (Pr{E_{priv}}\geq1 - (\sqrt[4]{\varepsilon’}+\varepsilon)). So there exists a code (C={x^n(m, k)}_{m\in M,k\in K}) such that (\overline{p}_e(C)\leq(\varepsilon’)^{3/4}) and (\forall m:oe(C_m)\leq\varepsilon + 4\sqrt{\varepsilon}+24\sqrt[4]{\varepsilon})

The process can be visualized in the following mermaid flowchart:

graph LR
    A[Start] --> B[Random Code Selection]
    B --> C{Is code good for comm?}
    C -- Yes --> D{Is each msg private?}
    C -- No --> E[Not a good code]
    D -- Yes --> F[Good private code found]
    D -- No --> E

2.4.5 Expurgation

We want to strengthen the code to have a low maximal probability of error. We expurgate codewords in two steps:
1. Expurgate the worst (\sqrt{\varepsilon’}) fraction of the codewords in each privacy amplification set.
2. Expurgate the worst (\sqrt{\varepsilon’}) fraction of the privacy amplification sets.

The expurgated sets (M’) and (K’) become a fraction (1-\sqrt{\varepsilon’}) of their original size. The expurgated code is (C’={x^n(m, k)} {m\in M’,k\in K’}), and the expurgated privacy amplification sets are (C_m’={x^n(m, k)} {k\in K’})

The fake density operator for each expurgated privacy amplification set is (\hat{\rho} {m’}^{E^n}\equiv\frac{1}{|C_m’|}\sum {k\in K’}\rho_{x^n(m,k)}^{E^n}). It can be shown that (\forall m\in M’), (|\hat{\rho}_{m’}^{E^n}-\hat{\rho}_m^{E^n}|_1\leq2\sqrt{\varepsilon’})

The expurgated code has good privacy: (\forall m\in M), (|\hat{\rho} m^{E^n}-\rho {E^n}|_1\leq\varepsilon + 4\sqrt{\varepsilon}+24\sqrt[4]{\varepsilon}+2\sqrt{\varepsilon’}) and reliable communication: (\forall m\in M,k\in K), (p_e(C,m,k)\leq\sqrt[4]{\varepsilon’})

The following table summarizes the properties of the original and expurgated codes:
| Code | Average Error | Maximal Error | Privacy |
| — | — | — | — |
| Original | (\overline{p} e(C)\leq(\varepsilon’)^{3/4}) | - | (\forall m:oe(C_m)\leq\varepsilon + 4\sqrt{\varepsilon}+24\sqrt[4]{\varepsilon}) |
| Expurgated | - | (p_e(C,m,k)\leq\sqrt[4]{\varepsilon’}) | (\forall m\in M), (|\hat{\rho}_m^{E^n}-\rho {E^n}|_1\leq\varepsilon + 4\sqrt{\varepsilon}+24\sqrt[4]{\varepsilon}+2\sqrt{\varepsilon’}) |

3. Applications and Implications

3.1 Quantum Key Distribution

The private classical capacity theorem is crucial for quantum key distribution. It determines the maximum rate at which two parties can generate a shared secret key. Since private classical communication ensures that no third - party can access the transmitted information, it provides a solid foundation for secure key generation.

3.2 Quantum Communication at Coherent Information Rate

Private classical communication can be used to achieve quantum communication at the coherent information rate. The intuition comes from the no - cloning theorem. If Alice can send private classical messages, Eve’s state is independent of the message. By creating a coherent version of the private classical code using superpositions, we can expect quantum information to be transmitted without Eve stealing the coherence.

The steps for using private classical communication for quantum communication are as follows:
1. Establish a private classical code with high privacy and reliability.
2. Create a coherent version of the private classical code by using superpositions of the private classical codewords.
3. Since Eve’s state is independent of the quantum message, the quantum information should appear at Bob’s end for decoding.

In conclusion, the concepts of coherent communication, trade - off coding, and private classical communication are fundamental in the field of quantum communication. They provide the theoretical basis for secure and efficient information transfer, with applications in quantum key distribution and quantum communication at the coherent information rate.