密码学中的state是什么意思（密码学中mod是什么意思）

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Charles Henry Bennett（生于1943年）是IBM Research的物理学家和信息理论家。Bennett最近在IBM工作的重点是重新审视信息的物理基础，将量子物理应用于信息交换的问题。 他在阐明物理和信息之间的相互联系方面发挥了重要作用，特别是在量子计算、元胞自动机和可逆计算等领域。 他与Gilles Brassard一起发现了量子密码学的概念，并且是现代量子信息理论的创始人之一。

视频

【墨子沙龙】Charles Bennett（上）_【墨子沙龙】Charles Henry Bennett（上）_腾讯视频

中文版

接下来我也将给大家介绍一下物理与数学之间的关系

但是我想要做的是，选择数学或物理学中的一块领域

它与火箭科学相比更难以理解

如果你不是有技术头脑的人

我会给你们介绍一些理解它的方法

或者如果你是一个有技术头脑的人

我将通过比喻的方式把它解释给大众

如果不懂量子物理学

量子物理是信息和通信的基础

我认为会是一件可怕的事

它是我们所生活的宇宙的基石之一

不只是科学家，每个人都应当懂得它

这就意味着我们需要找到解释它的正确方法

我们正处于一场信息革命中

这场革命源于许多人提出的一些非常杰出的抽象思想

但是我们会由这些人想到图灵【注：艾伦·麦席森·图灵，英国数学家、逻辑学家，被称为计算机科学之父，人工智能之父】

他将计算的理论置于抽象的基础上

让你觉得计算与电脑硬件是相互独立的

存在一个通用的计算的概念

存在一个可以模拟任何其他计算机的通用计算机

香农【注：克劳德·艾尔伍德·香农是美国数学家、信息论的创始人】

他做了一个更具有革命性的工作

用一种完全数学上的方式来考虑通信

也就是说，存在一种

独立于信息本身含义的通信理论

即使在如今，如果你在马路上和一个人说：

“我想和你交流但是不告诉你信息的含义。”

他们会认为你有点疯狂

这就引起了我们今天的信息革命

但是图灵和香农认为的信息载体

在物理学家看来是经典系统

就是说，原则上，这些态是确实可以分辨的

并且测量它们的时候，态不会改变

你是可以复制信息的

事实上，你可以很容易完成这件事

你遇到的唯一麻烦就是典型的法律问题

为了描述两个事物的联合状态

充分必要条件是描述每件事物的状态

这几乎是不言而喻的，无论如何

物理学家和化学家已经知道很久的是

原子和类似于原子与光子的这类小粒子和经典系统的行为是不一样的

只有对于物理学家和化学家来说，这是个感兴趣的问题

现在我们知道，这张PPT最后描写的粒子的性质

不仅对于物理和化学是重要的

同样它们对于理解什么是信息和通信也是重要的

换句话说，这是香农和图灵需要考虑的

如今每个人都理解经典信息

因为每个人每时每刻都在使用它

如果你去一个电脑商店，说：

“你有可以给我的电脑macbook用的那个程序吗？”

原则上，你可以使用图灵的观点：存在一台通用的计算机

所以我不用再向你解释经典信息了

但是量子信息是神秘的

你经常发现你处在这样一种情形下

人们说：“你是从事量子计算和量子信息（研究工作）的，

那是什么？”

我便开始给他们讲希尔伯特空间

【注：在数学中，希尔伯特空间是欧几里德空间的一个推广 ，其不再局限于有限维的情形】

以及诸如此类的东西

他们便在那里玩，对我的工作内容不是很感兴趣

所以我需要找到一个适当的方式来解释我的工作

我能想到的最好的解释方法是：

量子信息就像在梦里的信息一样

如果你试着去向别人描述你的梦

不久后，你不记得梦

你只记得你描述梦的时候说了什么

而且你无法向别人证明你梦到的内容

你可以在描述梦的时候说谎，而且不会被发现

除非你在向配偶描述梦的时候说谎

无论如何，和梦不同的是

尽管西格蒙德·弗洛伊德做出了最大的努力【注：西格蒙德·弗洛伊德，奥地利精神病医师、心理学家、精神分析学派创始人】

关于量子信息如何工作的理论是众所周知的

这就是我今天想要讲的内容

量子力学和量子信息的基本原理是

在任何一个物理系统中，存在最大数量的确定可区分态

对于非常简单的系统来说，就像我们一直讨论的偏振光子

存在两个确定可区分的态

在任何一对确定可区分态中间

存在其他中间态

中间态与任何一个确定可区分态之间都不是确定可区分的

这不意味着只有一些态是确定可区分的

任何一对态

可能的物理态对应空间中的一个方向

不是普通的三维空间

而是一个内部空间

其维度跟系统的最大可区分态数量一样多

因此对于一个光子来说，这个空间是二维的

如果方向是正交的，那么态是可区分的

如果方向不是正交的，任何测量都是不能区分他们的

就是这样，简而言之，这是量子力学

不是那么难理解

所以我们来看一些关于上面结论的证明

但是在讲之前，需要说明一下，虽然物理学家明白这一些

但是没想到它会与信息有关

现在我们知道，普通信息可以简化为比特

我们可以把任何无论多么复杂的东西进行数字化

将它变成许多0和1

整个数字化过程可以简化为

对这些0和1做“与”和“非”操作

如果你将比特做的更小更快更便宜

你会把计算机做得更加实用化

无论你用什么作为比特的载体都没关系

如果你有一种办法，可以实现0和1

与和非，你便准备好了

现在物理学家还没想到这个方法

但是你可以对量子信息做类似表述

把量子信息简化成量子比特

任何一种双态系统，例如一个偏振光子

对它做任何操作

对一个量子态做任何操作可以简化为对比特

做一次单比特门和双比特门操作

大约1995年才被发现

正如经典比特可以独立于它们的物理载体

而被用在数学或者通信上

所以量子比特和量子“门”在不同量子系统中是可替代的

这里有你们之前就已经看过的图解示例

我们知道我们可以用单个偏振光子来携带一比特的信息

如果我们像这样设置，水平偏振的光子将会直走穿过晶体

竖直偏振的光子将会偏离原路径

可以用两个单独的计数器对水平和竖直偏振光子计数

但是如果你入射对角偏振的光子

它不会偏移成一个中间量

而是一部分对角偏振的光子会进入这一束（水平偏振光走的路径）

变成水平偏振光

另一部分进入那一束（竖直偏振光走的路径），变成竖直偏振光

这是一件让爱因斯坦非常困惑的事

他说：“为什么在相同条件下准备好的事物

经历同样的操作，表现出来的却是随机的呢？”

他说

他不喜欢这个事实，看起来上帝好像在和宇宙玩骰子

稍后我们会做解释

但是我们现在讨论将装置旋转

在我旋转装置之前

我将要谈论这个奇怪的行为，这个概率性的行为

对于这件事最好的比喻就是，我从同事那里听到的故事

同事比尔·伍德斯从事本科生教学数十年

他说：“事实上这是一个有教育意义的比喻。”

还记得老式学校，学生不应该在课堂上说话

只应该回答老师的提问

被测量的量子系统就像这样

所以老师就像是测量改进的部分

在这种情况下，由晶体和探测器组成

学生就像光子一样

所以老师问：“你是竖直偏振还是水平偏振的？”

学生回答到：“我的偏振方向与水平方向夹角是55度。”

（老师说）“我在问你问题 , 你是竖直偏振还是水平偏振的？”

（学生说）“水平偏振，先生。”

（老师说）“你之前有其他方向的偏振吗？”

（学生说）“没有，先生，我一直都是水平偏振的。”

量子测量也是这样的情况

但是这样并没有让量子测量少一点神秘感

这只是看起来能让人们对量子测量有更多熟悉感

在我们解释量子测量这件事为什么发生之前

先解释一下量子测量是如何用于密码学的

我认为已经很好地解释了

你可以准备一连串竖直和水平偏振的光子

如果你知道它们是什么

一个光子可以携带一个比特的信息

他们可以确定的走另一条路

如果你入射对角偏振的光子，你同样可以确定区分他们

但是你必须用一个旋转过的测量装置

没有测量可以区分所有四种偏振态

这个基本限制使得量子货币和量子密码学成为可能

然后这些光子会被执行由Gilles (Gilles Brassard) 和我发展出的特定的操作

即我们把四种不同偏振的光子从Alice发送到Bob

然后我们对普通信息进行一个经典的讨论

这些信息是窃听者可以窃听的

最终的结果是，Alice和Bob之间有一串共享密钥串

如果有太多窃听，他们会把这串密钥丢弃

如果能以很高的置信度知道并没有窃听行为

他们会知道这个密钥只属于他们，其他任何人都不知晓

现在我开始讲另一种密码术，这种密码术建立在量子纠缠上

但是我需要解释什么是纠缠

但是这种不确定原理是无法摆脱的

如果你试着测量光子，你无法准确测量它的偏振

如果你不知道最开始的偏振是什么

因此不存在这样一种装置，你放进去一个光子

它会告诉你这个光子的偏振是什么

你无法克隆一个光子，因为如果你可以克隆它

你就可以造出成千上万的光子

然后测量光子，从统计数据中以任意精度确定光子的偏振

存在一种可以放大光子的装置

我手中就拿着一个，这个装置叫激光

但是激光的工作原理是

你知道“laser”这个单词意思是受激辐射光放大

这个意思是，你输入一个光子，会由几个光子输出

它们通常工作在振荡模式下

所以它可能是一个关于受激辐射的光振荡器

但是他们认为“loser”这个缩写并不是很好

因此无论如何，这是一个光放大器

但是有一个关于激光的事

光放大器的意思是如果你输入一个非常弱的信号

例如一个单光子

有时它会被放大，但有时不会被放大

所以你会得到一个与它一起输出的噪声光子

所以，光束会变得更亮

但是在分辨光子偏振上，与原始未放大光子相比

亮光束不会更有用

因此没有办法解决这个不确定原则

——The end——

了解更多量子密码学

旧时代密码术和密码学圣杯 （上）冷战中的“一次一密”| Gilles Brassard

公钥密码术的原理、发展及其在量子时代的困境 （中） | Gilles Brassard

量子密码学的诞生及其战场（下）| Gilles Brassard

量子密码学的实现（上）爱因斯坦的困惑| Artur Ekert

量子密码学的实现（下）像物理学家一样思考| Artur Ekert

英文版

So I'm going to be talking about the relation of physics to mathematics also.

But what I'd like to do is take an area of mathematics or physics,

which is considered to be really hard to understand, worse than rocket science.

And give you some tools for understanding it

if you're not a technically minded person.

Or if you are a technically minded person,

some metaphors are ways of explaining it to the general public.

Because I think the non understanding of quantum physics,

which is now lies at the root of information and communication,

is a terrible thing to not understand.

It's one of the fundamental things about the universe that we live in

that everyone should understand, not just scientists.

And that means that we need to find the right ways of explaining it.

We're in the middle of an information revolution.

This revolution is based on some really brilliant abstractions by many people.

But we associate them with Turing,

who put the theory of computing into an abstract basis,that is

that you can think about computing is independent of the notion of thehardware.

There is a universal notion of computing.

There is a universal computer that can simulate any other computer.

And Shannon,

who did something even more revolutionary,

that is to think about communication in an entirely mathematical way.

That is that there is a theory of communication.

that is independent of the meaning of the message.

Now,even today, if you try to say to a person on the street,

"I want to tell you about communication without telling you about meaning."

they would think you were a little crazy.

And yet that's what leads to the information revolution we had today.

But the information carriers, Turing and Shannon thought about, were viewed

as what a physicist would call a classical system.

And that is that their statesare in principle reliably distinguishable,

and are not disturbed by measuring them.

Youcan make a copy of information.

In fact, you can do it very easily.

And the only kind of trouble you can get into is a typically legal trouble.

Tospecify the joint state of two things,

it's necessary and sufficient to say the state of each one.

That almost goes without saying.

Well,anyway, physicists and chemists have known for a long time,

that atoms and small particles like atoms andphotons don't really behave that way.

But that was a matter of interest, really only for physicists and chemists.

And now we know that these last properties here

are important not only for physics and chemistry,

butfor understanding what information and communication are all about.

In other words, what Shannon and Turing should be thinking about.

Nowadays,everybody understands classical information,

cause we use it all the time.

If you go to a computer store and you say,

"Do you have that program for my mac?"

You're in principle, using the idea of Turing and there is a universal computer.

SoI don't have to explain classical information to you anymore.

But quantum information is mysterious.

And you often find yourself.

What is it?"

AndI start telling them about Hilbert space,

andso on like that.

And they get plays over there, not very interested.

SoI have to find a way of explaining it.

And the best way I can think of is to say that

quantum information is like the information in the dream.

If you try to describe your dream to someone else,

And you only remember what you said about it. And you can't prove tosomeone what you dreamed.

And you can lie about your dream and not get caught,

except maybe if you try to lie to your spouse about your dream.

Well, anyway, but unlike dreams,

despite the best efforts of Sigmund Freud,

there is a well-understood theory about how quantum information behaves.

And that's what I want to talk about today.

The basic principle of quantum mechanics and of quantum information is that

in any physical system, there is a maximum number of reliablydistinguishable states.

And for very simple system like these polarized photons, we've beentalking about,

there are two reliably distinguishable states.

And that between any pair of reliably distinguish states,

there are other intermediate states

that are not just reliably distinguishable from either of those.

Now, it doesn't mean that only some states are reliably distinguishable.

If any pair of states,

the possible physical states correspond to directions in space,

not ordinary three dimensional space.

But inner space with as many dimensions as

the system’s maximum number of reliably distinguishable states.

So for a photon, that's two.

And if the directions are perpendicular, then the states are distinguishable.

If they're not perpendicular, they are not distinguishable by any procedure at all.

So that's it. In a nutshell, that's quantummechanics.

So that's not so hard to understand.

So let's look at some of the manifestations of that.

But before saying that, this is something physicists have understood,

but they hadn't thought of it as having to do with information.

Now we know that ordinary information is reducible to bits,

We can take anything, however complicated, digitize it,

and produce it to a lot of zeros and ones.

And all processing of it can be reduced to

acting on these zeros and ones with ANDs and NOTs.

And that if you make the bits smaller andfaster and cheaper,

you just make the computer more and more useful.

It doesn't matter what you used to carry the bits.

If you have something that can you dozeros andones

and ANDs and NOTs, you're good to go.

Now,physicists haven't thought this way.

But you can make the analogous statements about quantum information.

Quantum information reducible to qubits

that is any kind of two-state system, such as a polarizedphoton.

And any processing of it,

any manipulation of a quantum state can be reduced to actions

on those one and two at a time.

That was only really discovered in about 1995.

And just as classical bits, can become independent of their physical embodiment

for their mathematical uses or for the uses of communication.

Soqubits and quantum gates are

fungibleamong the different quantum systems.

So here’s the example that you've seen this kind of diagram before.

But, so we know that we can use a single polarized photon to carry a bit of information.

If we set it up like this, the horizontal photons will go straight through(crystal).

The vertical photons will be deviated

and can count them on two separate counters.

But if you put in diagonal photons,

instead of being deviated by an intermediate amount,

they will some of them go into the this beam

and become horizontal

and some of them into that beam and become vertical.

That's something that disturbed Einstein a lot.

Hesaid,"How can something that is identically prepared

and then subjected to an identical treatment behave randomly?"

And that's what he said,

he didn't like the fact that god seems to be playing dice with the universe.

We'll explain that a little bit later.

Nowlet's say we rotate the apparatus.

Before I rotate the apparatus,

I'm just going to talk about this this strange behavior, thisprobabilistic behavior.

And the best metaphor for that is what I've heard fromourcolleague

BillWooders who’s taught undergraduates for many decades.

Hesaid, "Well, it's actually an educational metaphor."

Remember the old fashioned kind of school where the student was not supposed to speak upin class,

and was just supposed to answer the questions the teacher asks.

The quantum system being measured is like that.

So the teacher is the measuring improvement,

in this case, consisting of the crystal and the detectors.

And the student is like the photons.

So the teacher says,"Are you vertical or horizontal?"

And the student says, "I'm polarized about 55 degree angle from horizontal."

(The teacher says) "I believeI asked you a question, are you vertical or horizontal?"

(The student says)"Horizontal, sir."

(The student says) "No, sir, Iwas always horizontal."

So that's the way quantum measurement behaves.

That's not making it seems any less mysterious,

just makes it seems more familiar.

But before we explain why that happens,

this is how it can be used for cryptography.

I think it's been pretty well explained already.

You can prepare a stream of vertical and horizontal photons

if you know that’s what they are.

And then it can carry one bit in each one.

And they go reliably into one path of the other.

If you put in diagonal photons, you can also reliably distinguish those.

But you have to do it with a rotated measuring apparatus.

And no measuring can distinguish all four kinds.

And this is the fundamental limitation that gives rise to quantum moneyand quantum cryptography.

And then they're in particular procedures that Gilles and I developed

where we send these four kinds of photons from Alice to Bob.

And then we have a classical discussion of ordinary messages,

that eavesdroppers allowed to listen to.

And the end result is that they have a shared secret bit string, a key,

that if there has been too much eavesdropping, they will reject it.

But if there is not with high confidence,

they know that it is secret from everyone, but the two of them.

But I have to say what entanglement is.

But there's no escape from this uncertainty principle.

If you try to measure a photon, you can't measure its polarization exactly

if you don't know what it is to begin with.

So there's no equipment that you can put a photon in

thatit will come out and tell you what its polarization is.

You can't clone a photon, because if you could clone it,

you could make millions of copies.

And then measure and from the statistics, determine the polarization with arbitrary accuracy.

There is a device whichamplifies photons,

and holding one in my hand is called laser.

Butlaser’s work.

You know the word "laser" stands for light amplification by stimulatedemission of radiation.

That was the idea that you send one photon and you get several photons out.

They'r enormally used in the oscillatory modes.

So itcould be a light oscillator for involving stimulated emission of radiation.

But they thought that the acronym "loser" would be not very good.

So anyway, it's a light amplifier.

But the thing about laser,

this light amplifiers, is that if you put in a very weak signal,

say a single photon,

it sometimes gets amplified, but it sometimes also doesn't get amplified

and you get a noise photon that comes out along with it.

So the beam gets brighter.

But it's no more useful in distinguishing the polarization

than the original unamplified beam.

So there's no way around this uncertainty principle.

关于“墨子沙龙”

墨子沙龙是由中国科学技术大学上海研究院主办、上海市浦东新区科学技术协会及中国科大新创校友基金会协办的公益性大型科普论坛。沙龙的科普对象为对科学有浓厚兴趣、热爱科普的普通民众，力图打造具有中学生学力便可以了解当下全球最尖端科学资讯的科普讲坛。

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