1.2.1 多量子比特

Carlton Caves

Hilbert space is a big place.
希尔伯特空间很大。

Suppose we have two qubits. If these were two classical bits, then there would be four possible states, 00, 01, 10, and 11. Correspondingly, a two qubit system has four computational basis states denoted $|00\rang$ , $|01\rang$ , $|10\rang$ , $|11\rang$ . A pair of qubits can also exist in superpositions of these four states, so the quantum state of two qubits involves associating a complex coefficient – sometimes called an amplitude – with each computational basis state, such that the state vector describing the two qubits is
假设我们有两个量子位。如果这是两个经典比特，那么就会有四种可能的状态，00、01、10和11。相应地，一个双量子位系统有四个计算基态，分别为$|00\rang$、$|01\rang$、$|10\rang$、$|11\rang$。一对量子位也可以存在于这四个状态的叠加态中，因此两个量子位的量子态涉及到将一个复杂系数(有时称为振幅)与每个计算基态相关联，这样描述两个量子位的状态向量是

$$\tag{1.5} |ψ = α_{00}|00\rang + α_{01}|01\rang + α_{10}|10\rang + α_{11}|11\rang$$

Similar to the case for a single qubit, the measurement result $x$ (= 00, 01, 10 or 11) occurs with probability $|α_x|^2$, with the state of the qubits after the measurement being $|x\rang$ . The condition that probabilities sum to one is therefore expressed by the normalization condition that $\sum_{x∈\{0,1\}^2}|α_x|^2 = 1$, where the notation ‘$\{0, 1\}^2$ ’ means ‘the set of strings of length two with each letter being either zero or one’. For a two qubit system, we could measure just a subset of the qubits, say the first qubit, and you can probably guess how this works: measuring the first qubit alone gives 0 with probability $|α_{00}|^2 +|α_{01}|^2$ , leaving the post-measurement state
与单个量子位的情况类似，测量结果$x$(= 00, 01, 10或11)出现的概率为$|α_x|^2$，测量后量子位的状态为$|x\rang$。因此，概率和为1的条件可以用归一化条件$\sum_{x∈\{0,1\}^2}|α_x|^2 = 1$来表示，其中符号' ${0,1}^2 $ '表示'长度为2的字符串的集合，每个字母为0或1 '。对于两个量子位系统，我们可以只测量量子位的一个子集，比如第一个量子位，你可能会猜出这是如何工作的:单独测量第一个量子位给出0的概率为$|α_{00}|^2 +|α_{01}|^2$，留下测量后的状态

$$\tag{1.6} |\psi^{\prime}\rang=\frac{\alpha_{00}|00\rang+\alpha_{01}|01\rang}{\sqrt{|\alpha_{00}|^2+|\alpha_{01}|^2}}$$

Note how the post-measurement state is re-normalized by the factor $\sqrt{|\alpha_{00}|^2+|\alpha_{01}|^2}$ so that it still satisfies the normalization condition, just as we expect for a legitimate quantum state.
请注意，测量后的状态是如何通过因子$\sqrt{|\alpha_{00}|^2+|\alpha_{01}|^2}$重新归一化的，因此它仍然满足归一化条件，就像我们对合法量子态的期望一样。

An important two qubit state is the Bell state or EPR pair,
一个重要的两个量子比特态是贝尔态或EPR对，

$$\tag{1.7} \frac{|00\rang+|11\rang}{\sqrt2}$$

This innocuous-looking state is responsible for many surprises in quantum computation and quantum information. It is the key ingredient in quantum teleportation and superdense coding, which we’ll come to in Section 1.3.7 and Section 2.3, respectively, and the prototype for many other interesting quantum states. The Bell state has the property that upon measuring the first qubit, one obtains two possible results: 0 with probability 1/2, leaving the post-measurement state $|ϕ^{\prime} = |00\rang$ , and 1 with probability $1/2$, leaving $|ϕ^{\prime} = |11\rang$ . As a result, a measurement of the second qubit always gives the same result as the measurement of the first qubit. That is, the measurement outcomes are correlated. Indeed, it turns out that other types of measurements can be performed on the Bell state, by first applying some operations to the first or second qubit, and that interesting correlations still exist between the result of a measurement on the first and second qubit. These correlations have been the subject of intense interest ever since a famous paper by Einstein, Podolsky and Rosen, in which they first pointed out the strange properties of states like the Bell state. EPR’s insights were taken up and greatly improved by John Bell, who proved an amazing result: the measurement correlations in the Bell state are stronger than could ever exist between classical systems. These results, described in detail in Section 2.6, were the first intimation that quantum mechanics allows information processing beyond what is possible in the classical world.
这种看似无害的状态导致了量子计算和量子信息中的许多意外。它是量子隐形传态和超密编码的关键成分，我们将分别在1.3.7节和2.3节讲到，它也是许多其他有趣量子态的原型。贝尔态具有这样的性质:在测量第一个量子位时，可以得到两个可能的结果:0的概率为1/2，使测量后的状态为$|ϕ^{\prime} = |00\rang$; 1的概率为$1/2$，使$|ϕ^{\prime} = |11\rang$。因此，测量第二个量子位的结果总是与测量第一个量子位的结果相同。也就是说，测量结果是相关的。事实上，事实证明，其他类型的测量可以在贝尔状态上执行，通过首先对第一个或第二个量子位应用一些操作，并且在第一个和第二个量子位的测量结果之间仍然存在有趣的相关性。自从爱因斯坦、波多尔斯基和罗森发表了一篇著名的论文，首次指出了贝尔态等状态的奇怪性质以来，这些相关性一直是人们浓厚兴趣的主题。约翰·贝尔(John Bell)接受了EPR的见解，并对其进行了极大的改进，他证明了一个惊人的结果:贝尔态的测量相关性比经典系统之间可能存在的更强。这些结果，在2.6节中详细描述，是量子力学允许超越经典世界可能的信息处理的第一个暗示。

More generally, we may consider a system of $n$ qubits. The computational basis states of this system are of the form $|x_1x_2 ...x_n\rang$ , and so a quantum state of such a system is specified by $2^n$ amplitudes. For $n = 500$ this number is larger than the estimated number of atoms in the Universe! Trying to store all these complex numbers would not be possible on any conceivable classical computer. Hilbert space is indeed a big place. In principle, however, Nature manipulates such enormous quantities of data, even for systems containing only a few hundred atoms. It is as if Nature were keeping $2^{500}$ hidden pieces of scratch paper on the side, on which she performs her calculations as the system evolves. This enormous potential computational power is something we would very much like to take advantage of. But how can we think of quantum mechanics as computation?
更一般地说，我们可以考虑一个有n个量子比特的系统。该系统的计算基态为$|x_1x_2…X_n \ range $，所以这样一个系统的量子态是由$2^n$振幅指定的。对于$n = 500$，这个数字大于宇宙中原子的估计数量!试图在任何可以想象的经典计算机上存储所有这些复数是不可能的。希尔伯特空间确实很大。然而，从原则上讲，即使对于只有几百个原子的系统，大自然也能操纵如此大量的数据。这就好像大自然在一边保留了2^{500}$的隐藏草稿纸，随着系统的发展，她在上面进行计算。这种巨大的潜在计算能力是我们非常想利用的。但是我们怎么能把量子力学看作是计算呢?